Subgroup ($H$) information
| Description: | not computed |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Index: | \(2\) |
| Exponent: | not computed |
| Generators: |
$\langle(4,6,5)(7,31,8,32,9,33)(10,22)(11,23)(12,24)(13,26)(14,27)(15,25)(16,28,17,29,18,30) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^5:F_9:C_2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.Q_8.C_6.C_2^3$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $C_3^5:F_9:C_2$ | ||
| Minimal over-subgroups: | $C_3^5:F_9:C_2$ | ||
| Maximal under-subgroups: | $C_3^5.C_3^2.C_4$ | $C_3^4.C_3^2.D_6$ | $C_3^5:D_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^5:F_9:C_2$ |