Subgroup ($H$) information
| Description: | $(C_3^2\times \He_3).F_9$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Index: | $1$ |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, egh^{2}, cde^{2}fh, e^{2}fg^{2}h^{2}, g, a^{2}, bc^{2}d, dg^{2}, de^{2}h^{2}, a^{4}$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and monomial.
Ambient group ($G$) information
| Description: | $(C_3^2\times \He_3).F_9$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.C_2.D_4^2$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.C_3^4.C_2.D_4^2$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $W$ | $(C_3^2\times \He_3).F_9$, of order \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $(C_3^2\times \He_3).F_9$ | |
| Complements: | $C_1$ | |
| Maximal under-subgroups: | $C_3^5.C_3^2.C_4$ | $C_3^3.F_9$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $(C_3^2\times \He_3).F_9$ |