Subgroup ($H$) information
| Description: | $C_2^9$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(3,4)(5,6)(15,16)(17,18), (1,2)(5,6)(9,10)(11,12), (1,2)(5,6)(7,8)(9,10) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, abelian (hence 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: | $C_2^9.C_3:S_3^3:C_4$ |
| Order: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3:S_3^3:C_4$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_3\wr D_4$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3^4.C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\GL(9,2)$ |
| $W$ | $D_5^3.C_2^2$, of order \(4000\)\(\medspace = 2^{5} \cdot 5^{3} \) |
Related subgroups
| Centralizer: | $C_2^9$ |
| Normalizer: | $C_2^9.C_3:S_3^3:C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4^2:\POPlus(4,3).D_4$ |