Subgroup ($H$) information
| Description: | $C_2^8.(C_6\times S_3^2)$ |
| Order: | \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(11,12)(13,14)(15,16)(17,18)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34), (13,14) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $A_4^2.C_2\wr D_6$ |
| Order: | \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^8.C_3^3.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^8.C_3^3.C_2^6$ |
| $W$ | $C_2^8.(C_6\times S_3^2)$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^2.C_2\wr D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^8.(C_6\times S_3^2)$ |