Subgroup ($H$) information
| Description: | $C_4^2.(C_2^3\times S_4)$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(5,6)(7,8)(13,14)(15,16)(21,22)(23,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_4^3:C_2^2:S_4$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:A_4.C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_4^2:A_4.C_2^4.C_2^4$ |
| $\card{W}$ | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |