Subgroup ($H$) information
| Description: | $C_2^9$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(4,6)(5,24)(7,22)(8,20), (4,20)(5,22)(6,8)(7,24), (2,23)(3,17)(9,11)(14,21) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor or a semidirect factor 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: | $C_7^3:C_3^2:D_6$ |
| Order: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Automorphism Group: | $C_2\times C_7^3.\He_3.Q_8.C_6$ |
| Outer Automorphisms: | $C_2\times \SL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 \(56899584\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $\GL(9,2)$ |
| $W$ | $C_7^3:C_3^2:S_3$, of order \(18522\)\(\medspace = 2 \cdot 3^{3} \cdot 7^{3} \) |
Related subgroups
| Centralizer: | $C_2^{10}$ |
| 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 | $C_2^9.C_7^3:C_3^2:D_6$ |