Subgroup ($H$) information
| Description: | $C_4^2:S_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(2,6,3)(4,11,5)(7,14,10)(8,12,16)(9,15,13), (1,6)(2,3)(4,12)(5,13)(7,9) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is normal, maximal, a direct factor, nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_2^5:S_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^3:(C_2^2\times S_4)$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_4^3:(C_2\times S_4)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_4^3:(C_2\times S_4)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_4^2:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2^5:S_4$ |