A curious non-abelian continuous structure defined by three different group extensions

I am puzzled by the following structure (assume it is a group), called it $G$, contains and satisfies the following ingredients (in particular we can focus on the case that $d=2$). The $G$ contains generators including

  1. $mathbb{Z}/2 equiv mathbb{Z}_2^A$

  2. $SO(d)$

  3. $mathbb{Z}/4 equiv mathbb{Z}_4^B$

The subindices $A$ and $B$ are to specify they are different groups. The group operations of $mathbb{Z}_2^A$, $SO(d)$, and $mathbb{Z}_4^B$ satisfy:

  • (1) $mathbb{Z}_2^A$ and $SO(d)$ do not commute:

They obey $mathbb{Z}_2^A rtimes SO(d) = Spin(d)$, where $frac{Spin(d)}{mathbb{Z}_2^A}=SO(d)$. Namely, we have
$ 1 to mathbb{Z}_2^A to Spin(d) to SO(d) to 1$.

  • (2) $SO(d)$ and $mathbb{Z}_4^B$ do not commute:

They obey $SO(d) rtimes mathbb{Z}_4^B =E(d)$, in fact $E(d)$ obeys that
the quotien group $frac{E(d)}{SO(d)}=mathbb{Z}_4^B$ and $frac{E(d)}{mathbb{Z}_2^B}=O(d)$
and $frac{mathbb{Z}_4^B}{mathbb{Z}_2^B}=mathbb{Z}_2’$ where $mathbb{Z}_2’$ is an another mod 2 abelian group. Namely, we have three short exact sequences
$ 1 to SO(d) to E(d) to mathbb{Z}_4^B to 1$, and
$ 1 to mathbb{Z}_2^B to E(d) to O(d) to 1$, and
$ 1 to mathbb{Z}_2^B to mathbb{Z}_4^B to mathbb{Z}_2′ to 1$.

We can also write $E(d)$ as
={ (M, j) in (O(d), mathbb{Z}_4^B) ; vert ; det M = j^2}$$

where $j in mathbb{Z}_4^B$ here can be written as ${1,i,-1,-i}$ with $i^4=+1.$

  • (3) $mathbb{Z}_4^B$ and $mathbb{Z}_2^A$ do not commute:

They obey $mathbb{Z}_4^B rtimes mathbb{Z}_2^A = D_8$ is a dihedral group of order 8. Namely, we have
$ 1 to mathbb{Z}_4^B to D_8 to mathbb{Z}_2^A to 1$.

In summary,
$G$ contain three group generators
mathbb{Z}_2^A, SO(d),mathbb{Z}_4^B

and group operations defined by
mathbb{Z}_2^A rtimes SO(d) = Spin(d),;$$

SO(d) rtimes mathbb{Z}_4^B =E(d),;$$

$$mathbb{Z}_4^B rtimes mathbb{Z}_2^A = D_8.

The curious part is that the semi direct products $rtimes$ are given in a way that $ mathbb{Z}_2^A, SO(d),mathbb{Z}_4^B$ can be a normal subgroup repsectively in (1), (2), (3); but $ mathbb{Z}_2^A, SO(d),mathbb{Z}_4^B$ can be a quotient subgroup respectively in (3), (1), (2). So their group operations are cyclic in some way.

So what are the precise constructions/descriptions of the group $G$? Does this generate some familiar groups?

p.s. any comments/partial answers/Refs are welcome!

extensions – Command “show” failed: Package authorizenet/magento-module-authorizenet not found in /var/www/html/composer.json

Im new in magento, i have this error when I enter in System -> Web Setup Wizard -> Extension Manager

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My error text:

Command “show” failed: Package authorizenet/magento-module-authorizenet not found in /var/www/html/composer.json

My magento version: ver.2.3.4

Capture of my composer.json in root Magento directory

enter image description here

My capture composer.json text

“authorizenet/magento-module-authorizenet”: “^1.0”

Thanks for Read :]

Not able to install extensions from mangeto connect & mage terminal. Magento 1.9

I am working on my eCommerce website called cravestore.in. My magento version is 1.9.4. I am not able to install any extensions from Magento Connect & Mage terminal both. I don’t know how to fix this. I am getting package dependencies & skipping error. Also suggest how to get the extension key from Magento Marketplace. Attached are the screenshots.

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Can Chrome Extensions Send Data to Remote Servers?

Suppose an extension has a scary list of permissions like that below (“Site access: On all sites”):

enter image description here

Does this also give the extension permission to send my data to the author’s servers via XHR?

I’ve read the documentation here but lack some background knowledge, so I am not sure in my interpretation:

Cross-Origin XMLHttpRequest

After reading this, it seemed like the extension isn’t allowed to send my data somewhere unless it has lines like the below in the manifest – is this correct?

"permissions": (

[FREE] 4 Good Chrome Extensions For Marketers | Proxies-free

[FREE] 4 Good Chrome Extensions For Marketers


I Found 3 Good Chrome Extensons:


Hidden Content:

You must reply before you can see the hidden data contained here.


Hidden Content:

You must reply before you can see the hidden data contained here.

email Grabber:

Hidden Content:

You must reply before you can see the hidden data contained here.


Autofill – automatic form filler

Hidden Content:

You must reply before you can see the hidden data contained here.

Free Email Extractor:

Hidden Content:

You must reply before you can see the hidden data contained here.


How do I uninstall these Firefox extensions?

Today I tried to use profiler.firefox.com which is some webapp that Mozilla makes to help you profile webpages in Firefox.

I have a simple webpage I wanted to profile which just has a small script and almost no html. I captured a profile and then in the profiler UI if I click the Firefox 78 tab a the top it shows I have a bunch of extensions installed.

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I never installed most of these extensions. I never installed a Wikipedia extension, nor Bing, nor Amazon, nor eBay, nor DuckDuckGo.

Checking the adds-on UI they don’t show up there

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How do I get rid of these extensions?

spfx extensions – Random bug with settings menu not loading on SP Online

I’m getting a random bug when my Settings menu (Gear icon in the top right corner) does not appear sometimes on my SP Online site. I found that when it happens I have an error in my console (see below).

When I refresh the page it is all good, if I refresh 10 times, I’m getting this problem 2-3 times. Looks like some race condition is involved. I’ve traced the error and it looks like a bug in SP JavaScript code.
It all looks like:

enter image description here

Here are all details about the error:

The console error:
This error I see in console

Here is the piece of code where the bug sits:
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So line return e.Id === n.newWorkloadSettingLink.Id fails because e sometimes null. I traced the source of e and it looks like it an iteration from _spPageContextInfo.MenuData.SettingsData array + merged with something else (+5 more elements) where one element is null in my case (item with index 3).

enter image description here

If I’m fast enough (can include script which runs on a page before the error call) and run the following block of code on Init in my Application Customizer extension, error disappears:

    const links = _spPageContextInfo.MenuData.SettingsData;
    for (let i=0; i < links.length; i++) {
      if (!links(i)) {
        links.splice(i, 1); // Removing item if it has no value

But this is kind of hack, and obviously, I need some help to understand how to resolve this issue for good. Does anyone have any idea why do I have one element in the array set to null or why Microsoft has this bug?

PS.: I play with SP on my own non-commertial research project and don’t have any MS support subscription. Anyway I don’t believe they will fix this bug for me in shot time as they are a bit slow sometimes. My last hope that someone from the community already saw/solved something like this 🙂
Thanks everyone in advance!

python – “Failed to load extensions” error appears while opening chrome-webdriver

I am a beginner on web automation and was trying to automate my chrome web driver to open youtube…I am
using selenium and my code is supposed to work just fine…here is the code

from selenium import webdriver
path="C:Program Files (x86)chromedriver.exe"
driver = webdriver.Chrome(path)

When I run this code it shows me a messagebox that says:”Failed to load extensions from
3017internal.Manifest file is missing or unreadable.”

Each time I run the code and this messagebox appears,it keeps generating these scope folders on my
“C:UsersuserAppDataLocalTemp” path…the folders may look like:”scoped_dir8792_193633017″,

After quitting the messagebox the chrome driver opens and display the good old “Aw snap” error…
and after quitting the chrome webdriver the following exception is raised…

  File "C:/Users/user/Desktop/Anobot.py/project2.py", line 3, in <module>
    driver = webdriver.Chrome(path)
  File "C:UsersuserAppDataLocalProgramsPythonPython38libsite-packagesseleniumwebdriverchromewebdriver.py", line 76, in __init__
  File "C:UsersuserAppDataLocalProgramsPythonPython38libsite-packagesseleniumwebdriverremotewebdriver.py", line 157, in __init__
    self.start_session(capabilities, browser_profile)
  File "C:UsersuserAppDataLocalProgramsPythonPython38libsite-packagesseleniumwebdriverremotewebdriver.py", line 252, in start_session
    response = self.execute(Command.NEW_SESSION, parameters)
  File "C:UsersuserAppDataLocalProgramsPythonPython38libsite-packagesseleniumwebdriverremotewebdriver.py", line 321, in execute
  File "C:UsersuserAppDataLocalProgramsPythonPython38libsite-packagesseleniumwebdriverremoteerrorhandler.py", line 242, in check_response
    raise exception_class(message, screen, stacktrace)
selenium.common.exceptions.WebDriverException: Message: unknown error: Chrome failed to start: exited normally.
  (unknown error: DevToolsActivePort file doesn't exist)
  (The process started from chrome location C:Program Files (x86)GoogleChromeApplicationchrome.exe is no longer running, so ChromeDriver is assuming that Chrome has crashed.)

And yes…I have also checked for malware in my device but there were none.