Subgroup ($H$) information
| Description: | $C_2^4:A_4^2$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Index: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,18,9)(2,17,10)(5,14,21)(6,13,22), (3,12,20)(4,11,19)(7,23,16)(8,24,15) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^8.C_6\wr D_4$ |
| Order: | \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \) |
| 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: | $C_{12}^2:D_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_3:(C_2^4.C_2^5.C_2^2)$ |
| Outer Automorphisms: | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3^4.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^8.A_5^2:\SOPlus(4,2)$, of order \(66355200\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5^{2} \) |
| $W$ | $A_4^2\wr C_2.C_2^2$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2^4$ |
| Normalizer: | $C_2^8.C_6\wr D_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^8.C_6\wr D_4$ |