Subgroup ($H$) information
| Description: | $C_6^4.C_6^2:C_{12}$ |
| Order: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Index: | \(2\) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(2,12,4,16,15,13,14,3,11)(5,18,6)(7,9,17)(20,21,22)(23,24)(25,30)(27,29,28) \!\cdots\! \rangle$
|
| 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: | $C_6^4.(C_6^2:C_4\times S_3)$ |
| Order: | \(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $(C_2\times C_6^3).C_3^2.C_6^2.C_2^5$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.(C_6^2:C_4\times S_3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.(A_4^2:C_4\times D_6)$ |