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C2×D9373 #49154199

order37492
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD18746
direct productC2×D9373
unambiguous semidirect product(C2×C7)⋊D1339
unambiguous semidirect product(C2×C13)⋊D721
unambiguous semidirect product(C2×C103)⋊D91

Splittings

Fitting, socleC2×C7×C13×C103 ⋊ C2
derived, HallC7×C13×C103 ⋊ C22
HallC13×C103 ⋊ C2×D7
HallC7×C103 ⋊ C2×D13
SylowC103 ⋊ C2×D91
HallC7×C13 ⋊ C2×D103
SylowC13 ⋊ C2×D721
SylowC7 ⋊ C2×D1339

Subgroups

subgroupG/H
center, hypercenterC2D9373
commutator, cosocle, hypocenterC7×C13×C103C22
FrattiniC1G
Fitting, socleC2×C7×C13×C103C2
LCSG ▷ C7×C13×C103
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 125
transitive degreeμt(G) = 18746

A minimal faithful permutation representation — 2 generators on 125 points:

(1,2)(3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(4,9)(5,8)(6,7)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,125)(25,124)(26,123)(27,122)(28,121)(29,120)(30,119)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Conjugacy classes & character table

9376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37492, 27);