Subgroup ($H$) information
| Description: | $A_4$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(680\)\(\medspace = 2^{3} \cdot 5 \cdot 17 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,14)(2,6)(3,4)(5,15)(7,10)(8,11)(9,12)(13,17), (1,12)(2,11)(3,15)(4,5)(6,8)(7,13)(9,14)(10,17), (1,4,11)(2,14,15)(3,6,12)(5,8,9)(10,17,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $\SOMinus(4,4)$ |
| Order: | \(8160\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 17 \) |
| Exponent: | \(1020\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 17 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,16).C_4$, of order \(16320\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_2\times A_4$ | ||
| Normal closure: | $\SL(2,16)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $A_5$ | $C_2^2:A_4$ | $C_2\times A_4$ |
| Maximal under-subgroups: | $C_2^2$ | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $340$ |
| Möbius function | $-1$ |
| Projective image | $\SOMinus(4,4)$ |