C₃×#511793 #72248283

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₃×#511793

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#32898
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴×C₃
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₂³×C₃×D₄ ◁ G

4952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#511793
maximal subgroup of	D₃×#511793
maximal normal subgroup of	D₃×#511793
maximal quotient of	C₃×#72213329

GAP

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

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

C3×#511793 #72248283