D₃×#12727 #72271959

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

D₃×#12727

Splittings

Fitting	C₃×#12727 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12727

Subgroups

	subgroup	G/H
center	C₂	D₃×#1782
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12727	C₂
socle	C₂×C₃	C₂×#1782
prime radical	D₃×#3508	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#12727	D₃
hypocenter	C₃	C₂×#12727
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #518 ◁ #12727

5244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

(2,3);(1,2,3)(6,13,15,14)(9,18,12,19);(4,6)(5,9)(7,14,17,13)(8,15)(10,19,16,18)(11,12);(4,5)(6,12,15,9)(7,16)(8,11)(10,17)(13,18,14,19)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511853, D₃×#493480
maximal normal subgroup of	D₃×#511853, D₃×#493480

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)(6,13,15,14)(9,18,12,19), (4,6)(5,9)(7,14,17,13)(8,15)(10,19,16,18)(11,12), (4,5)(6,12,15,9)(7,16)(8,11)(10,17)(13,18,14,19));

D3×#12727 #72271959