Subgroup ($H$) information
| Description: | $C_2^7$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,4)(2,7)(3,5)(6,8)(11,12)(13,14), (2,7)(6,8), (2,6)(7,8), (1,4)(2,6)(3,5) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, abelian (hence 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: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.(S_3\times \GL(3,2))$, of order \(129024\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $\GL(7,2)$, of order \(163849992929280\)\(\medspace = 2^{21} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \cdot 127 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times \GL(3,2)$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(128\)\(\medspace = 2^{7} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^7$ | ||||
| Normalizer: | $C_2^5:S_4$ | ||||
| Complements: | $S_3$ | ||||
| Minimal over-subgroups: | $C_2^5:A_4$ | $C_2^5:D_4$ | |||
| Maximal under-subgroups: | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $3$ |
| Projective image | $C_2^4:S_4$ |