Subgroup ($H$) information
| Description: | $\SOPlus(4,2)^2.C_2$ |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Index: | \(24200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(2,21,11)(5,15,18)(9,12,14), (1,20,8,10)(13,22,17,16), (4,19)(5,14)(6,7) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2\times M_{11}\wr C_2$ |
| Order: | \(250905600\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times M_{11}\wr C_2$, of order \(250905600\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_4^2.C_2^4$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\card{W}$ | 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 | $6050$ |
| Möbius function | not computed |
| Projective image | not computed |