Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
5 & 0 \\
0 & 5
\end{array}\right), \left(\begin{array}{rr}
12 & 0 \\
0 & 12
\end{array}\right)$
|
| 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: | $\SL(2,13):C_2$ |
| Order: | \(4368\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \PGL(2,13)$, of order \(4368\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\SL(2,13):C_2$ | |||
| Normalizer: | $\SL(2,13):C_2$ | |||
| Minimal over-subgroups: | $C_{52}$ | $C_{28}$ | $C_{12}$ | $C_2\times C_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $-1092$ |
| Projective image | $\PSL(2,13)$ |