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C2×D16215 #72302134

order64860
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD32430
direct productC2×D16215

Splittings

Fitting, socleC2×C3×C5×C23×C47 ⋊ C2
derived, HallC3×C5×C23×C47 ⋊ C22
HallC5×C23×C47 ⋊ C2×D3
HallC3×C23×C47 ⋊ C2×D5
HallC23×C47 ⋊ C2×D15
HallC3×C5×C47 ⋊ C2×D23
HallC3×C5×C23 ⋊ C2×D47
HallC5×C47 ⋊ C2×D69
HallC3×C47 ⋊ C2×D115
HallC5×C23 ⋊ C2×D141
HallC3×C23 ⋊ C2×D235
SylowC47 ⋊ C2×D345
SylowC23 ⋊ C2×D705
HallC3×C5 ⋊ C2×D1081
SylowC5 ⋊ C2×D3243
SylowC3 ⋊ C2×D5405

Subgroups

subgroupG/H
center, hypercenterC2D16215
commutator, cosocle, hypocenterC3×C5×C23×C47C22
FrattiniC1G
Fitting, socleC2×C3×C5×C23×C47C2
LCSG ▷ C3×C5×C23×C47
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 80
transitive degreeμt(G) = 32430

A minimal faithful permutation representation — 2 generators on 80 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);(4,5)(7,10)(8,9)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Conjugacy classes & character table

16218 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), (4,5)(7,10)(8,9)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58));