Properties

Label 40500.e.3.a1
Order $ 2^{2} \cdot 3^{3} \cdot 5^{3} $
Index $ 3 $
Normal Yes

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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: $a^{3}, b^{3}, c^{3}d^{9}, b^{10}c^{10}d^{6}, c^{10}, d^{3}, a^{6}, c^{10}d^{10}$ Copy content Toggle raw display
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

Centralizer:$C_3$
Normalizer:$C_{15}\wr C_3:C_4$
Complements:$C_3$
Minimal over-subgroups:$C_{15}\wr C_3:C_4$
Maximal under-subgroups:$(C_5\times C_{15}^2):C_6$$C_3\times C_5^3:(C_3:C_4)$$C_5\wr C_3:C_{12}$$C_{15}^2:C_3:C_4$$\He_3:F_5$

Other information

Number of conjugacy classes in this autjugacy class$1$
Möbius function$-1$
Projective image$C_{15}^2:C_{15}:C_4$