Subgroup ($H$) information
| Description: | $C_2^{13}.A_{14}$ |
| Order: | \(357082280755200\)\(\medspace = 2^{23} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(720720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) |
| Generators: |
$\langle(3,4)(5,6)(9,10)(11,19,21,16,13,30,26,27,18,12,20,22,15,14,29,25,28,17) \!\cdots\! \rangle$
|
| Derived length: | $0$ |
The subgroup is nonabelian and perfect (hence nonsolvable).
Ambient group ($G$) information
| Description: | $C_2^{14}.A_{15}.C_2$ |
| Order: | \(21424936845312000\)\(\medspace = 2^{25} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| Exponent: | \(720720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(21424936845312000\)\(\medspace = 2^{25} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | Group of order \(714164561510400\)\(\medspace = 2^{24} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $15$ |
| Möbius function | not computed |
| Projective image | not computed |