C₂×D₃×#1418 #72255376

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

Aliases

direct product

C₂×D₃×#1418

Splittings

Fitting	C₂×C₃×#1418 ⋊ C₂
Sylow	C₃ ⋊ C₂²×#1418

Subgroups

	subgroup	G/H
center	C₂²	D₃×#357
commutator	C₃×C₁₆	C₂³×C₄
Frattini	C₂×C₁₆	C₂³×D₃
Fitting	C₂×C₃×#1418	C₂
socle	C₂²×C₃	C₂×#357
prime radical	C₂×D₃×#508	C₂
cosocle	C₂×C₃×C₁₆	C₂⁴
hypercenter	C₂×#1418	D₃
hypocenter	C₃	C₂²×#1418
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#1418

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 37 points:

(2,3);(1,2,3)(7,12,16,26)(8,17)(10,23)(11,22)(13,14,28,31)(15,32,29,25)(18,21)(19,33)(20,36)(24,27,37,30)(34,35);(4,5);(6,7,8,13,18,27,36,25,11,15,20,24,10,14,17,26,9,16,19,28,23,30,35,32,22,29,34,37,21,31,33,12)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×#13658
maximal normal subgroup of	C₂×D₃×#13658

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)(7,12,16,26)(8,17)(10,23)(11,22)(13,14,28,31)(15,32,29,25)(18,21)(19,33)(20,36)(24,27,37,30)(34,35), (4,5), (6,7,8,13,18,27,36,25,11,15,20,24,10,14,17,26,9,16,19,28,23,30,35,32,22,29,34,37,21,31,33,12));

C2×D3×#1418 #72255376