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D₃×#12687 #72227516

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Splittings

Fitting	C₃×#12687 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12687

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×#99
commutator	C₂⁴×C₃	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12687	C₂
socle	C₂²×C₃	C₂²×#99
prime radical	C₂²×D₃×#120	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#12687	D₃
hypocenter	C₃	C₂×#12687
LCS	G ▷ C₂⁴×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #12687

22592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 15 points:

(2,3);(1,2,3)(8,10)(11,15);(5,7)(9,12)(13,14);(4,5)(6,7)(8,9,10,13,11,14,15,12)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#144366, D₃×#511790, D₃×#139750, …
maximal normal subgroup of	D₃×#144366, D₃×#511790, D₃×#139750, …
maximal quotient of	D₃×#147555, C₂×D₃×#12687
automorphism group of	#870, #4543

GAP

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

gap> G := Group((2,3), (1,2,3)(8,10)(11,15), (5,7)(9,12)(13,14), (4,5)(6,7)(8,9,10,13,11,14,15,12));