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C2×#16809822⋊C3 #34117683

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

Aliases

direct productC2×#16809822⋊C3
unambiguous semidirect product(C2×#16809822)⋊C3

Splittings

Fitting, socle, HallC2×C11×C61 ⋊ C3×C5
derived, HallC11×C61 ⋊ C2×C3×C5
HallC2×#16809822 ⋊ C3
HallC2×C11×C61⋊C3 ⋊ C5
Hall#16809822 ⋊ 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#16809822⋊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:

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