Finite Group Databasebeta

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C5×D7×C3⋊D39 #30655679

order16380
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC5×D7×C3⋊D39
unambiguous semidirect product(C3×D7)⋊(C5×D39)
unambiguous semidirect product(C13×D7)⋊(C5×C3⋊D3)

Splittings

Fitting, socle, HallC32×C5×C7×C13 ⋊ C22
derived, HallC32×C7×C13 ⋊ C22×C5
HallC32×C5×C13 ⋊ C2×D7
HallC5×C7×C13 ⋊ C2×C3⋊D3
HallC32×C5×C7 ⋊ C2×D13
HallC32×C13 ⋊ C2×C5×D7
HallC7×C13 ⋊ C2×C5×C3⋊D3
HallC5×C13 ⋊ D7×C3⋊D3
HallC32×C7 ⋊ C2×C5×D13
HallC32×C5 ⋊ D7×D13
HallC5×C7 ⋊ C2×C3⋊D39
SylowC13 ⋊ C5×D7×C3⋊D3
SylowC32 ⋊ C5×D7×D13
SylowC7 ⋊ C2×C5×C3⋊D39

Subgroups

subgroupG/H
center, hypercenterC5D7×C3⋊D39
commutator, cosocle, hypocenterC32×C7×C13C22×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C13C22
LCSG ▷ C32×C7×C13
UCSC1 ◁ C5

240 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 4095

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

1500 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16380, 729);