Subgroup ($H$) information
| Description: | $C_{15}^2:C_{15}$ |
| Order: | \(3375\)\(\medspace = 3^{3} \cdot 5^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$b^{10}c^{4}d^{6}, d^{3}, b^{3}, c^{10}, c^{5}d^{10}, c^{3}$
|
| Derived length: | $2$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_{15}\wr C_3:C_4$ |
| Order: | \(40500\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}^2).C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_4\times C_5^2.C_3^3.C_{12}.C_2^3$ |
| $W$ | $C_{15}^2:C_3:C_4$, of order \(2700\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{15}^2:C_{15}:C_4$ |