D₃×#12767 #72271897

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

Aliases

direct product

D₃×#12767

Splittings

Fitting	C₃×#12767 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12767

Subgroups

	subgroup	G/H
center	C₂	D₃×#1790
commutator	C₂×C₃×D₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12767	C₂
socle	C₂×C₃	C₂×#1790
prime radical	D₃×#3543	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#12767	D₃
hypocenter	C₃	C₂×#12767
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ #12767

5660 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)(6,13,14,15)(9,18,12,19);(4,6)(5,9,11,12)(7,15,17,13)(8,14)(10,19)(16,18);(4,5)(6,12)(7,16)(8,11)(9,14)(10,17)(13,19)(15,18)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#493635, D₃×#511853, D₃×#511811
maximal normal subgroup of	D₃×#493635, D₃×#511853, D₃×#511811

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

D3×#12767 #72271897