D₃×#133839 #72231821

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₃×#133839

Splittings

Fitting	C₃×#133839 ⋊ C₂
Sylow	C₃ ⋊ C₂×#133839

Subgroups

	subgroup	G/H
center	C₂	D₃×#8541
commutator	C₂²×C₃×D₄	C₂³×C₄
Frattini	C₂³×D₄	C₂³×D₃
Fitting	C₃×#133839	C₂
socle	C₂×C₃	C₂×#8541
prime radical	D₃×#60399	C₂
cosocle	C₂³×C₃×D₄	C₂⁴
hypercenter	#133839	D₃
hypocenter	C₃	C₂×#133839
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #133839

26110 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:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×B(5,𝔽₂)
maximal normal subgroup of	D₃×B(5,𝔽₂)
maximal quotient of	D₃×#72219159

GAP

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

gap> G := Group((1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,18,19), (18,19), (1,2,5,8)(3,6,10,14)(4,11,12,7)(9,15,16,13), (3,9)(6,13,14,15)(7,11)(10,16));

D3×#133839 #72231821