Subgroup ($H$) information
| Description: | $C_2^8$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(13,16)(14,15), (5,8)(6,7), (1,2)(3,4)(13,15)(14,16), (13,15)(14,16), (5,6) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, abelian (hence 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: | $C_2^8.S_3^2\wr C_2$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3^2\wr C_2$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_3\wr D_4$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2\wr C_2.C_4.D_4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\GL(8,2)$, of order \(5348063769211699200\)\(\medspace = 2^{28} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 17 \cdot 31 \cdot 127 \) |
| $W$ | $C_2^{10}$, of order \(1024\)\(\medspace = 2^{10} \) |
Related subgroups
| Centralizer: | $C_2^8$ |
| Normalizer: | $C_2^8.S_3^2\wr C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^8.S_3^2\wr C_2$ |