Subgroup ($H$) information
| Description: | $C_2\times D_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(6,7)(9,10)(12,13)(14,19)(16,17)(18,20), (2,3)(8,11)(9,10), (1,4)(2,3), (1,2)(3,4)(6,7)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_6^4:D_6$ |
| Order: | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^2.C_3^4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed | |||
| Normalizer: | not computed | |||
| Normal closure: | $C_6^4:D_6$ | |||
| Core: | $C_2$ | |||
| Minimal over-subgroups: | $S_3\times D_4$ | $S_3\times D_4$ | ||
| Maximal under-subgroups: | $D_4$ | $D_4$ | $D_4$ | $D_4$ |
Other information
| Number of subgroups in this autjugacy class | $162$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |