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C2×C7×AGL(1,𝔽31) #27189111

order13020
automorphism?
homologyAbelianization (H1) = C22×C3×C5×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×AGL(1,𝔽31)
unambiguous semidirect product(D31⋊C3)⋊(C2×C5×C7)
unambiguous semidirect product(D31⋊C5)⋊(C2×C3×C7)
unambiguous semidirect product(C2×D31⋊C3)⋊(C5×C7)
unambiguous semidirect product((C31⋊C3)⋊C5)⋊(C22×C7)
unambiguous semidirect product(C2×D31⋊C5)⋊(C3×C7)
unambiguous semidirect product(C2×(C31⋊C3)⋊C5)⋊(C2×C7)
unambiguous semidirect product(C2×C7×D31⋊C3)⋊C5

Splittings

Fitting, socleC2×C7×C31 ⋊ C2×C3×C5
derived, SylowC31 ⋊ C22×C3×C5×C7
HallC2×C7×D31⋊C5 ⋊ C3
HallC7×(C31⋊C3)⋊C5 ⋊ C22
HallC2×C7×D31⋊C3 ⋊ C5
HallC7×ΓL(1,𝔽32) ⋊ C22×C3
HallC2×C7×D31 ⋊ C3×C5
HallC7×C31⋊C3 ⋊ C22×C5
HallC2×D31⋊C5 ⋊ C3×C7
Hall(C31⋊C3)⋊C5 ⋊ C22×C7
HallC2×D31⋊C3 ⋊ C5×C7
HallC7×C31 ⋊ C22×C3×C5
HallΓL(1,𝔽32) ⋊ C22×C3×C7
HallC2×D31 ⋊ C3×C5×C7
HallC31⋊C3 ⋊ C22×C5×C7

Subgroups

subgroupG/H
center, hypercenterC2×C7AGL(1,𝔽31)
commutator, cosocle, hypocenterC31C22×C3×C5×C7
FrattiniC1G
Fitting, socleC2×C7×C31C2×C3×C5
LCSG ▷ C31
UCSC1 ◁ C2×C7

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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 78);