C₃×#383145 #72248385

order	1536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₃×#383145

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#2668
commutator	C₂²×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#3003	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

4984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72248402, D₃×#383145
maximal subgroup of	#72248402, D₃×#383145, C₃×#72215520
maximal normal subgroup of	#72248402, D₃×#383145, C₃×#72215520

GAP

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

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

C3×#383145 #72248385