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C2×#16809821⋊C3 #34117685

order20130
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×#16809821⋊C3
unambiguous semidirect product(C2×#16809821)⋊C3

Splittings

Fitting, socle, HallC2×C11×C61 ⋊ C3×C5
derived, HallC11×C61 ⋊ C2×C3×C5
HallC2×#16809821 ⋊ C3
HallC2×C11×C61⋊C3 ⋊ C5
Hall#16809821 ⋊ C2×C3
HallC11×C61⋊C3 ⋊ C2×C5
HallC2×C61⋊C3 ⋊ C11⋊C5
HallC61⋊C3 ⋊ C2×C11⋊C5
HallC2×C61 ⋊ C3×C11⋊C5
SylowC61 ⋊ C2×C3×C11⋊C5
HallC2×C11 ⋊ (C61⋊C3)⋊C5
SylowC11 ⋊ C2×(C61⋊C3)⋊C5

Subgroups

subgroupG/H
center, hypercenterC2#16809821⋊C3
commutator, cosocle, hypocenterC11×C61C2×C3×C5
FrattiniC1G
Fitting, socleC2×C11×C61C3×C5
LCSG ▷ C11×C61
UCSC1 ◁ C2

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 74
transitive degreeμt(G) = 1342

A minimal faithful permutation representation — 2 generators on 74 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20130, 14);