C₃×#375718 #72271730

order	1536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

direct product

C₃×#375718

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#14169
commutator	C₂²×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃×C₄ ◁ G

2234 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)(4,5,8,11,9,13,22,27)(6,14,16,24,17,25,21,10)(7,12,20,26,19,29,30,35)(15,28,18,32,31,34,33,23);(4,6)(5,10,13,24)(7,18,19,33)(8,21)(9,17)(11,25,27,14)(12,28)(15,30,31,20)(16,22)(23,26)(29,34)(32,35);(5,11)(8,22)(10,14)(12,26)(13,27)(15,31)(16,21)(20,30)(23,34)(24,25)(28,32)(29,35);(4,7,9,19)(5,12,13,29)(6,15)(8,20,22,30)(10,23)(11,26,27,35)(14,28)(16,18)(17,31)(21,33)(24,32)(25,34)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72216507
maximal normal subgroup of	C₃×#72216507

GAP

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

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

C3×#375718 #72271730