Subgroup ($H$) information
| Description: | $C_2^{12}$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Index: | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(11,12)(17,18)(25,26)(31,32), (19,20)(21,22)(23,24)(25,26)(27,28)(29,30) \!\cdots\! \rangle$
|
| 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, a semidirect factor, or almost simple has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{16}.C_6^2.S_3^2$ |
| Order: | \(84934656\)\(\medspace = 2^{20} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6^2:\POPlus(4,3)$ |
| Order: | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_{55}:F_{11}$, of order \(2985984\)\(\medspace = 2^{12} \cdot 3^{6} \) |
| Outer Automorphisms: | $C_6^2:D_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
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 \(8153726976\)\(\medspace = 2^{25} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | Group of order \(644\!\cdots\!000\)\(\medspace = 2^{66} \cdot 3^{8} \cdot 5^{3} \cdot 7^{4} \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 31^{2} \cdot 73 \cdot 89 \cdot 127 \) |
| $\card{W}$ | 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 |