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C5×D3×D623 #49153183

order37380
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D3×D623
unambiguous semidirect product(D3×C7)⋊(C5×D89)
unambiguous semidirect product(D3×C89)⋊(C5×D7)

Splittings

Fitting, socle, HallC3×C5×C7×C89 ⋊ C22
derived, HallC3×C7×C89 ⋊ C22×C5
HallC5×C7×C89 ⋊ C2×D3
HallC3×C5×C89 ⋊ C2×D7
HallC7×C89 ⋊ C2×C5×D3
HallC5×C89 ⋊ D3×D7
HallC3×C89 ⋊ C2×C5×D7
HallC3×C5×C7 ⋊ C2×D89
SylowC89 ⋊ C5×D3×D7
HallC5×C7 ⋊ D3×D89
HallC3×C7 ⋊ C2×C5×D89
HallC3×C5 ⋊ C2×D623
SylowC7 ⋊ C5×D3×D89
SylowC3 ⋊ C2×C5×D623

Subgroups

subgroupG/H
center, hypercenterC5D3×D623
commutator, cosocle, hypocenterC3×C7×C89C22×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C89C22
LCSG ▷ C3×C7×C89
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 104
transitive degreeμt(G) = 9345

A minimal faithful permutation representation — 2 generators on 104 points:

(1,2,3)(4,5,6,7,8)(10,11)(12,13)(14,15)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104);(2,3)(9,10,12,14,15,13,11)(16,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20,18)

Conjugacy classes & character table

4695 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37380, 77);