D₃×#10951 #72271747

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

Aliases

direct product

D₃×#10951

Splittings

Fitting	C₃×#10951 ⋊ C₂
Sylow	C₃ ⋊ C₂×#10951

Subgroups

	subgroup	G/H
center	C₄	D₃×#386
commutator	C₂×C₃×C₄	C₂²×C₄²
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₃×#10951	C₂
socle	C₂×C₃	C₂×#1437
prime radical	C₂²×D₃×#41	C₂²
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	#10951	D₃
hypocenter	C₃	C₂×#10951
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #10951

5700 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)(4,7,10,21)(5,12,13,31)(6,16,18,32)(8,23,9,22)(11,26,27,35)(14,19,24,30)(15,29,28,34)(17,33,25,20);(2,3);(4,6)(5,11,13,27)(7,20,21,33)(8,17)(9,25)(10,18)(12,29)(14,28,24,15)(16,22,32,23)(19,26)(30,35)(31,34);(4,5,9,14,10,13,8,24)(6,15,17,27,18,28,25,11)(7,19,23,12,21,30,22,31)(16,26,20,34,32,35,33,29)

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

D3×#10951 #72271747