D₃×#144364 #72231831

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

D₃×#144364

Splittings

Fitting	C₃×#144364 ⋊ C₂
Sylow	C₃ ⋊ C₂×#144364

Subgroups

	subgroup	G/H
center	C₂	D₃×#10336
commutator, cosocle	C₂³×C₃×D₄	C₂⁴
Frattini	C₂³×D₄	C₂³×D₃
Fitting	C₃×#144364	C₂
socle	C₂×C₃	C₂×#10336
hypercenter	#144364	D₃
hypocenter	C₃	C₂×#144364
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #144364

13582 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)(11,16)(12,15)(17,18,19);(18,19);(1,2,5,8)(3,6,10,14)(4,11)(7,12)(9,13)(15,16);(3,9)(6,13,14,16)(7,11)(10,15)

Conjugacy classes & character table

114 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×#72219163

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

D3×#144364 #72231831