Subgroup ($H$) information
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(19,21)(20,22), (1,9,6)(2,3,15)(4,7,18)(5,8,17)(10,16,12)(11,13,14), (2,3,15)(10,16,12)(11,13,14)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), the socle, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3^4:D_6\wr C_2$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2\times C_3^4:D_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^4.Q_8.C_6.C_2^4.C_2$ |
| Outer Automorphisms: | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.C_3^4.C_2^6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |