Subgroup ($H$) information
| Description: | $C_{12}.D_8$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(5\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, b^{4}, c^{45}, b, c^{30}, c^{40}, b^{6}c^{45}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{12}.D_{40}$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2^4.C_2^6.C_2)$ |
| $\operatorname{Aut}(H)$ | $C_2^3\times D_4^2$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(S)$ | $C_2^3\times D_4^2$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^2:C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $-1$ |
| Projective image | $D_{10}:C_4$ |