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C5×#15690442⋊C2 #44501291

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

Aliases

direct productC5×#15690442⋊C2
unambiguous semidirect product#15690442⋊(C2×C5)
unambiguous semidirect product#15690446⋊(C2×C5)
unambiguous semidirect product#15690450⋊(C2×C5)

Splittings

Fitting, socle, HallC5×C7×C73 ⋊ C22×C3
derived, HallC7×C73 ⋊ C22×C3×C5
HallC5×D7×D73 ⋊ C3
HallC5×#13095276 ⋊ C22
HallD7×D73 ⋊ 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, hypercenterC5#15690442⋊C2
commutator, cosocle, hypocenterC7×C73C22×C3×C5
FrattiniC1G
Fitting, socleC5×C7×C73C22×C3
LCSG ▷ C7×C73
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 2555

A minimal faithful permutation representation — 2 generators on 85 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30660, 54);