D₃×#13048 #72248496

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

D₃×#13048

Splittings

Fitting	C₃×#13048 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13048

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator	C₂×C₃×D₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#13048	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	C₂×D₃×H(5,𝔽₂)	C₂²
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#13048	D₃
hypocenter	C₃	C₂×#13048
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #13048

4524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(22,23);(1,3,4,2)(10,18)(14,16)(21,22,23);(1,2,4,3)(5,7)(6,10,11,18)(8,16,17,14)(9,15)(12,20)(13,19);(5,6)(7,13,15,20)(8,12)(9,11)(10,14,18,16)(17,19)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#494600
maximal normal subgroup of	D₃×#494600
maximal quotient of	D₃×#139777

GAP

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

gap> G := Group((22,23), (1,3,4,2)(10,18)(14,16)(21,22,23), (1,2,4,3)(5,7)(6,10,11,18)(8,16,17,14)(9,15)(12,20)(13,19), (5,6)(7,13,15,20)(8,12)(9,11)(10,14,18,16)(17,19));

D3×#13048 #72248496