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C2×D3×#1426 #72255372

order1536
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×D3×#1426

Splittings

FittingC2×C3×#1426 ⋊ C2
SylowC3 ⋊ C22×#1426

Subgroups

subgroupG/H
centerC22D3×#367
commutatorC3×C16C23×C4
FrattiniC2×C16C23×D3
FittingC2×C3×#1426C2
socleC22×C3C2×#367
prime radicalC23×D3SD32
cosocleC2×C3×C16C24
hypercenterC2×#1426D3
hypocenterC3C22×#1426
LCSG ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C2×#1426

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 192

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

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×#13658
maximal normal subgroup ofC2×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)(7,12,14,24)(8,17)(9,21)(10,22)(13,27,25,28)(15,32,30,29)(16,33,31,26)(18,23)(19,37)(20,36)(34,35), (4,5), (6,7,8,13,18,28,37,12,10,16,19,29,9,15,17,26,11,14,20,25,21,27,35,24,22,31,34,32,23,30,36,33));