Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,5)(2,6)(3,4)(7,8), (1,3)(2,7)(4,5)(6,8)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_2^5:S_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
Quotient group ($Q$) structure
| Description: | $C_2^3\times S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_4\times C_2^3:\GL(3,2)$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $C_2^3:\GL(3,2)$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3.C_6.C_2^3$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^4:D_4$ | ||||||
| Normalizer: | $C_2^5:S_4$ | ||||||
| Minimal over-subgroups: | $A_4$ | $C_2^3$ | $C_2^3$ | $C_2^3$ | $C_2^3$ | $D_4$ | $C_2\times C_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $768$ |
| Projective image | $C_2^5:S_4$ |