Subgroup ($H$) information
| Description: | $C_4^2.C_2^5$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(3,4)(5,6)(15,16)(17,18), (1,23)(2,24)(5,6)(9,10)(11,13)(12,14)(17,18)(21,22) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is a direct factor, a semidirect factor, metacyclic, monomial, or rational 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: | $C_2^8.S_4$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $D_5^3.C_2^2$, of order \(196608\)\(\medspace = 2^{16} \cdot 3 \) |
| Outer Automorphisms: | $D_4\times C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian, solvable, and rational. 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 \(201326592\)\(\medspace = 2^{26} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | Group of order \(32469952757760\)\(\medspace = 2^{35} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\card{W}$ | \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^6$ |
| 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 |