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C2×D8721 #72273141

order34884
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD17442
direct productC2×D8721

Splittings

FittingC2×C17×C19×C27 ⋊ C2
derived, HallC17×C19×C27 ⋊ C22
HallC19×C27 ⋊ C2×D17
HallC17×C27 ⋊ C2×D19
HallC17×C19 ⋊ C2×D27
SylowC27 ⋊ C2×D323
SylowC19 ⋊ C2×D459
SylowC17 ⋊ C2×D513

Subgroups

subgroupG/H
center, hypercenterC2D8721
commutator, cosocle, hypocenterC17×C19×C27C22
FrattiniC9C2×D969
FittingC2×C17×C19×C27C2
socleC2×C3×C17×C19D9
LCSG ▷ C17×C19×C27
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 65
transitive degreeμt(G) = 17442

A minimal faithful permutation representation — 2 generators on 65 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);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Conjugacy classes & character table

8724 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), (4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57));