Subgroup ($H$) information
| Description: | $C_3^3:S_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(10,15,14)(11,16,12)(13,18,17), (1,7)(2,6)(3,5)(8,9)(10,12)(11,14)(13,18) \!\cdots\! \rangle$
|
| 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: | $C_3^4:D_4.D_4$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2^2\wr C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr S_3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:(Q_8^2:C_2^2)$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\AGL(4,3)$, of order \(1965150720\)\(\medspace = 2^{9} \cdot 3^{10} \cdot 5 \cdot 13 \) |
| $W$ | $C_3^4:D_4.D_4$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_3^4:D_4.D_4$ |