Subgroup ($H$) information
| Description: | $C_3^4:C_{12}$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{3}, ef^{2}gh^{2}, dh^{2}, a^{2}b^{6}c^{2}d^{2}fh, fh^{2}, b^{6}c^{2}d^{2}fh, cef^{2}h^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^6.\PSU(3,2)$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| 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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^3.C_4.C_2^4$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:C_{24}:D_6$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $W$ | $C_3^3:\PSU(3,2)$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $162$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_3^6.\PSU(3,2)$ |