Subgroup ($H$) information
| Description: | not computed |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,8)(2,20)(5,11)(6,24)(7,14)(12,18)(13,19)(16,22), (1,2,8,20)(5,6,11,24) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary). Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^8.D_4^2$ |
| Order: | \(16384\)\(\medspace = 2^{14} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $5$ |
| Derived length: | $3$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(16777216\)\(\medspace = 2^{24} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_2^6:F_7$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^8.D_4^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |