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C2×D13717 #72290738

order54868
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD27434
direct productC2×D13717

Splittings

Fitting, socleC2×C11×C29×C43 ⋊ C2
derived, HallC11×C29×C43 ⋊ C22
HallC29×C43 ⋊ C2×D11
HallC11×C43 ⋊ C2×D29
HallC11×C29 ⋊ C2×D43
SylowC43 ⋊ C2×D319
SylowC29 ⋊ C2×D473
SylowC11 ⋊ C2×D1247

Subgroups

subgroupG/H
center, hypercenterC2D13717
commutator, cosocle, hypocenterC11×C29×C43C22
FrattiniC1G
Fitting, socleC2×C11×C29×C43C2
LCSG ▷ C11×C29×C43
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 27434

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);(4,13)(5,12)(6,11)(7,10)(8,9)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(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

13720 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), (4,13)(5,12)(6,11)(7,10)(8,9)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(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));