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C5×#13049955⋊C2 #27189140

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

Aliases

direct productC5×#13049955⋊C2

Splittings

Fitting, socle, HallC5×C7×C31 ⋊ C22×C3
derived, HallC7×C31 ⋊ C22×C3×C5
HallC5×D7×D31 ⋊ C3
HallC5×#10620253 ⋊ C22
HallD7×D31 ⋊ C3×C5
Hall#10620253 ⋊ C22×C5
HallC5×C31 ⋊ C2×AGL(1,𝔽7)
HallC5×C7 ⋊ C2×D31⋊C3
SylowC31 ⋊ C2×C5×AGL(1,𝔽7)
SylowC7 ⋊ C2×C5×D31⋊C3

Subgroups

subgroupG/H
center, hypercenterC5#13049955⋊C2
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:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 107);