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C2×D13889 #72291103

order55556
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD27778
direct productC2×D13889

Splittings

Fitting, socleC2×C17×C19×C43 ⋊ C2
derived, HallC17×C19×C43 ⋊ C22
HallC19×C43 ⋊ C2×D17
HallC17×C43 ⋊ C2×D19
HallC17×C19 ⋊ C2×D43
SylowC43 ⋊ C2×D323
SylowC19 ⋊ C2×D731
SylowC17 ⋊ C2×D817

Subgroups

subgroupG/H
center, hypercenterC2D13889
commutator, cosocle, hypocenterC17×C19×C43C22
FrattiniC1G
Fitting, socleC2×C17×C19×C43C2
LCSG ▷ C17×C19×C43
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 27778

A minimal faithful permutation representation — 2 generators on 81 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);(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Conjugacy classes & character table

13892 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), (4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61));