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C2×C7×D1335 #49153232

order37380
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×D1335
unambiguous semidirect product(C2×C3)⋊(C7×D445)
unambiguous semidirect product(C2×C5)⋊(C7×D267)
unambiguous semidirect product(C2×C89)⋊(C7×D15)

Splittings

Fitting, socleC2×C3×C5×C7×C89 ⋊ C2
derived, HallC3×C5×C89 ⋊ C22×C7
HallC3×C5×C7×C89 ⋊ C22
HallC5×C7×C89 ⋊ C2×D3
HallC3×C7×C89 ⋊ C2×D5
HallC7×C89 ⋊ C2×D15
HallC5×C89 ⋊ C2×D3×C7
HallC3×C89 ⋊ C2×C7×D5
HallC3×C5×C7 ⋊ C2×D89
SylowC89 ⋊ C2×C7×D15
HallC5×C7 ⋊ C2×D267
HallC3×C7 ⋊ C2×D445
HallC3×C5 ⋊ C2×C7×D89
SylowC5 ⋊ C2×C7×D267
SylowC3 ⋊ C2×C7×D445

Subgroups

subgroupG/H
center, hypercenterC2×C7D1335
commutator, cosocle, hypocenterC3×C5×C89C22×C7
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C89C2
LCSG ▷ C3×C5×C89
UCSC1 ◁ C2×C7

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 106
transitive degreeμt(G) = 18690

A minimal faithful permutation representation — 2 generators on 106 points:

(4,5)(7,8)(9,10)(11,12,13,14,15,16,17)(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)(105,106);(1,2)(3,4,5)(6,7,9,10,8)(18,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,106,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20)

Conjugacy classes & character table

9366 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37380, 126);