C₃⋊#1084 #66412

order	324
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃²

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃×H(3,𝔽₃)	C₄
Fitting, prime radical, cosocle	C₂×C₃×H(3,𝔽₃)	C₂
socle	C₂×C₃²	C₃⋊D₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃

148 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 104);