Subgroup ($H$) information
| Description: | $C_2^2:A_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(8,10)(9,13)(11,15)(12,14), (4,5,6)(8,10,15)(12,13,14), (8,15)(9,12)(10,11)(13,14), (4,7)(5,6)(8,15)(10,11), (4,5)(6,7)(8,11)(10,15)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3\times C_2^5:S_4$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_6.C_2^5$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $\AGammaL(2,4)$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^3:A_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $C_3\times C_2^3:A_4$ | ||
| Normal closure: | $C_2^4:A_4$ | ||
| Core: | $C_2^2$ | ||
| Minimal over-subgroups: | $C_2^4:A_4$ | $C_2^4:C_3^2$ | $C_2^3:A_4$ |
| Maximal under-subgroups: | $C_2^4$ | $A_4$ | $A_4$ |
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3\times C_2^5:S_4$ |