Subgroup ($H$) information
| Description: | $C_2^2:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(177147\)\(\medspace = 3^{11} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,30,33,23)(2,28,31,24)(3,29,32,22)(4,20,36,26)(5,21,34,25)(6,19,35,27) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^7.(C_3^2\wr C_2.D_4)$ |
| Order: | \(2834352\)\(\medspace = 2^{4} \cdot 3^{11} \) |
| 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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(102036672\)\(\medspace = 2^{6} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^2:C_4$ |
| Normal closure: | $C_3^7.(C_3^2\wr C_2.D_4)$ |
| Core: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $177147$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.(C_3^2\wr C_2.D_4)$ |