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C2×C3×#16809823 #34117677

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

Aliases

direct productC2×C3×#16809823

Splittings

Fitting, socle, HallC2×C3×C11×C61 ⋊ C5
derived, HallC11×C61 ⋊ C2×C3×C5
HallC3×C11×C61 ⋊ C2×C5
HallC2×C11×C61 ⋊ C3×C5
HallC2×C3×C61 ⋊ C11⋊C5
HallC3×C61 ⋊ C2×C11⋊C5
HallC2×C61 ⋊ C3×C11⋊C5
HallC2×C3×C11 ⋊ C61⋊C5
SylowC61 ⋊ C2×C3×C11⋊C5
HallC3×C11 ⋊ C2×C61⋊C5
HallC2×C11 ⋊ C3×C61⋊C5
SylowC11 ⋊ C2×C3×C61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2×C3#16809823
commutator, cosocle, hypocenterC11×C61C2×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C11×C61C5
LCSG ▷ C11×C61
UCSC1 ◁ C2×C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 77
transitive degreeμt(G) = 4026

A minimal faithful permutation representation — 2 generators on 77 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20130, 6);