C₃⋊(C₃⋊#168) #67107

order	360
isoclinism	isoclinic class of C₃⋊D₁₅
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₃⋊D₁₅
commutator, hypocenter	C₃²×C₅	C₈
Fitting, cosocle	C₃²×C₄×C₅	C₂
socle, prime radical	C₂×C₃²×C₅	C₄
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₄

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 3 generators on 19 points:

(2,3)(4,5)(6,7,8,10,9,11,12,13)(15,16)(18,19);(1,2,4,5,3)(17,18,19);(14,15,16)

Polycyclic presentation:

2^3,3^2,5|a^2=b,b^2=c,d^a=d^2,e^a=e^2,f^a=f^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 37);