Subgroup ($H$) information
| Description: | $(C_5\times C_{15}^2).C_6$ |
| Order: | \(6750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{3} \) |
| Index: | $1$ |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$a^{3}d^{5}, c^{3}d^{3}, c^{10}, d^{10}, a^{2}, d^{3}, b$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, monomial, and metabelian.
Ambient group ($G$) information
| Description: | $(C_5\times C_{15}^2).C_6$ |
| Order: | \(6750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}^2).C_{12}^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $(C_5\times C_{15}^2).C_{12}^2.C_2^2$ |
| $W$ | $C_3\times C_5^3:C_6$, of order \(2250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{3} \) |
Related subgroups
| Centralizer: | $C_3$ | ||||
| Normalizer: | $(C_5\times C_{15}^2).C_6$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $C_{15}^2.C_{15}$ | $C_3^2\times C_5^2:D_5$ | $C_5^3:C_{18}$ | $C_{15}^2.C_6$ | $C_{45}:C_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3\times C_5^3:C_6$ |