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C2×C5×#13049954 #27189197

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

Aliases

direct productC2×C5×#13049954

Splittings

Fitting, socleC2×C5×C7×C31 ⋊ C2×C3
derived, HallC7×C31 ⋊ C22×C3×C5
HallC2×C5×C7×D31 ⋊ C3
HallC5×#10620252 ⋊ C22
HallC5×C7×C31 ⋊ C22×C3
HallC2×C7×D31 ⋊ C3×C5
Hall#10620252 ⋊ C22×C5
HallC2×C5×D31 ⋊ ΓL(1,𝔽8)
HallC5×C31 ⋊ C22×ΓL(1,𝔽8)
HallC2×D31 ⋊ C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×D31⋊C3
SylowC31 ⋊ C22×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C5×D31⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C5#13049954
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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 164);