C₃⋊Dic₁₅ #4372

order	180
isoclinism	isoclinic class of C₃⋊D₁₅
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×AGL(2,𝔽₃) (order 17280) · Inn(G) = C₃⋊D₁₅ · Out(G) = C₂×#985
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

C₃⋊Dic₁₅

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

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 3 generators on 15 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 17);