C₃×#147573 #72248113

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#147573

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1781
commutator	C₂³×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×#41	C₂³×C₃
prime radical	C₃×#35512	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×H(5,𝔽₂) ◁ G

3750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72248142, D₃×#147573
maximal subgroup of	#72248142, D₃×#147573, C₃×#72215601
maximal normal subgroup of	#72248142, D₃×#147573, C₃×#72215601

GAP

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

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

C3×#147573 #72248113