Subgroup ($H$) information
| Description: | $C_5^2\times C_{15}$ |
| Order: | \(375\)\(\medspace = 3 \cdot 5^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$c^{10}, c^{3}d^{9}, d^{3}, b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and elementary for $p = 5$ (hence hyperelementary).
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: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4.\PSL(3,5)$ |
| $W$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-3$ |
| Projective image | $C_3\times C_5^3:C_6$ |