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C2×C5×#10620321⋊C2 #27190187

order13140
automorphism?
homologyAbelianization (H1) = C22×C5×C9 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×#10620321⋊C2
unambiguous semidirect product#10620321⋊(C22×C5)
unambiguous semidirect product(C2×#10620321)⋊(C2×C5)

Splittings

derived, SylowC73 ⋊ C22×C5×C9
HallC5×#10620321 ⋊ C22
HallC2×C5×D73 ⋊ C9
Hall#10620321 ⋊ C22×C5
HallC5×C73 ⋊ C22×C9
HallC2×D73 ⋊ C5×C9

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5D73⋊C3
commutator, hypocenterC73C22×C5×C9
FrattiniC3C2×C5×D73⋊C3
Fitting, socleC2×C3×C5×C73C2×C3
prime radicalC2×C3×C5×D73C3
cosocleC3×C73C22×C3×C5
LCSG ▷ C73
UCSC1 ◁ C2×C3×C5

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 89
transitive degreeμt(G) = 6570

A minimal faithful permutation representation — 2 generators on 89 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,59)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,51,57,37,42,60)(34,61,62,49,48,56)(40,65,64,53,54,52)(50,70,71,63,58,72)(55,73,68,66,67,69)(74,78,82,76,81,77,79,75,80)(83,84,85,86,87);(1,2,5,11,19,32,35,22,7,6,12,21,34,50,64,45,15,26,14,25,40,55,68,62,28,43,44,27,42,58,67,71,47,60,65,61,46,31,48,54,57,59,72,73,70,51,18,29,39,41,56,69,66,53,36,10,16,24,38,52,63,49,33,20,4,8,13,23,37,30,17,9,3)(88,89)

Conjugacy classes & character table

540 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 21);