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C2×D15093 #72295490

order60372
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD30186
direct productC2×D15093

Splittings

FittingC2×C13×C27×C43 ⋊ C2
derived, HallC13×C27×C43 ⋊ C22
HallC27×C43 ⋊ C2×D13
HallC13×C43 ⋊ C2×D27
HallC13×C27 ⋊ C2×D43
SylowC43 ⋊ C2×D351
SylowC27 ⋊ C2×D559
SylowC13 ⋊ C2×D1161

Subgroups

subgroupG/H
center, hypercenterC2D15093
commutator, cosocle, hypocenterC13×C27×C43C22
FrattiniC9C2×D1677
FittingC2×C13×C27×C43C2
socleC2×C3×C13×C43D9
LCSG ▷ C13×C27×C43
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 30186

A minimal faithful permutation representation — 2 generators on 85 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);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Conjugacy classes & character table

15096 conjugacy classes. The character table is too large to display.

GAP

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

gap> G := Group((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), (4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65));