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C2×C5×C31×AGL(1,𝔽7) #27189191

order13020
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C5×C31 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×C31×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×C5×C31)
unambiguous semidirect productΓL(1,𝔽8)⋊(C22×C5×C31)
unambiguous semidirect product(C2×D7)⋊(C3×C5×C31)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C2×C5×C31)
unambiguous semidirect product(C2×C5×D7)⋊(C3×C31)

Splittings

Fitting, socleC2×C5×C7×C31 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C3×C5×C31
HallC2×C5×D7×C31 ⋊ C3
HallC5×ΓL(1,𝔽8)×C31 ⋊ C22
HallC5×C7×C31 ⋊ C22×C3
HallC2×D7×C31 ⋊ C3×C5
HallΓL(1,𝔽8)×C31 ⋊ C22×C5
HallC7×C31 ⋊ C22×C3×C5
HallC2×C5×D7 ⋊ C3×C31
HallC5×ΓL(1,𝔽8) ⋊ C22×C31
HallC5×C7 ⋊ C22×C3×C31
HallC2×D7 ⋊ C3×C5×C31
HallΓL(1,𝔽8) ⋊ C22×C5×C31

Subgroups

subgroupG/H
center, hypercenterC2×C5×C31AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C22×C3×C5×C31
FrattiniC1G
Fitting, socleC2×C5×C7×C31C2×C3
LCSG ▷ C7
UCSC1 ◁ C2×C5×C31

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:

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

Conjugacy classes & character table

2170 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 158);