Subgroup ($H$) information
| Description: | not computed |
| Order: | \(354294\)\(\medspace = 2 \cdot 3^{11} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(4,5,6)(10,11,12)(16,18,17)(22,24,23)(28,30,29)(34,35,36), (7,8,9)(10,12,11) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is normal, nonabelian, and supersolvable (hence solvable and monomial). Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.(C_3^2\wr C_2.D_4)$ |
| Order: | \(2834352\)\(\medspace = 2^{4} \cdot 3^{11} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(102036672\)\(\medspace = 2^{6} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
Related subgroups
| Centralizer: | $C_3^2$ |
| Normalizer: | $C_3^7.(C_3^2\wr C_2.D_4)$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^7.(C_3^2\wr C_2.D_4)$ |