Subgroup ($H$) information
| Description: | $C_4^2:D_6$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(5\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, c^{2}, c^{3}, b^{3}, b^{4}, d^{5}, b^{6}c^{3}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $(C_2\times D_{60}):C_4$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}:(C_2^3.C_2^6.C_2^2)$ |
| $\operatorname{Aut}(H)$ | $D_4^2:C_2^2\times S_3$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_4^2:C_2^2\times S_3$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times D_{12}$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $-1$ |
| Projective image | $C_{10}:D_{12}$ |