C₃⋊(C₃⋊C₃₂) #63417

order	288
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₃₂ · Schur multiplier (H₂) = C₃

Aliases

C₃⋊(C₃⋊C₃₂)

C₃² ⋊ C₃₂

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	C₃⋊D₃
commutator, hypocenter	C₃²	C₃₂
Fitting, cosocle	C₃²×C₁₆	C₂
socle, prime radical	C₂×C₃²	C₁₆
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₆

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2,3,7,4,8,11,17,5,9,12,18,14,20,23,27,6,10,13,19,15,21,24,28,16,22,25,29,26,30,31,32)(34,35)(37,38);(36,37,38);(33,34,35)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 65);