Subgroup ($H$) information
| Description: | $C_3^7:C_2^2$ |
| Order: | \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| Index: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(19,20,21)(22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,36,35), (1,3,2)(4,6,5) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.C_6^4:D_6$ |
| Order: | \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
| 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: | $C_3\wr S_3\times D_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $(C_6\times \He_3).C_2^5$ |
| Outer Automorphisms: | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.A_4.C_3.C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_5^2\wr C_2:\OD_{16}$, of order \(967458816\)\(\medspace = 2^{14} \cdot 3^{10} \) |
| $\card{W}$ | \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | $C_3^4$ |
| Normalizer: | $C_3^6.C_6^4:D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.C_6^4:D_6$ |