Subgroup ($H$) information
| Description: | $C_{12}:C_2^4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,10,3,4)(5,7)(11,12)(13,14), (2,4)(3,10)(11,13)(12,14), (2,3)(4,10), (4,10)(6,8)(11,14)(12,13), (2,3), (1,7,5)(2,3)(4,10), (11,13)(12,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.
Ambient group ($G$) information
| Description: | $S_5^2:D_4$ |
| Order: | \(115200\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times A_5^2).D_4^2$, of order \(460800\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^9.(D_6\times S_4)$, of order \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \) |
| $W$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_{12}:C_2^5$ | ||
| Normal closure: | $C_2^2\times S_5^2$ | ||
| Core: | $C_2$ | ||
| Minimal over-subgroups: | $C_2\times D_4\times S_5$ | $C_2^2:D_6^2$ | $C_{12}:C_2^5$ |
Other information
| Number of subgroups in this autjugacy class | $600$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $S_5^2:C_2^2$ |