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C2×#13049959⋊C5 #27189115

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

Aliases

direct productC2×#13049959⋊C5
unambiguous semidirect product(C2×#13049959)⋊C5

Splittings

Fitting, socleC2×C7×C31 ⋊ C2×C3×C5
derived, HallC7×C31 ⋊ C22×C3×C5
HallC2×D7×ΓL(1,𝔽32) ⋊ C3
Hall#10620253⋊C5 ⋊ C22
HallC2×#13049959 ⋊ C5
HallC7×ΓL(1,𝔽32) ⋊ C22×C3
HallC2×D7×C31 ⋊ C3×C5
Hall#10620253 ⋊ C22×C5
HallΓL(1,𝔽32) ⋊ C2×AGL(1,𝔽7)
SylowC31 ⋊ C2×C5×AGL(1,𝔽7)
HallC2×D7 ⋊ (C31⋊C3)⋊C5
SylowC7 ⋊ C22×(C31⋊C3)⋊C5

Subgroups

subgroupG/H
center, hypercenterC2#13049959⋊C5
commutator, cosocle, hypocenterC7×C31C22×C3×C5
FrattiniC1G
Fitting, socleC2×C7×C31C2×C3×C5
LCSG ▷ C7×C31
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 434

A minimal faithful permutation representation — 2 generators on 40 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 82);