Subgroup ($H$) information
| Description: | $C_2^2\times \He_3$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$c^{3}, d^{2}, d^{3}, c^{2}, d^{4}e$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3\times \He_3):S_4$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| 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: | $C_3\times S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_3:A_4.C_6.C_2$ |
| $\operatorname{Aut}(H)$ | $S_3\times C_3^2:\GL(2,3)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $W$ | $C_3^2:S_3$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_6^2$ | |||
| Normalizer: | $(C_3\times \He_3):S_4$ | |||
| Minimal over-subgroups: | $C_3^2:C_6^2$ | $\He_3:A_4$ | $\He_3:A_4$ | $C_6^2:C_6$ |
| Maximal under-subgroups: | $C_2\times \He_3$ | $C_6^2$ | $C_6^2$ |
Other information
| Möbius function | $-3$ |
| Projective image | $(C_3\times \He_3):S_4$ |