Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(350\)\(\medspace = 2 \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(3\) |
| Generators: |
$c^{70}$
|
| 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), a direct factor, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{15}:D_{35}$ |
| Order: | \(1050\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_5:D_{35}$ |
| Order: | \(350\)\(\medspace = 2 \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Automorphism Group: | $C_5^2:C_4.S_5\times F_7$ |
| Outer Automorphisms: | $C_3\times \GL(2,5)$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_5^2:C_4.S_5\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(504000\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{3} \cdot 7 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{15}:D_{35}$ | |||||||
| Normalizer: | $C_{15}:D_{35}$ | |||||||
| Complements: | $C_5:D_{35}$ | |||||||
| Minimal over-subgroups: | $C_{21}$ | $C_{15}$ | $C_{15}$ | $C_{15}$ | $C_{15}$ | $C_{15}$ | $C_{15}$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $875$ |
| Projective image | $C_5:D_{35}$ |