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C2×D15405 #72296539

order61620
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD30810
direct productC2×D15405

Splittings

Fitting, socleC2×C3×C5×C13×C79 ⋊ C2
derived, HallC3×C5×C13×C79 ⋊ C22
HallC5×C13×C79 ⋊ C2×D3
HallC3×C13×C79 ⋊ C2×D5
HallC3×C5×C79 ⋊ C2×D13
HallC13×C79 ⋊ C2×D15
HallC5×C79 ⋊ C2×D39
HallC3×C79 ⋊ C2×D65
HallC3×C5×C13 ⋊ C2×D79
SylowC79 ⋊ C2×D195
HallC5×C13 ⋊ C2×D237
HallC3×C13 ⋊ C2×D395
HallC3×C5 ⋊ C2×D1027
SylowC13 ⋊ C2×D1185
SylowC5 ⋊ C2×D3081
SylowC3 ⋊ C2×D5135

Subgroups

subgroupG/H
center, hypercenterC2D15405
commutator, cosocle, hypocenterC3×C5×C13×C79C22
FrattiniC1G
Fitting, socleC2×C3×C5×C13×C79C2
LCSG ▷ C3×C5×C13×C79
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 102
transitive degreeμt(G) = 30810

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

Conjugacy classes & character table

15408 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)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (4,5)(7,10)(8,9)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64));