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C2×D16359 #72304293

order65436
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD32718
direct productC2×D16359

Splittings

Fitting, socleC2×C3×C7×C19×C41 ⋊ C2
derived, HallC3×C7×C19×C41 ⋊ C22
HallC7×C19×C41 ⋊ C2×D3
HallC3×C19×C41 ⋊ C2×D7
HallC3×C7×C41 ⋊ C2×D19
HallC19×C41 ⋊ C2×D21
HallC3×C7×C19 ⋊ C2×D41
HallC7×C41 ⋊ C2×D57
HallC7×C19 ⋊ C2×D123
HallC3×C41 ⋊ C2×D133
HallC3×C19 ⋊ C2×D287
SylowC41 ⋊ C2×D399
HallC3×C7 ⋊ C2×D779
SylowC19 ⋊ C2×D861
SylowC7 ⋊ C2×D2337
SylowC3 ⋊ C2×D5453

Subgroups

subgroupG/H
center, hypercenterC2D16359
commutator, cosocle, hypocenterC3×C7×C19×C41C22
FrattiniC1G
Fitting, socleC2×C3×C7×C19×C41C2
LCSG ▷ C3×C7×C19×C41
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 32718

A minimal faithful permutation representation — 2 generators on 72 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);(4,5)(7,12)(8,11)(9,10)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

16362 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), (4,5)(7,12)(8,11)(9,10)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53));