We describe abstract groups using standard building blocks:

- $C_n$ denotes the cyclic group of order $n$.
- $S_n$ denotes the symmetric group on $n$ letters.
- $A_n$ denotes the alternating group on $n$ letters.
- $D_n$ denotes the dihedral group of order $2n$.
- $Q_n$ denotes the quaternion group of order $n$.
- $\mathrm{GL}(n,q)$ denotes the general linear group of degree $n$ over the finite field of order $q$.
- $\mathrm{SL}(n,q)$ denotes the special linear group of degree $n$ over the finite field of order $q$.
- $SD_n$ denotes the semidihedral group or quasidihedral group of order $n=2^k$.
- $OD_n$ denotes the other dihedral group (or modular maximal-cyclic group) of order $n=2^k$. It is the non-trivial semidirect product $C_{2^{k-1}} : C_2$ which is not isomorphic to either $SD_n$ or $D_{2^{k-1}}$.
- $F_q$ denotes the Frobenius group for a prime power $q$. It is the group of affine linear transformations of the finite field $\mathbb{F}_q$. In other words, $F_q$ is a semidirect product $\mathbb{F}_q : \mathbb{F}_q^{\times}$.
- $He_p$ denotes the Heisenberg group, the unique non-abelian group of order $p^3$ and exponent $p$ for an odd prime $p$.

Groups $A$ and $B$ may be used to construct a larger group:

- $A\times B$ for the direct product of $A$ and $B$
- $A:B$ for the semidirect product of $A$ and $B$ (with normal subgroup $A$)
- $A.B$ an extension with normal subgroup $A$ and quotient isomorphic to $B$
- $A\wr B$ for the wreath product of $A$ and $B$

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Jones on 2022-06-29 13:11:40

**Referred to by:**

- lmfdb/groups/abstract/main.py (line 935)
- lmfdb/groups/abstract/main.py (line 1056)
- lmfdb/groups/abstract/main.py (line 1072)
- lmfdb/groups/abstract/main.py (line 1080)
- lmfdb/groups/abstract/templates/abstract-show-group.html (line 21)
- lmfdb/groups/abstract/templates/abstract-show-subgroup.html (lines 10-12)
- lmfdb/groups/abstract/templates/abstract-show-subgroup.html (line 38)
- lmfdb/groups/abstract/templates/abstract-show-subgroup.html (line 60)
- lmfdb/groups/glnC/main.py (line 115)
- lmfdb/groups/glnQ/main.py (line 106)

**History:**(expand/hide all)

- 2023-07-11 11:07:07 by David Roe
- 2023-07-11 01:49:11 by Manami Roy
- 2023-07-11 01:48:50 by Manami Roy
- 2023-07-11 01:47:59 by Manami Roy
- 2023-07-11 01:46:51 by Manami Roy
- 2023-07-11 01:46:05 by Manami Roy
- 2023-07-11 01:42:13 by Manami Roy
- 2023-07-11 01:41:48 by Manami Roy
- 2023-07-11 01:38:07 by Manami Roy
- 2023-07-11 01:34:14 by Manami Roy
- 2023-07-11 01:33:11 by Manami Roy
- 2023-07-11 01:28:07 by Manami Roy
- 2023-07-11 01:27:50 by Manami Roy
- 2023-07-11 01:26:55 by Manami Roy
- 2023-07-11 01:24:31 by Manami Roy
- 2023-07-11 01:23:39 by Manami Roy
- 2022-06-29 13:11:40 by John Jones (Reviewed)
- 2022-06-29 13:11:15 by John Jones
- 2022-06-29 13:08:56 by John Jones
- 2020-12-08 06:50:42 by Manami Roy
- 2020-12-04 09:04:41 by John Jones
- 2020-12-04 09:01:22 by John Jones
- 2020-12-04 09:00:44 by John Jones
- 2020-12-04 08:59:53 by John Jones
- 2020-12-02 14:26:59 by Manami Roy
- 2020-12-02 14:26:32 by Manami Roy
- 2020-12-02 13:32:33 by Manami Roy
- 2020-12-01 20:37:41 by Manami Roy
- 2020-12-01 19:54:21 by Manami Roy
- 2020-11-20 15:47:14 by Manami Roy

**Differences**(show/hide)