Subgroup ($H$) information
| Description: | $C_2^3:\GL(2,\mathbb{Z}/4)$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(15,17), (3,9)(5,8)(6,7)(10,11,12,13)(14,15,16,17), (3,5)(8,9)(10,12)(11,13) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $S_3^3.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3.C_4$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $A_4^2.C_2^5.C_2^2$ |
| $W$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^3:\GL(2,\mathbb{Z}/4)$ |
| Normal closure: | $S_3^3.\GL(2,\mathbb{Z}/4)$ |
| Core: | $C_2^4$ |
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5^4:(C_2^2\times C_4)$ |