Subgroup ($H$) information
| Description: | $C_3^2:S_3^2$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(10,15,12), (12,15)(13,14), (1,2,4)(11,14,13), (2,4)(5,6)(10,15)(13,14), (1,2,4)(3,6,5), (11,13,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_3^3:S_3^2:C_4$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $D_6$ |
| 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: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^5.D_4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:\GL(2,3)\wr C_2$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\wr D_4$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_3^2\wr C_2.D_4$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3$ | |||
| Normalizer: | $C_3^3:S_3^2:C_4$ | |||
| Minimal over-subgroups: | $C_3^3:S_3^2$ | $C_3:S_3^3$ | $C_3^3:(C_4\times S_3)$ | |
| Maximal under-subgroups: | $C_3^2\wr C_2$ | $C_3^3:S_3$ | $C_3:S_3^2$ | $C_3:S_3^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-6$ |
| Projective image | $C_3^3:S_3^2:C_4$ |