Subgroup ($H$) information
| Description: | $C_2^5$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,4)(2,15)(3,10)(5,8)(6,14)(7,13)(9,11)(12,16), (1,11)(2,7)(3,5)(4,9)(6,16) \!\cdots\! \rangle$
|
| 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: | $C_2^4:F_8:C_6$ |
| Order: | \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $F_8:C_3$ |
| Order: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Automorphism Group: | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:F_8:C_6$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $\GL(5,2)$, of order \(9999360\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $W$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2^5$ | |||
| Normalizer: | $C_2^4:F_8:C_6$ | |||
| Complements: | $F_8:C_3$ | |||
| Minimal over-subgroups: | $C_2^2\times F_8$ | $C_2^3\times A_4$ | $D_4\times C_2^3$ | |
| Maximal under-subgroups: | $C_2^4$ | $C_2^4$ | $C_2^4$ | $C_2^4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-56$ |
| Projective image | $C_2^3:F_8:C_6$ |