Subgroup ($H$) information
| Description: | $C_5^7:D_7$ |
| Order: | \(1093750\)\(\medspace = 2 \cdot 5^{7} \cdot 7 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(6,7,8,9,10)(16,17,18,19,20)(21,22,23,24,25)(26,30,29,28,27)(31,34,32,35,33) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, solvable, and an A-group. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_5^7:(C_2\times F_7)$ |
| Order: | \(6562500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{7} \cdot 7 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^6.C_{35}.C_3.C_4^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_5^6.C_{217}.C_{30}.C_2^3.C_2$ |
| $\card{W}$ | 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 |