What is the meaning of symbol " : " in demonstrating the group in GAP? For example C_8:(C_2×C_2).

Question

I was calculating the Sylow 2-Subgroups of Mathieu group M(12) through GAP. I am facing a problem, some of the Sylow 2-Subgroups are represented as C_8:(C_2×C_2) with colon(" : "). I don't know about the meaning of this. If we see the group order it is 32 of this group. In the same way every Sylow 2-Subgroups of this representation is having order of multiples like this 8x2x2=32.

I don't use GAP, but it wasn't too hard to find in the manual that : means semidirect products. That's assuming you're using some kind of structured description function in GAP. Note that something that would make it easier to answer your question is if you give the actual code you're inputting into GAP, a so-called "minimal working example" so that others can see exactly what you're seeing. — Kyle Miller, Jan 18 '23 at 17:22
Yes, I saw this, too. It is $C_8 \rtimes (C_2\times C_2)$, and this page (among others) has the groups of small order listed with names and labels. — Dietrich Burde, Jan 18 '23 at 17:53
After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $\checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?. — Shaun, Jan 18 '23 at 18:16
This question is about "Atlas notation", so I've closed it as a duplicate of an older question on this notation (it is on page xx, I believe, of the ATLAS of Finite Groups). Derek Holt's answer there completely answers this question. — user1729, Jan 19 '23 at 15:19

Shaun · Answer 1 · 2023-02-14T22:04:49.273

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It means a semidirect product. In general, we can write $G = N:Q$ for $G = N\rtimes Q$, (or $G=Q\ltimes N$) for a semidirect product, where $N$ is normal in $G$.

edited Feb 14 '23 at 22:04

answered Jan 18 '23 at 17:25

Shaun

47,747

Is the normal group always written first in the colon notation? – Dietrich Burde Jan 18 '23 at 17:55
I believe so, yes, @DietrichBurde. – Shaun Jan 18 '23 at 18:00
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I see, thank you! – Dietrich Burde Jan 18 '23 at 18:02

What is the meaning of symbol " : " in demonstrating the group in GAP? For example C_8:(C_2×C_2).

1 Answers1