Subgroup ($H$) information
| Description: | $C_2^4\times C_4$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$b^{3}, d^{2}ghijk, d^{2}, hijk, f^{2}gjk$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^6.C_2\wr S_3$ |
| Order: | \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^4:S_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^6:(S_3\times \GL(3,2))$, of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1158}:C_{16}$, of order \(786432\)\(\medspace = 2^{18} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^4.A_8$, of order \(10321920\)\(\medspace = 2^{15} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^6.C_2^4$ |
| Normalizer: | $C_2^6.C_2\wr S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_4^2.C_2^4:S_4$ |