Subgroup ($H$) information
| Description: | not computed |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Index: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | not computed |
| Generators: |
$\langle(3,4)(19,20)(25,26)(27,28), (23,24)(27,28), (21,22)(23,24)(25,26)(27,28) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, abelian (hence metabelian and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{13}.\PSL(2,13)$ |
| Order: | \(8945664\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence nonsolvable).
Quotient group ($Q$) structure
| Description: | $\PSL(2,13)$ |
| Order: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(546\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Automorphism Group: | $\PGL(2,13)$, of order \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $0$ |
The quotient is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{14}.\PSL(2,13).C_2$, of order \(35782656\)\(\medspace = 2^{17} \cdot 3 \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $\PSL(2,13)$, of order \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
Related subgroups
| Centralizer: | $C_2^{13}$ |
| Normalizer: | $C_2^{13}.\PSL(2,13)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |