Subgroup ($H$) information
| Description: | $C_7^5:D_5$ |
| Order: | \(168070\)\(\medspace = 2 \cdot 5 \cdot 7^{5} \) |
| Index: | $1$ |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$ad^{5}e^{6}f^{3}, ef^{3}, b^{5}, cd^{5}e^{6}f^{5}, b^{21}c^{4}d^{3}, f, def^{4}$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), nonabelian, a Hall subgroup, and an A-group. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_7^5:D_5$ |
| Order: | \(168070\)\(\medspace = 2 \cdot 5 \cdot 7^{5} \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5.C_{120}.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_7^5.C_{120}.C_6.C_2^3$ |
| $W$ | $C_7^5:D_5$, of order \(168070\)\(\medspace = 2 \cdot 5 \cdot 7^{5} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_7^5:D_5$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5:D_5$ |