Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(2250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$c^{10}$
|
| 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), the Frattini subgroup, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), stem, a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $(C_5\times C_{15}^2):C_6$ |
| Order: | \(6750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3\times C_5^3:C_6$ |
| Order: | \(2250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $C_5^3.C_{12}^2.C_2^3$ |
| Outer Automorphisms: | $D_6\times \OD_{16}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $(C_5\times C_{15}^2):C_6$ | |||||
| Normalizer: | $(C_5\times C_{15}^2):C_6$ | |||||
| Minimal over-subgroups: | $C_{15}$ | $C_{15}$ | $C_{15}$ | $C_3^2$ | $C_3^2$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-375$ |
| Projective image | $C_3\times C_5^3:C_6$ |