Subgroup ($H$) information
| Description: | $C_3^5:\PSU(3,2)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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: | $4$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable. Whether it is monomial 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 \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.C_3^4.(Q_8.A_4).D_6$, of order \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| $W$ | $C_3^5:F_9:C_2$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $C_3^5:F_9:C_2$ | ||
| Complements: | $C_2$ | ||
| Minimal over-subgroups: | $C_3^5:F_9:C_2$ | ||
| Maximal under-subgroups: | $C_3^5.C_3^2.C_4$ | $C_3^5.C_3^2.C_4$ | $C_3^5:Q_8$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^5:F_9:C_2$ |