D₃×#13186 #72258679

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₃×#13186

Splittings

Fitting	C₃×#13186 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13186

Subgroups

	subgroup	G/H
center	C₂²	D₃×#476
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13186	C₂
socle	C₂²×C₃	C₂×#476
prime radical	C₂³×D₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13186	D₃
hypocenter	C₃	C₂×#13186
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #13186

1492 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:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495850, D₃×#136361
maximal normal subgroup of	D₃×#495850, D₃×#136361

GAP

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

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

D3×#13186 #72258679