D₃×#12725 #72271956

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

Aliases

direct product

D₃×#12725

Splittings

Fitting	C₃×#12725 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12725

Subgroups

	subgroup	G/H
center	C₂	D₃×#1782
commutator	C₂⁴×C₃	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12725	C₂
socle	C₂×C₃	C₂×#1782
prime radical	D₃×#3490	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#12725	D₃
hypocenter	C₃	C₂×#12725
LCS	G ▷ C₂⁴×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #518 ◁ #12725

5624 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)(9,18)(12,19)(14,17);(4,6,7,14)(5,9,10,12)(8,17)(11,19)(13,15)(16,18);(4,5)(6,12,14,9)(7,10)(8,16)(11,15)(13,18,17,19)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511853, D₃×#493474
maximal normal subgroup of	D₃×#511853, D₃×#493474

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

D3×#12725 #72271956