C₃⋊(C₃⋊(C₃⋊C₈)) #6247

order	216
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	Aut(G) order 1213056
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃³

Aliases

C₃³ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₈
Fitting, cosocle	C₃³×C₄	C₂
socle, prime radical	C₂×C₃³	C₄
LCS	G ▷ C₃³
UCS	C₁ ◁ C₄

112 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 4 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 84);