C₃×#495573 #72270198

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#495573

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#21408
commutator	C₂×C₈	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴×C₃
prime radical	C₃×#60698	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#2875 ◁ G

1564 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)(6,10)(7,11)(8,13)(9,12)(14,30)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(21,29)(22,32)(31,35)(33,34);(4,6,7,16,21,15,33,32,9,17,20,31,8,18,19,14)(5,10,11,24,29,23,34,22,12,25,28,35,13,26,27,30);(6,14)(7,19)(8,21)(10,22)(11,27)(13,29)(15,31)(16,18)(17,32)(20,33)(23,24)(25,30)(26,35)(28,34);(6,15,17,18)(7,20)(10,23,25,26)(11,28)(14,31,32,16)(19,33)(22,24,30,35)(27,34)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72270277, D₃×#495573
maximal subgroup of	#72270277, D₃×#495573
maximal normal subgroup of	#72270277, D₃×#495573

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

C3×#495573 #72270198