Subgroup ($H$) information
| Description: | $C_2\times C_6^4.S_3^2$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Index: | \(2\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,3,5,2,4,7)(6,8)(11,15,13)(14,16,17)(18,20,24,23,26,21)(19,22,25)(28,30) \!\cdots\! \rangle$
|
| 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: | $C_6^4.(S_3\times S_4)$ |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^4.C_3^3.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^8.D_5^2.C_2^3$, of order \(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \) |
| $W$ | $C_2\times C_6^4:D_6$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.(S_3\times S_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times C_6^4:D_6$ |