Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(3,5)(4,6)(7,11)(8,10)(9,12), (7,11)(8,10)(9,12), (1,2)(3,4)(5,6)(7,11)(8,12)(9,10)\rangle$
|
| 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), and rational.
Ambient group ($G$) information
| Description: | $C_2\times C_3^3:S_4^2$ |
| Order: | \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_3^3.A_4^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | not computed | ||||
| Normalizer: | not computed | ||||
| Normal closure: | $C_2\times C_3^3:S_4^2$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_2\times D_6$ | ||||
| Maximal under-subgroups: | $C_2^2$ | $C_2^2$ | $C_2^2$ | $C_2^2$ | $C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $648$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_2\times C_3^3:S_4^2$ |