Subgroup ($H$) information
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$b^{3}d^{3}g^{2}, gh^{2}, b^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\He_3^2.(C_2\times D_4)$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.C_3^4.D_4.C_2^3$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3\times C_{12}$ | ||
| Normalizer: | $C_3:D_{12}$ | ||
| Normal closure: | $C_3^2.C_3^4.C_4$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\He_3:C_4$ | $C_3\times C_{12}$ | $D_{12}$ |
| Maximal under-subgroups: | $C_6$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $162$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $\He_3^2.(C_2\times D_4)$ |