C₂³×D₃² #64392

order	288
isoclinism	isoclinic class of D₃²
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product	C₂³×D₃²
unambiguous semidirect product	(C₃×D₃)⋊C₂⁴
unambiguous semidirect product	(C₃⋊D₃)⋊C₂⁴

Fitting, socle	C₂³×C₃² ⋊ C₂²
derived, Sylow	C₃² ⋊ C₂⁵

	subgroup	G/H
center, hypercenter	C₂³	D₃²
commutator, cosocle, hypocenter	C₃²	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³

1563 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 5 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1040);