Subgroup ($H$) information
| Description: | $C_2\times F_5$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(12,13), (2,10,7,4,8)(3,9,5,11,6), (2,8,7,4)(3,5,9,6)(12,13), (2,7)(3,9)(4,8)(5,6)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\times M_{11}$ |
| Order: | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $M_{11}$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $C_2\times F_5$ | |||
| Normal closure: | $C_2\times M_{11}$ | |||
| Core: | $C_2$ | |||
| Minimal over-subgroups: | $A_6.C_2^2$ | $C_2\times S_5$ | ||
| Maximal under-subgroups: | $D_{10}$ | $F_5$ | $F_5$ | $C_2\times C_4$ |
Other information
| Number of subgroups in this conjugacy class | $396$ |
| Möbius function | $1$ |
| Projective image | $M_{11}$ |