Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(2,3)(4,5)(7,18)(8,17)(9,16)(10,12)(13,15), (1,6)(2,5)(3,4)(7,9)(10,13)(11,14)(12,15)(16,18)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_3^5:(C_2\times S_4)$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3^2\times S_3^3):D_6$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^3$ | |||||||
| Normalizer: | $C_2^3$ | |||||||
| Normal closure: | $C_3^5:(C_2\times S_4)$ | |||||||
| Core: | $C_1$ | |||||||
| Minimal over-subgroups: | $D_6$ | $D_6$ | $D_6$ | $D_6$ | $D_6$ | $D_6$ | $D_6$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $1458$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:(C_2\times S_4)$ |