Subgroup ($H$) information
| Description: | $C_2^6$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(3,4)(5,6)(15,16)(17,18), (19,20)(21,22), (1,2)(11,12)(13,14)(23,24), (15,16) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{10}.C_2^7:S_4$ |
| Order: | \(3145728\)\(\medspace = 2^{20} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^5.C_2\wr S_4$ |
| Order: | \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2^8.A_4.C_2^6.C_2^6.C_2$, of order \(25165824\)\(\medspace = 2^{23} \cdot 3 \) |
| Outer Automorphisms: | $C_2^8.D_4$, of order \(2048\)\(\medspace = 2^{11} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 \(201326592\)\(\medspace = 2^{26} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $\GL(6,2)$, of order \(20158709760\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $\card{W}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^9.C_2^6.C_2^2$ |
| Normalizer: | $C_2^{10}.C_2^7:S_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |