Subgroup ($H$) information
| Description: | $C_2^8$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(19,20)(23,24), (3,4)(5,6)(7,8)(11,12), (19,20)(21,22), (9,10)(11,12), (7,8) \!\cdots\! \rangle$
|
| 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: | $A_4^2\wr C_2.C_2^2.C_2^3$ |
| Order: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^4:C_4^2:C_2^2$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^4.C_2^3.C_2^5.C_2^4$ |
| Outer Automorphisms: | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3^4.C_2.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\GL(8,2)$ |
| $\card{W}$ | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2^9$ |
| Normalizer: | $A_4^2\wr C_2.C_2^2.C_2^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4^2\wr C_2.C_2^2.C_2^3$ |