Subgroup ($H$) information
| Description: | $C_3:D_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, b, b^{2}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $(C_2\times C_{24}):D_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2\times C_2^6.C_2^6.C_2)$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{W}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $C_6:D_4$ | |||
| Normal closure: | $C_{12}:D_4$ | |||
| Core: | $C_6$ | |||
| Minimal over-subgroups: | $C_6:D_4$ | |||
| Maximal under-subgroups: | $C_2\times C_6$ | $D_6$ | $C_3:C_4$ | $D_4$ |
Other information
| Number of subgroups in this autjugacy class | $64$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | not computed |