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C2×#10682656 #72255385

order1536
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×#10682656

Splittings

FittingC2×C3×#1398 ⋊ C2
SylowC3 ⋊ C2×#7389

Subgroups

subgroupG/H
centerC22#67516
commutatorC3×C16C22×C8
Frattini#346C22×D3
FittingC2×C3×#1398C2
socleC22×C3#1333
prime radicalC2×#4904C4
cosocleC3×#346C23
hypercenterC2×#1398D3
hypocenterC3C2×#7389
LCSG ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ C2×#1398

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×#7389, C2×#72222500, C2×#72222524
maximal normal subgroup ofC2×D3×#7389, C2×#72222500, C2×#72222524

GAP

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

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