Subgroup ($H$) information
| Description: | $C_2^4:\GL(3,2)$ |
| Order: | \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,13)(2,3)(4,14)(5,6)(7,11)(8,9)(10,16)(12,15)(17,20)(18,19), (1,6)(2,16) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_2^5:(C_2\times \GL(3,2))$ |
| Order: | \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_2^4:\GL(3,2)$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $W$ | $C_2^4:\GL(3,2)$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $C_2^5:\GL(3,2)$ |