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C5×AGL(1,𝔽7)×D31 #27189141

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

Aliases

direct productC5×AGL(1,𝔽7)×D31
unambiguous semidirect productD7⋊(C3×C5×D31)

Splittings

Fitting, socle, HallC5×C7×C31 ⋊ C22×C3
derived, HallC7×C31 ⋊ C22×C3×C5
HallC5×D7×D31 ⋊ C3
HallC5×ΓL(1,𝔽8)×C31 ⋊ C22
HallD7×D31 ⋊ C3×C5
HallΓL(1,𝔽8)×C31 ⋊ C22×C5
HallC5×C31 ⋊ C2×AGL(1,𝔽7)
HallC5×ΓL(1,𝔽8) ⋊ C2×D31
HallC5×C7 ⋊ C2×C3×D31
SylowC31 ⋊ C2×C5×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×C5×D31
SylowC7 ⋊ C2×C3×C5×D31

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽7)×D31
commutator, cosocle, hypocenterC7×C31C22×C3×C5
FrattiniC1G
Fitting, socleC5×C7×C31C22×C3
LCSG ▷ C7×C31
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 1085

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 108);