Subgroup ($H$) information
| Description: | $C_3^2:S_3$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$b^{6}, df, ef, f$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^2:\SU(3,2)$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $S_3^2$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $\SOPlus(4,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3^2\times S_3^2):\SD_{16}$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $W$ | $\PSU(3,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^3$ | |||||
| Normalizer: | $C_3^2:\SU(3,2)$ | |||||
| Minimal over-subgroups: | $C_3^3:S_3$ | $C_3^3:S_3$ | $C_3^3:S_3$ | $\He_3:C_4$ | $\He_3:C_4$ | $\He_3:C_4$ |
| Maximal under-subgroups: | $\He_3$ | $C_3\times S_3$ |
Other information
| Möbius function | $18$ |
| Projective image | $C_3^4:Q_8$ |