Subgroup ($H$) information
| Description: | $C_6.C_2^4$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(9,11,10), (3,7)(5,8), (2,6)(3,7), (1,5)(2,3)(4,8)(6,7), (1,2)(3,5)(4,6)(7,8), (1,4)(2,6)(3,7)(5,8)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^2\times Q_8:A_4$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:S_3.C_2^6.S_3^3$ |
| $\operatorname{Aut}(H)$ | $S_4^2:C_2^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $A_4^2:C_2^3$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $W$ | $C_2^2:A_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3\times C_6$ | ||
| Normalizer: | $C_3^2\times Q_8:A_4$ | ||
| Complements: | $C_3^2$ | ||
| Minimal over-subgroups: | $C_3\times Q_8:A_4$ | $D_4:C_6^2$ | |
| Maximal under-subgroups: | $D_4:C_6$ | $C_6\times D_4$ | $D_4:C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $3$ |
| Projective image | $C_2^4:C_3^2$ |