Subgroup ($H$) information
| Description: | $C_3:S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(9,12,13)(10,14,11), (10,14,11), (11,14)(12,13)(15,18)(16,17)\rangle$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $(S_3^2\times A_4^2).D_4$ |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times A_4^2:D_4$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $A_4^2.C_2^4.C_2^3$ |
| Outer Automorphisms: | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{50}).D_{25}$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $W$ | $\SOPlus(4,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2^2\times A_4^2$ |
| Normalizer: | $(S_3^2\times A_4^2).D_4$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $(S_3^2\times A_4^2).D_4$ |