Subgroup ($H$) information
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{63} & 0 \\ 0 & \alpha^{63} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{84} & 0 \\ 0 & \alpha^{84} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{126} & 0 \\ 0 & \alpha^{126} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Fitting subgroup, the radical, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_8.\PSL(2,13)$ |
| Order: | \(8736\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \PSL(2,13).C_2$, of order \(8736\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_8.\PSL(2,13)$ | |||
| Normalizer: | $C_8.\PSL(2,13)$ | |||
| Minimal over-subgroups: | $C_{104}$ | $C_{56}$ | $C_{24}$ | $C_2\times C_8$ |
| Maximal under-subgroups: | $C_4$ |
Other information
| Möbius function | $-1092$ |
| Projective image | $\PSL(2,13)$ |