Subgroup ($H$) information
| Description: | $C_6^2:C_9$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{4}, d^{3}, c^{3}, d^{2}, b^{3}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is the commutator subgroup (hence characteristic and normal), maximal, a semidirect factor, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^2:D_9$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.(S_3\times S_4)$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_6^2.C_3^3.D_6$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3:S_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^2$ | ||||
| Normalizer: | $C_6^2:D_9$ | ||||
| Complements: | $C_2$ | ||||
| Minimal over-subgroups: | $C_6^2:D_9$ | ||||
| Maximal under-subgroups: | $C_3\times C_6^2$ | $C_3^2.A_4$ | $C_3^2.A_4$ | $C_3^2.A_4$ | $C_3^2:C_9$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_3^2.S_4$ |