Subgroup ($H$) information
| Description: | $C_2.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(6,7)(14,15), (1,5)(3,4)(8,9)(12,13), (6,7)(10,11), (1,5)(3,4)(6,7)(10,11) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^3:F_5\times S_4$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^6:D_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $20$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $C_2^2\times F_5\times S_4$ |