Subgroup ($H$) information
| Description: | $C_3^6.C_2^7:S_4$ |
| Order: | \(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(31,34,35)(32,33,36), (1,3,6)(2,4,5)(13,16,18)(14,15,17)(19,24,22)(20,23,21) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is normal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.(C_2^9.S_4)$ |
| Order: | \(8957952\)\(\medspace = 2^{12} \cdot 3^{7} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(573308928\)\(\medspace = 2^{18} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.C_2^6.A_4.C_2^6.C_2^2$, of order \(143327232\)\(\medspace = 2^{16} \cdot 3^{7} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.(C_2^9.S_4)$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |