Subgroup ($H$) information
| Description: | $\He_3:C_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$ab, d^{2}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $\He_3:C_6^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times C_3^4.C_3^4.C_2^3$, of order \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $\operatorname{res}(S)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $W$ | $\He_3$, of order \(27\)\(\medspace = 3^{3} \) |
Related subgroups
| Centralizer: | $C_6^2$ | |
| Normalizer: | $\He_3:C_6^2$ | |
| Complements: | $C_2\times C_6$ $C_2\times C_6$ $C_2\times C_6$ | |
| Minimal over-subgroups: | $\He_3:C_3^2$ | $\He_3:C_6$ |
| Maximal under-subgroups: | $\He_3$ | $C_3\times C_9$ |
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $9$ |
| Möbius function | $-2$ |
| Projective image | $C_3^2:C_6^2$ |