Subgroup ($H$) information
| Description: | $C_2^5.C_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,3)(2,5)(4,8)(6,7), (10,12), (3,8)(5,7), (1,5,3,6,4,7,8,2)(9,12)(10,11)(14,15), (1,4)(2,6)(3,8)(5,7), (9,11)(10,12), (2,6)(5,7)\rangle$
|
| Nilpotency class: | $3$ |
| 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: | $(D_6\times C_2^4):C_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^7.C_2^6.C_2^2)$ |
| $\operatorname{Aut}(H)$ | $(C_2^2\times D_4^2).D_4^2$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\card{W}$ | \(64\)\(\medspace = 2^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | not computed |