C₃×#123403 #72226103

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₁₆

Aliases

direct product

C₃×#123403

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1712
commutator, Frattini, cosocle	C₈²	C₂³×C₃
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5)(6,9)(7,16)(8,12)(10,13)(11,14)(15,18)(17,19)(21,25)(22,26)(23,29)(27,35)(28,33)(30,34)(31,32);(5,9)(6,13)(7,14)(10,11)(12,18)(15,19)(16,17)(21,25)(22,26)(23,29)(27,35)(28,33)(30,34)(31,32);(20,21,26,32,23,27,30,33,24,28,34,35,29,31,22,25)

Conjugacy classes & character table

402 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×#72213956
maximal normal subgroup of	C₃×#72213956

GAP

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

gap> G := Group((1,2,3)(4,5)(6,9)(7,16)(8,12)(10,13)(11,14)(15,18)(17,19)(21,25)(22,26)(23,29)(27,35)(28,33)(30,34)(31,32), (5,9)(6,13)(7,14)(10,11)(12,18)(15,19)(16,17)(21,25)(22,26)(23,29)(27,35)(28,33)(30,34)(31,32), (20,21,26,32,23,27,30,33,24,28,34,35,29,31,22,25));

C3×#123403 #72226103