# Abstract groups downloaded from the LMFDB on 05 September 2026.
# Search link: https://www.lmfdb.org/Groups/Abstract/?family=Q
# Query "{'label': {'$in': ['1.1', '2.1', '4.1', '8.4', '16.9', '32.20', '64.54', '128.163', '256.541', '512.2044']}}" returned 10 groups, sorted by order.

# Each entry in the following data list has the form:
#    [Label, 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"	1	1	1	"1.1"	"Trivial"
"2.1"	"C_2"	2	2	2	"2.1"	"Cyclic"
"4.1"	"C_4"	4	4	4	"4.1"	"Cyclic"
"8.4"	"Q_8"	8	4	5	"2.1"	"Nilpotent - 2"
"16.9"	"Q_{16}"	16	8	7	"2.1"	"Nilpotent - 3"
"32.20"	"Q_{32}"	32	16	11	"2.1"	"Nilpotent - 4"
"64.54"	"Q_{64}"	64	32	19	"2.1"	"Nilpotent - 5"
"128.163"	"Q_{128}"	128	64	35	"2.1"	"Nilpotent - 6"
"256.541"	"Q_{256}"	256	128	67	"2.1"	"Nilpotent - 7"
"512.2044"	"Q_{512}"	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:
#    <ul>
#     <li> $C_n$ denotes the cyclic group of order $n$.
#     <li> $S_n$ denotes the symmetric group on $n$ letters.
#     <li> $A_n$ denotes the alternating group on $n$ letters.
#     <li> $D_n$ denotes the dihedral group of order $2n$.
#     <li> $Q_n$ denotes the (generalized) quaternion group of order $n$.
#     <li> $F_q$ 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  $\mathbb{F}_q : \mathbb{F}_q^{\times}$.
#     <li> Sporadic Groups (refer to this <a href="https://en.wikipedia.org/wiki/Sporadic_group">wikipage</a>):
#    <ul>
#     <li> $M_n$ for $n=11,12, 22,23, 24$ denotes the Mathieu groups
#     <li> $\Ru$ denotes the Rudvalis group
#     <li> $\McL$ denotes the McLaughlin group
#     <li> $\He$ denotes the Held group
#     <li> $\HS$ denotes the Higman–Sims group
#     <li> $J_1$, $J_2$, $J_3$ denote the Janko groups
#     <li> $\Co_2$ and $\Co_3$ denote the Conway groups
#    </ul>
#     <li> Here is a list of finite groups of Lie type.
#     <li> $\SD_n$ denotes the semidihedral group or quasidihedral group of order $n=2^k$.
#     <li> $\OD_n$ 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 $D_{2^{k-1}}$.
#     <li> $\He_p$ denotes the Heisenberg group, the unique non-abelian group of order $p^3$ and exponent $p$ for an odd prime $p$.
#    </ul>

#    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 $B$.


# 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

#    <ul>
#    <li> 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 $d_i>0$.</li>

#    <li> If $G$ is nilpotent but not abelian then we write  "Nilpotent - $n$", where $n$ is the nilpotency class of $G$.</li>

#    <li> If $G$ is solvable but not nilpotent then we write  "Solvable - $n$", where $n$ is the derived length of $G$.</li>

#    <li> If $G$ is simple we write "Simple".

#     <li> For non-simple, non-solvable groups we write "Non-solvable - $n$", where $n$ is the number of Jordan factors of $G$.</li>
#    </ul>