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C2×#10715810 #72255362

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

Aliases

direct productC2×#10715810

Splittings

FittingC2×C3×#2177 ⋊ C2
SylowC3 ⋊ C2×#13658

Subgroups

subgroupG/H
centerC22#70885
commutatorC3×C16C23×C4
FrattiniC2×C16C23×D3
FittingC2×C3×#2177C2
socleC22×C3#2177
prime radicalC2×#71458C2
cosocleC2×C3×C16C24
hypercenterC2×#2177D3
hypocenterC3C2×#13658
LCSG ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ C2×#2177

900 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:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×#13658, C2×#72222516, C2×#72222524
maximal normal subgroup ofC2×D3×#13658, C2×#72222516, C2×#72222524

GAP

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

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