Subgroup ($H$) information
| Description: | $C_2^9.C_7^3:C_3^2:D_6$ |
| Order: | \(18966528\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 7^{3} \) |
| Index: | $1$ |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(2,4,9,20)(3,6,14,8)(5,11,22,23)(7,17,24,21)(10,16,18,13,19,15,12), (1,3,8,18,23,22,12,21,6,16,2,4,10,9,5,13,14,7,19,17,24) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a semidirect factor, nonabelian, and a Hall subgroup. Whether it is a direct factor or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.C_7^3:C_3^2:D_6$ |
| Order: | \(18966528\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{10}.C_7^3:C_3\wr S_3$, of order \(56899584\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2^{10}.C_7^3:C_3\wr S_3$, of order \(56899584\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 7^{3} \) |
| $W$ | $C_2^9.C_7^3:C_3^2:S_3$, of order \(9483264\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 7^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^9.C_7^3:C_3^2:D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^9.C_7^3:C_3^2:S_3$ |