Subgroup ($H$) information
| Description: | $C_5^7:D_7$ |
| Order: | \(1093750\)\(\medspace = 2 \cdot 5^{7} \cdot 7 \) |
| Index: | \(3\) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$h, gh^{4}, b^{21}, ef^{2}g^{4}h^{4}, ce^{2}f^{2}gh^{2}, fg^{3}h^{4}, a^{3}, b^{5}cd^{2}e^{2}f^{4}g^{3}, de^{3}f^{4}h^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Hall subgroup, solvable, and an A-group. Whether it is a direct factor or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_5^7:F_7$ |
| Order: | \(3281250\)\(\medspace = 2 \cdot 3 \cdot 5^{7} \cdot 7 \) |
| 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: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| 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, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^6.C_{35}.C_3.C_4^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_5^6.C_{217}.C_{30}.C_2^3.C_2$ |
| $W$ | $C_5^7:F_7$, of order \(3281250\)\(\medspace = 2 \cdot 3 \cdot 5^{7} \cdot 7 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_5^7:F_7$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5^7:F_7$ |