Subgroup ($H$) information
| Description: | $C_4^2:C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ac^{3}, b, d^{21}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_2\times C_{12}).D_{28}$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times C_{42}).C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \) |
| $\card{W}$ | \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $42$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |