D₃×#20370 #72271745

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

D₃×#20370

Splittings

Fitting	C₃×#20370 ⋊ C₂
Sylow	C₃ ⋊ C₂×#20370

Subgroups

	subgroup	G/H
center	C₄	C₂²×C₄²×D₃
commutator	C₃×C₄	C₂³×C₄²
Frattini	C₂²×C₄	C₂⁴×D₃
Fitting	C₃×#20370	C₂
socle	C₂×C₃	C₂²×#376
prime radical	C₂⁴×D₃	#41
cosocle	C₂²×C₃×C₄	C₂⁵
hypercenter	#20370	D₃
hypocenter	C₃	C₂×#20370
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #20370

5484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 35 points:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

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

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

D3×#20370 #72271745