Subgroup ($H$) information
| Description: | $C_6^4.C_6:D_{12}$ |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,11,7)(10,13,17)(19,27,23)(21,29,25), (3,11,7)(4,12,16)(6,18,14)(10,13,17) \!\cdots\! \rangle$
|
| 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: | $C_6^4.C_6^2:D_4$ |
| Order: | \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.D_6^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6^4.C_3.D_6^2.C_2^2$ |
| $W$ | $C_3^4.C_2\wr S_3^2$, of order \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.C_6^2:D_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.C_2\wr S_3^2$ |