D₃×#10950 #72271767

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

D₃×#10950

Splittings

Fitting	C₃×#10950 ⋊ C₂
Sylow	C₃ ⋊ C₂×#10950

Subgroups

	subgroup	G/H
center	C₄	D₃×#386
commutator	C₂×C₃×C₄	C₂²×C₄²
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₃×#10950	C₂
socle	C₂×C₃	C₂×#1437
prime radical	C₂²×D₃×#41	C₂²
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	#10950	D₃
hypocenter	C₃	C₂×#10950
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #10950

5636 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)(4,7)(5,12)(6,15)(8,23)(9,22)(10,21)(11,25)(13,19)(14,29)(16,32)(17,20)(18,31)(24,30)(26,35)(27,34)(28,33);(2,3);(4,6)(5,11)(7,20,21,32)(8,16)(9,17)(10,18)(12,28,29,35)(13,26)(14,27)(15,22,31,23)(19,25,30,34)(24,33);(4,5,9,13,10,14,8,24)(6,11,17,26,18,27,16,33)(7,19,23,12,21,30,22,29)(15,28,32,34,31,35,20,25)

Conjugacy classes & character table

174 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)(4,7)(5,12)(6,15)(8,23)(9,22)(10,21)(11,25)(13,19)(14,29)(16,32)(17,20)(18,31)(24,30)(26,35)(27,34)(28,33), (2,3), (4,6)(5,11)(7,20,21,32)(8,16)(9,17)(10,18)(12,28,29,35)(13,26)(14,27)(15,22,31,23)(19,25,30,34)(24,33), (4,5,9,13,10,14,8,24)(6,11,17,26,18,27,16,33)(7,19,23,12,21,30,22,29)(15,28,32,34,31,35,20,25));

D3×#10950 #72271767