Subgroup ($H$) information
| Description: | $C_2^{10}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(19,20)(21,22)(25,26)(33,34)(35,36), (3,4)(5,6)(7,8)(13,14)(15,16)(17,18) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is 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: | $C_2^{16}.C_6^2.\PSU(3,2).D_4$ |
| Order: | \(1358954496\)\(\medspace = 2^{24} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $A_4^2\wr C_2.C_2^2.D_4$ |
| Order: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Automorphism Group: | $C_2^8.S_3\wr D_4$, of order \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(43486543872\)\(\medspace = 2^{29} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | Group of order \(366\!\cdots\!200\)\(\medspace = 2^{45} \cdot 3^{6} \cdot 5^{2} \cdot 7^{3} \cdot 11 \cdot 17 \cdot 31^{2} \cdot 73 \cdot 127 \) |
| $\card{W}$ | 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 |