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C2×C5×#13049958 #27189194

order13020
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×#13049958

Splittings

Fitting, socleC2×C5×C7×C31 ⋊ C2×C3
derived, HallC7×C31 ⋊ C22×C3×C5
HallC2×C5×D7×C31 ⋊ C3
HallC5×#10620252 ⋊ C22
HallC5×C7×C31 ⋊ C22×C3
HallC2×D7×C31 ⋊ C3×C5
Hall#10620252 ⋊ C22×C5
HallC5×C31 ⋊ C2×AGL(1,𝔽7)
HallC2×C5×D7 ⋊ C31⋊C3
HallC5×C7 ⋊ C22×C31⋊C3
SylowC31 ⋊ C2×C5×AGL(1,𝔽7)
HallC2×D7 ⋊ C5×C31⋊C3
SylowC7 ⋊ C22×C5×C31⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C5#13049958
commutator, cosocle, hypocenterC7×C31C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C31C2×C3
LCSG ▷ C7×C31
UCSC1 ◁ C2×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 2170

A minimal faithful permutation representation — 2 generators on 45 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,21)(11,15,18)(13,19,16)(17,24,27)(20,26,25)(22,28,23)(29,31,30)(33,36,37,34,35,38)(39,40,41,42,43);(1,2,4,8,15,3,6,11,17,24,7,13,20,21,27,14,22,29,28,12,18,25,30,31,19,5,9,16,23,26,10)(32,33,35,38,37,36,34)(44,45)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 161);