Subgroup ($H$) information
| Description: | $C_5\times C_{15}^2$ |
| Order: | \(1125\)\(\medspace = 3^{2} \cdot 5^{3} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$c^{10}d^{10}, b^{2}, c^{3}d^{12}, d^{10}, d^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the Fitting subgroup (hence nilpotent, solvable, supersolvable, and monomial), a semidirect factor, and abelian (hence metabelian and an A-group).
Ambient group ($G$) information
| Description: | $C_{15}^2:(C_6\times F_5)$ |
| Order: | \(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{12}$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(2,3)\times C_4.\PSL(3,5)$ |
| $W$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{15}^2:(C_6\times F_5)$ |