C₂³×C₃⋊D₃ #3853

order	144
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) order 580608
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

C₂³×C₃⋊D₃

Fitting, socle	C₂³×C₃² ⋊ C₂
derived, Sylow	C₃² ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂³	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³

402 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 4 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 196);