C₃×#144388 #72255292

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#144388

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂³×C₃
prime radical	C₃×#23900	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

2844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4,6)(3,5)(7,8)(9,10)(11,17,19,14)(12,21)(13,16)(15,22)(18,24,20,23)(25,26,27);(5,8)(11,18)(14,23)(17,24)(19,20);(1,3,4,7)(2,5)(6,8)(9,11,12,18)(10,14,15,23)(13,19,22,20)(16,17,21,24)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72270525, D₃×#144388
maximal subgroup of	C₃×#72213293, #72270525, D₃×#144388
maximal normal subgroup of	C₃×#72213293, #72270525, D₃×#144388

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,4,6)(3,5)(7,8)(9,10)(11,17,19,14)(12,21)(13,16)(15,22)(18,24,20,23)(25,26,27), (5,8)(11,18)(14,23)(17,24)(19,20), (1,3,4,7)(2,5)(6,8)(9,11,12,18)(10,14,15,23)(13,19,22,20)(16,17,21,24));

C3×#144388 #72255292