Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{132} & 0 \\ 0 & \alpha^{132} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{24} & 0 \\ 0 & \alpha^{24} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $\SL(2,13):C_{14}$ |
| Order: | \(30576\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{2} \cdot 13 \) |
| 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: | $C_2\times \PSL(2,13)$ |
| Order: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| 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: | $1$ |
The quotient is nonabelian, an A-group, and nonsolvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6\times \PSL(2,13).C_2$, of order \(26208\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\SL(2,13):C_{14}$ | |||||
| Normalizer: | $\SL(2,13):C_{14}$ | |||||
| Minimal over-subgroups: | $C_{182}$ | $C_7\times C_{14}$ | $C_{42}$ | $C_{28}$ | $C_2\times C_{14}$ | $C_{28}$ |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
Other information
| Möbius function | $1092$ |
| Projective image | $C_2\times \PSL(2,13)$ |