Subgroup ($H$) information
| Description: | $C_3^3:S_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(12,16,15), (11,14,13), (1,4,5)(2,6,3)(11,13,14), (2,6,3)(11,14,13)(12,16,15), (2,6)(4,5)(11,13)(12,15)\rangle$
|
| 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: | $C_6^2:\SOPlus(4,2)$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| 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: | $C_2\times D_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8.C_6.C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $\AGL(4,3)$, of order \(1965150720\)\(\medspace = 2^{9} \cdot 3^{10} \cdot 5 \cdot 13 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^4:\GL(2,3):D_4$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_3^4:D_4$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||||
| Normalizer: | $C_6^2:\SOPlus(4,2)$ | ||||
| Minimal over-subgroups: | $C_3^3:D_6$ | $C_3^4:C_4$ | $C_3^3:D_6$ | $C_3^2:S_3^2$ | $C_3^4:C_4$ |
| Maximal under-subgroups: | $C_3^4$ | $C_3^2:S_3$ | $C_3^2:S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_6^2:\SOPlus(4,2)$ |