Subgroup ($H$) information
| Description: | $C_2^5:D_4$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,8)(2,16)(3,12)(4,19)(5,15)(6,10)(7,18)(9,13), (1,5)(3,7)(8,15)(12,18) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{10}.D_4$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| 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: | $C_2^5$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\GL(5,2)$, of order \(9999360\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| Outer Automorphisms: | $\GL(5,2)$, of order \(9999360\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| 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), 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 \(6442450944\)\(\medspace = 2^{31} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^{12}.C_2^6.C_3.C_2^4.\PSL(2,7)$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |