Subgroup ($H$) information
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(2,7)(6,8), (2,8)(6,7), (1,5)(2,8)(3,4)(6,7), (1,4)(2,7)(3,5)(6,8)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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: | $C_2^2\times D_6$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3\times C_2^3:\GL(3,2)$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $C_2^3:\GL(3,2)$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, 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 \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3:D_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^6$ | |||
| Normalizer: | $C_2^5:S_4$ | |||
| Complements: | $C_2^2\times D_6$ | |||
| Minimal over-subgroups: | $C_2^2:A_4$ | $C_2^5$ | $C_2^2\wr C_2$ | $C_2^2\wr C_2$ |
| Maximal under-subgroups: | $C_2^3$ | $C_2^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-192$ |
| Projective image | $C_2^5:S_4$ |