Subgroup ($H$) information
| Description: | $C_5^7:\PGL(2,7)$ |
| Order: | \(26250000\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{7} \cdot 7 \) |
| Index: | \(2\) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(21,24,22,25,23)(36,38,40,37,39), (1,11,2,12,3,13,4,14,5,15)(6,23,7,24,8,25,9,21,10,22) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and nonsolvable. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_5^7:(C_2\times \PGL(2,7))$ |
| Order: | \(52500000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{7} \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^7:(C_4\times \PGL(2,7))$, of order \(105000000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{7} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_5^7:(C_4\times \PGL(2,7))$, of order \(105000000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{7} \cdot 7 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |