Subgroup ($H$) information
Description: | $C_{15}^2:C_{15}:C_4$ |
Order: | \(13500\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{3} \) |
Index: | \(3\) |
Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Generators: |
$a^{3}, b^{3}, c^{3}d^{9}, b^{10}c^{10}d^{6}, c^{10}, d^{3}, a^{6}, c^{10}d^{10}$
|
Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and monomial (hence solvable).
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: | $C_3$ |
Order: | \(3\) |
Exponent: | \(3\) |
Automorphism Group: | $C_2$, of order \(2\) |
Outer Automorphisms: | $C_2$, of order \(2\) |
Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
$\operatorname{Aut}(G)$ | $(C_5\times C_{15}^2).C_6^2.C_2^4$ |
$\operatorname{Aut}(H)$ | $F_5\times C_5^2.\He_3.C_2^3.C_2$ |
$W$ | $C_{15}^2:C_{15}:C_4$, of order \(13500\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{3} \) |
Related subgroups
Other information
Number of conjugacy classes in this autjugacy class | $1$ |
Möbius function | $-1$ |
Projective image | $C_{15}^2:C_{15}:C_4$ |