Subgroup ($H$) information
| Description: | $C_2^5:\GL(3,2)$ |
| Order: | \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| Index: | \(3\) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(4,7)(5,6), (8,9)(10,13)(11,14)(12,15), (8,14)(9,10)(11,12)(13,15), (4,5) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_6\times C_2^4:\GL(3,2)$ |
| Order: | \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^4.\PSL(2,7)\times S_3$ |
| $\operatorname{Aut}(H)$ | $C_2^4.\PSL(2,7)\times S_3$ |
| $W$ | $C_2^3:\GL(3,2)$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3\times C_2^3:\GL(3,2)$ |