Finite Group Databasebeta

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C5×D3003 #44473248

order30030
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3003
unambiguous semidirect productC3⋊(C5×D1001)
unambiguous semidirect productC7⋊(C5×D429)
unambiguous semidirect productC13⋊(C5×D231)

Splittings

Fitting, socle, HallC3×C5×C7×C11×C13 ⋊ C2
derived, HallC3×C7×C11×C13 ⋊ C2×C5
HallC5×C7×C11×C13 ⋊ D3
HallC3×C5×C11×C13 ⋊ D7
HallC3×C5×C7×C13 ⋊ D11
HallC3×C5×C7×C11 ⋊ D13
HallC7×C11×C13 ⋊ C5×D3
HallC5×C11×C13 ⋊ D21
HallC5×C7×C13 ⋊ D33
HallC3×C11×C13 ⋊ C5×D7
HallC5×C7×C11 ⋊ D39
HallC3×C7×C13 ⋊ C5×D11
HallC3×C7×C11 ⋊ C5×D13
HallC3×C5×C13 ⋊ D77
HallC3×C5×C11 ⋊ D91
HallC11×C13 ⋊ C5×D21
HallC3×C5×C7 ⋊ D143
HallC7×C13 ⋊ C5×D33
HallC7×C11 ⋊ C5×D39
HallC5×C13 ⋊ D231
HallC5×C11 ⋊ D273
HallC3×C13 ⋊ C5×D77
HallC5×C7 ⋊ D429
HallC3×C11 ⋊ C5×D91
HallC3×C7 ⋊ C5×D143
HallC3×C5 ⋊ D1001
SylowC13 ⋊ C5×D231
SylowC11 ⋊ C5×D273
SylowC7 ⋊ C5×D429
SylowC3 ⋊ C5×D1001

Subgroups

subgroupG/H
center, hypercenterC5D3003
commutator, cosocle, hypocenterC3×C7×C11×C13C2×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C11×C13C2
LCSG ▷ C3×C7×C11×C13
UCSC1 ◁ C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 15015

A minimal faithful permutation representation — 2 generators on 39 points:

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

Conjugacy classes & character table

7515 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30030, 33);