Subgroup ($H$) information
| Description: | $C_2^9$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(21,22)(23,24), (17,18)(19,20)(21,22) \!\cdots\! \rangle$
|
| 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: | $C_2^{12}.(A_4:S_4^2.A_4)$ |
| Order: | \(339738624\)\(\medspace = 2^{22} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^5.A_4^3:A_4$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | Group of order \(3822059520\)\(\medspace = 2^{20} \cdot 3^{6} \cdot 5 \) |
| Outer Automorphisms: | $C_5^3:C_{20}.Q_8$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(86973087744\)\(\medspace = 2^{30} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\GL(9,2)$, of order \(699\!\cdots\!200\)\(\medspace = 2^{36} \cdot 3^{5} \cdot 5^{2} \cdot 7^{3} \cdot 17 \cdot 31 \cdot 73 \cdot 127 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |