Subgroup ($H$) information
| Description: | $C_{56}$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{48} & 0 \\ 0 & \alpha^{48} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{132} & 0 \\ 0 & \alpha^{132} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{150} & 0 \\ 0 & \alpha^{150} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{159} & 0 \\ 0 & \alpha^{159} \\ \end{array}\right)$
|
| 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, and cyclic (hence elementary ($p = 2,7$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_{56}.\PGL(2,13)$ |
| Order: | \(122304\)\(\medspace = 2^{6} \cdot 3 \cdot 7^{2} \cdot 13 \) |
| 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: | $\PGL(2,13)$ |
| Order: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Automorphism Group: | $\PGL(2,13)$, of order \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian, almost simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times C_6\times \PSL(2,13).C_2$, of order \(104832\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{56}.\PGL(2,13)$ | ||||
| Normalizer: | $C_{56}.\PGL(2,13)$ | ||||
| Minimal over-subgroups: | $C_{728}$ | $C_7\times C_{56}$ | $C_{168}$ | $C_2\times C_{56}$ | $C_2\times C_{56}$ |
| Maximal under-subgroups: | $C_{28}$ | $C_8$ |
Other information
| Möbius function | $1092$ |
| Projective image | $\PGL(2,13)$ |