Subgroup ($H$) information
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(14880\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 31 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
15 & 0 \\
0 & 15
\end{array}\right), \left(\begin{array}{rr}
8 & 0 \\
0 & 8
\end{array}\right)$
|
| 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, the socle, and cyclic (hence elementary ($p = 2,5$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_5\times \SL(2,31)$ |
| Order: | \(148800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 31 \) |
| Exponent: | \(14880\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 31 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $\PSL(2,31)$ |
| Order: | \(14880\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 31 \) |
| Exponent: | \(7440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 31 \) |
| Automorphism Group: | $\PGL(2,31)$, of order \(29760\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 31 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $0$ |
The quotient is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times \PSL(2,31).C_2$, of order \(119040\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 31 \) |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_5\times \SL(2,31)$ | |||
| Normalizer: | $C_5\times \SL(2,31)$ | |||
| Minimal over-subgroups: | $C_{310}$ | $C_5\times C_{10}$ | $C_{30}$ | $C_{20}$ |
| Maximal under-subgroups: | $C_5$ | $C_2$ |
Other information
| Möbius function | $29760$ |
| Projective image | $\PSL(2,31)$ |