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C5×D9×D119 #36339245

order21420
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D9×D119
unambiguous semidirect product(C7×D9)⋊(C5×D17)
unambiguous semidirect product(C17×D9)⋊(C5×D7)

Splittings

Fitting, HallC5×C7×C9×C17 ⋊ C22
derived, HallC7×C9×C17 ⋊ C22×C5
HallC5×C9×C17 ⋊ C2×D7
HallC5×C7×C17 ⋊ C2×D9
HallC5×C7×C9 ⋊ C2×D17
HallC9×C17 ⋊ C2×C5×D7
HallC7×C17 ⋊ C2×C5×D9
HallC5×C17 ⋊ D7×D9
HallC7×C9 ⋊ C2×C5×D17
HallC5×C9 ⋊ C2×D119
HallC5×C7 ⋊ D9×D17
SylowC17 ⋊ C5×D7×D9
SylowC9 ⋊ C2×C5×D119
SylowC7 ⋊ C5×D9×D17

Subgroups

subgroupG/H
center, hypercenterC5D9×D119
commutator, cosocle, hypocenterC7×C9×C17C22×C5
FrattiniC3C5×D3×D119
FittingC5×C7×C9×C17C22
socleC3×C5×C7×C17C2×D3
LCSG ▷ C7×C9×C17
UCSC1 ◁ C5

120 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 38
transitive degreeμt(G) = 5355

A minimal faithful permutation representation — 2 generators on 38 points:

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

Conjugacy classes & character table

1830 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 77);