# gps_conj_classes downloaded from the LMFDB on 15 July 2026.
# Search link: https://www.lmfdb.org/Groups/Abstract/?group=100.10&order=4&search_type=ConjugacyClasses
# Query "{'order': 4, 'group_order': 100, 'group_counter': 10}" returned 2 gps_conj_classes, sorted by group.

# Each entry in the following data list has the form:
#    [Group, Label, Order, Size, Centralizer, Powers, Representative]
# For more details, see the definitions at the bottom of the file.



"100.10"	"4A1"	4	25	"25.a1.a1"	"2A,4A1"	"ab^{4}c^{4}"
"100.10"	"4A-1"	4	25	"25.a1.a1"	"2A,4A-1"	"a^{3}b^{3}c^{4}"


# Group --
#    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 $B$.


# Label --
#    The **label** of a rational conjugacy class of a group $G$ has the format $NA$ where $N$ is the order of elements in the conjugacy class, $A$ is a sequence of capital letters.


#    The **label** of a complex conjugacy class of a group $G$ has the format $NAj$ where $NA$ is the label of the division containing it, (so $N$ is the order of elements in the conjugacy class), and $j$ is an enumeration of class within that rational conjugacy class ($j$ is omitted if there is a single class within rational conjugacy classes).


# Order --
#    All elements in a particular conjugacy class are of the same order in the group.


# Size --
#    The **size** of a conjugacy class of a finite group is the cardinality of the set of elements conjugate to each other.

#    By the Orbit-Stabilizer Theorem, the size of a conjugacy class divides the order of the group. Since conjugacy classes partition a group, the sum of the sizes of the conjugacy classes equals the size of the group.

#    A conjugacy class is of size $1$ if and only if the element is in the center of the group.




# Centralizer --
#    If $G$ is a group and $H$ is a subgroup of $G$, then the **centralizer** of $H$ in $G$ is
#    \[ C_G(H) = \{g\in G\mid gh=hg \text{ for all } h\in H\}.\]


# Powers --
#    If $G$ is a group, $g\in G$ is an element of a conjugacy class $P$, and $n$ is a positive integer, then $nP$ denotes the conjugacy class of the element $g^n$.


# Representative --
#    A representative of each conjugacy class is given in different formats depending on how the abstract group is stored in the database.

#    For *permutation groups* the representative is given as a permutation of degree given in the Construction section of this group's main page.

#    For *matrix groups* the representatives are given by matrices in the group described in the Construction section of this group's main page.

#    For solvable groups, the representatives are given as words in the generators from the presentation given in the Construction section of this group's main page.