Subgroup ($H$) information
| Description: | $C_3^5:S_3$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(7,8,9)(10,11,12)(13,14,15), (1,3,2)(4,6,5)(10,15,12,14,11,13), (13,14,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^5:D_6$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times (C_3\times \PSU(3,2)).S_3^3$ |
| $\operatorname{Aut}(H)$ | $C_3^3:C_3^2.C_6.C_2^2.S_3^3$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_3^3$ | ||||
| Normalizer: | $C_3^5:D_6$ | ||||
| Complements: | $C_2$ $C_2$ | ||||
| Minimal over-subgroups: | $C_3^5:D_6$ | ||||
| Maximal under-subgroups: | $C_3^5:C_3$ | $C_3^4:S_3$ | $C_3^4:S_3$ | $S_3\times C_3^4$ | $C_3^4:S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^3:S_3^2$ |