# Abstract groups downloaded from the LMFDB on 08 September 2026.
# Search link: https://www.lmfdb.org/Groups/Abstract/?family=SD
# Query "{'label': {'$in': ['1.1', '2.1', '4.2', '8.2', '16.8', '32.19', '64.53', '128.162', '256.540', '512.2043']}}" returned 10 groups, sorted by order.
# Each entry in the following data list has the form:
# [Label, Name, Family name, Order, Exponent, $\card{\mathrm{conj}(G)}$, Center, Type - length]
# For more details, see the definitions at the bottom of the file.
"1.1" "C_1" NULL 1 1 1 "1.1" "Trivial"
"2.1" "C_2" NULL 2 2 2 "2.1" "Cyclic"
"4.2" "C_2^2" NULL 4 2 4 "4.2" "Abelian - 2"
"8.2" "C_2\\times C_4" NULL 8 4 8 "8.2" "Abelian - 2"
"16.8" "\\SD_{16}" NULL 16 8 7 "2.1" "Nilpotent - 3"
"32.19" "\\SD_{32}" NULL 32 16 11 "2.1" "Nilpotent - 4"
"64.53" "\\SD_{64}" NULL 64 32 19 "2.1" "Nilpotent - 5"
"128.162" "\\SD_{128}" NULL 128 64 35 "2.1" "Nilpotent - 6"
"256.540" "\\SD_{256}" NULL 256 128 67 "2.1" "Nilpotent - 7"
"512.2043" "\\SD_{512}" NULL 512 256 131 "2.1" "Nilpotent - 8"
# Label --
# The **label** for a group $G$ has the form $\mathtt{N.i}$, where $\mathtt{N}$ is the order of the group and $\mathtt{i}$ is the small group id that distinguishes groups of the same order (as determined in GAP). If a group is not in GAP Small Groups database, we replace $\mathtt{i}$ with an incrementing letter code, assigning labels to groups as they are added to our database.
# Subgroups are labeled using the ambient group, index, and an ordering of subgroups up to automorphism and then up to conjugacy with the same invariants.
#Name (tex_name) --
# 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 (generalized) quaternion group of order $n$.
#
- $F_q$ 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 $\mathbb{F}_q : \mathbb{F}_q^{\times}$.
#
- Sporadic Groups (refer to this wikipage):
#
# - $M_n$ for $n=11,12, 22,23, 24$ denotes the Mathieu groups
#
- $\Ru$ denotes the Rudvalis group
#
- $\McL$ denotes the McLaughlin group
#
- $\He$ denotes the Held group
#
- $\HS$ denotes the Higman–Sims group
#
- $J_1$, $J_2$, $J_3$ denote the Janko groups
#
- $\Co_2$ and $\Co_3$ denote the Conway groups
#
# - Here is a list of finite groups of Lie type.
#
- $\SD_n$ 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 $D_{2^{k-1}}$.
#
- $\He_p$ 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 $B$.
#Family name (family_name) --
# The currently supported families of groups are:
# - $C_n$ : Cyclic
# - $S_n$ : Symmetric
# - $D_n$ : Dihedral
# - $A_n$ : Alternating
# - $Q_n$ : (Generalized) Quaternion
# - $F_n$ : Frobenius
# - $\SD_n$ : Semi-dihedral
# - $\OD_n$ : Other-dihedral
# - $\He_n$ : Heisenberg
# - $\Dic_n$ : Dicyclic
# - $M_{11}, M_{12}, J_1, M_{22}, J_2, M_{23}, \HS, J_3, M_{24}, \McL, \He, \Ru, \Co_3, \Co_2$ : Sporadic
# - $\GL(n,q)$ : General linear
# - $\SL(n,q)$ : Special linear
# - $\PGL(n,q)$ : Projective general linear
# - $\PSL(n,q)$ : Projective special linear
# - $\Sp(n,q)$ : Symplectic
# - $\PSp(n,q)$ : Projective symplectic
# - $\GO(n,q)$ : General orthogonal
# - $\GOPlus(n,q)$ : General orthogonal (plus)
# - $\GOMinus(n,q)$ : General orthogonal (minus)
# - $\SO(n,q)$ : Special orthogonal
# - $\SOPlus(n,q)$ : Special orthogonal (plus)
# - $\SOMinus(n,q)$ : Special orthogonal (minus)
# - $\PGO(n,q)$ : Projective orthogonal
# - $\PGOPlus(n,q)$ : Projective orthogonal (plus)
# - $\PGOMinus(n,q)$ : Projective orthogonal (minus)
# - $\PSO(n,q)$ : Projective special orthogonal
# - $\PSOPlus(n,q)$ : Projective special orthogonal (plus)
# - $\PSOMinus(n,q)$ : Projective special orthogonal (minus)
# - $\GU(n,q)$ : General unitary
# - $\SU(n,q)$ : Special unitary
# - $\PGU(n,q)$ : Projective unitary
# - $\PSU(n,q)$ : Projective special unitary
# - $\Omega(n,q)$ : Omega
# - $\OmegaPlus(n,q)$ : Omega (plus)
# - $\OmegaMinus(n,q)$ : Omega (minus)
# - $\POmega(n,q)$ : Projective omega
# - $\POmegaPlus(n,q)$ : Projective omega (plus)
# - $\POmegaMinus(n,q)$ : Projective omega (minus)
# - $\Spin(n,q)$ : Spin
# - $\SpinPlus(n,q)$ : Spin (plus)
# - $\SpinMinus(n,q)$ : Spin (minus)
# - $\AGL(n,q)$ : Affine general linear
# - $\ASL(n,q)$ : Affine special linear
# - $\ASp(n,q)$ : Affine symplectic
# - $\AGammaL(n,q)$ : Affine general linear automorphism
# - $\ASigmaL(n,q)$ : Affine special linear automorphism
# - $\PGammaL(n,q)$ : Projective general linear automorphism
# - $\PSigmaL(n,q)$ : Projective special linear automorphism
# - $\ASigmaSp(n,q)$ : Affine symplectic automorphism
# - $\PSigmaSp(n,q)$ : Projective symplectic automorphism
# - $\PGammaU(n,q)$ : Projective unitary automorphism group
# - $\CSp(n,q)$ : Conformal symplectic
# - $\CO(n,q)$ : Conformal orthogonal
# - $\COPlus(n,q)$ : Conformal orthogonal (plus)
# - $\COMinus(n,q)$ : Conformal orthogonal (minus)
# - $\CSO(n,q)$ : Conformal special orthogonal
# - $\CSOPlus(n,q)$ : Conformal special orthogonal (plus)
# - $\CSOMinus(n,q)$ : Conformal special orthogonal (minus)
# - $\CU(n,q)$ : Conformal unitary
# - $\CSU(n,q)$ : Conformal special unitary
# - $A(n,q), B(n,q), C(n,q), D(n,q), E(n,q), F(4,q), G(2,q)$ : Chevalley
# - ${}^2A(n,q), {}^2B(n,2^{2k+1}), {}^2D(n,q), {}^3D(4,q), {}^2E(6,q), {}^2F(4,2), {}^2G(2, 3)$: Twisted Chevalley
# - $W(A_n), W(B_n), W(D_n), W(E_n), W(F_4), H_n, I_2(n)$: Coxeter
# Order --
# The **order** of a group is its cardinality as a set.
# Exponent --
# The **exponent of a group** $G$ is the smallest positive integer $n$ such that $g^n=e$ for all $g\in G$. If no such positive integer exists, then the exponent of the group is infinite.
#$\card{\mathrm{conj}(G)}$ (number_conjugacy_classes) --
# If $G$ is a group and $x\in G$, the **conjugacy class** of $x$ is the set $\{gxg^{-1}\mid g\in G\}$. These sets partition $G$, and the set of conjugacy classes is denoted by $\mathrm{conj}(G)$.
# Since conjugation by fixed $g\in G$ is an automorphism of $G$, all conjugate elements have the same order in the group.
#Center (center_label) --
# If $G$ is a group, the **center** of $G$ is the set
# \[ Z(G) = \{g\in G\mid gh=hg \text{ for all } h\in G\}.\]
# It is a normal subgroup of $G$.
#Type - length (type) --
# For a finite group $G$, we write **Type - length** as follows
#
# - If $G$ is cyclic we write "Cyclic". For abelian groups that are not cyclic we write "Abelian - $n$", where $n$ is the number of cyclic groups in the unique cyclic decomposition of $G$ as $C_{d_1} \times C_{d_2}\times \cdots \times C_{d_n}$ with $d_i\mid d_{i+1}$ and $d_i>0$.
# - If $G$ is nilpotent but not abelian then we write "Nilpotent - $n$", where $n$ is the nilpotency class of $G$.
# - If $G$ is solvable but not nilpotent then we write "Solvable - $n$", where $n$ is the derived length of $G$.
# - If $G$ is simple we write "Simple".
#
- For non-simple, non-solvable groups we write "Non-solvable - $n$", where $n$ is the number of Jordan factors of $G$.
#