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C2×C5×#15690450 #44501366

order30660
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×#15690450

Splittings

Fitting, socleC2×C5×C7×C73 ⋊ C2×C3
derived, HallC7×C73 ⋊ C22×C3×C5
HallC2×C5×D511 ⋊ C3
HallC5×#13095276 ⋊ C22
HallC5×C7×C73 ⋊ C22×C3
HallC2×D511 ⋊ C3×C5
Hall#13095276 ⋊ C22×C5
HallC5×C73 ⋊ C2×AGL(1,𝔽7)
SylowC73 ⋊ C2×C5×AGL(1,𝔽7)
HallC5×C7 ⋊ C2×D73⋊C3
SylowC7 ⋊ C2×C5×D73⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C5#15690450
commutator, cosocle, hypocenterC7×C73C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C73C2×C3
LCSG ▷ C7×C73
UCSC1 ◁ C2×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 87
transitive degreeμt(G) = 5110

A minimal faithful permutation representation — 2 generators on 87 points:

(3,4,5,6,7)(9,12,13,10,11,14)(16,20,21,17,18,22)(19,28,29,23,24,38)(25,41,42,31,32,55)(26,40,36,34,30,27)(33,60,61,44,45,73)(35,58,59,47,43,52)(37,39,57,49,50,53)(46,65,71,51,56,74)(48,75,76,63,62,70)(54,79,78,67,68,66)(64,84,85,77,72,86)(69,87,82,80,81,83);(1,2)(8,9,11,14,13,12,10)(15,16,19,25,33,46,49,36,21,20,26,35,48,64,78,59,29,40,28,39,54,69,82,76,42,57,58,41,56,72,81,85,61,74,79,75,60,45,62,68,71,73,86,87,84,65,32,43,53,55,70,83,80,67,50,24,30,38,52,66,77,63,47,34,18,22,27,37,51,44,31,23,17)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30660, 129);