Subgroup ($H$) information
| Description: | $C_3^5:S_3$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Index: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(10,12,11)(13,14,15)(22,24,23)(25,27,26)(28,30,29)(31,33,32), (1,2,3)(4,5,6) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence 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: | $C_3^{12}.C_2^6.C_{12}.C_2$ |
| Order: | \(816293376\)\(\medspace = 2^{9} \cdot 3^{13} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^6.C_2^5:S_4$ |
| Order: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_3^6.C_2^7:S_4$, of order \(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(13060694016\)\(\medspace = 2^{13} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $\AGL(6,3)$, of order \(61330486826476707840\)\(\medspace = 2^{13} \cdot 3^{21} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13^{2} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |