Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(7\) |
| Generators: |
$c^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $7$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{210}.S_3$ |
| Order: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_6.D_{15}$ |
| Order: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Automorphism Group: | $C_{30}:C_{12}:\GL(2,3)$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| Outer Automorphisms: | $\GL(2,3):C_2^2$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6\times \AGL(2,3)\times F_5$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{210}.S_3$ | |||||
| Normalizer: | $C_{210}.S_3$ | |||||
| Complements: | $C_6.D_{15}$ | |||||
| Minimal over-subgroups: | $C_{35}$ | $C_{21}$ | $C_{21}$ | $C_{21}$ | $C_{21}$ | $C_{14}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_6.D_{15}$ |