Subgroup ($H$) information
| Description: | $C_3^5.(S_3\times M_{11})$ |
| Order: | \(11547360\)\(\medspace = 2^{5} \cdot 3^{8} \cdot 5 \cdot 11 \) |
| Index: | $1$ |
| Exponent: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$\langle(1,3,2)(7,9,8)(10,12,11)(13,15,14)(19,20,21)(22,23,24)(25,26,27)(28,30,29) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, and nonsolvable. Whether it is a direct factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^5.(S_3\times M_{11})$ |
| Order: | \(11547360\)\(\medspace = 2^{5} \cdot 3^{8} \cdot 5 \cdot 11 \) |
| Exponent: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.(S_3\times M_{11})$, of order \(11547360\)\(\medspace = 2^{5} \cdot 3^{8} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_3^5.(S_3\times M_{11})$, of order \(11547360\)\(\medspace = 2^{5} \cdot 3^{8} \cdot 5 \cdot 11 \) |
| $W$ | $C_3^5.(S_3\times M_{11})$, of order \(11547360\)\(\medspace = 2^{5} \cdot 3^{8} \cdot 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_3^5.(S_3\times M_{11})$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^5.(S_3\times M_{11})$ |