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C5×D9×D209 #49178384

order37620
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D9×D209
unambiguous semidirect product(D9×C19)⋊(C5×D11)

Splittings

Fitting, HallC5×C9×C11×C19 ⋊ C22
derived, HallC9×C11×C19 ⋊ C22×C5
HallC5×C11×C19 ⋊ C2×D9
HallC5×C9×C19 ⋊ C2×D11
HallC5×C9×C11 ⋊ C2×D19
HallC11×C19 ⋊ C2×C5×D9
HallC9×C19 ⋊ C2×C5×D11
HallC9×C11 ⋊ C2×C5×D19
HallC5×C19 ⋊ D9×D11
HallC5×C11 ⋊ D9×D19
HallC5×C9 ⋊ C2×D209
SylowC19 ⋊ C5×D9×D11
SylowC11 ⋊ C5×D9×D19
SylowC9 ⋊ C2×C5×D209

Subgroups

subgroupG/H
center, hypercenterC5D9×D209
commutator, cosocle, hypocenterC9×C11×C19C22×C5
FrattiniC3C5×D3×D209
FittingC5×C9×C11×C19C22
socleC3×C5×C11×C19C2×D3
LCSG ▷ C9×C11×C19
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) = 44
transitive degreeμt(G) = 9405

A minimal faithful permutation representation — 2 generators on 44 points:

(1,2,3,4,5)(6,11,9,8,13,10,12,7,14)(16,17)(18,19)(20,21)(22,23)(24,25)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44);(7,9)(8,12)(10,13)(11,14)(15,16,18,20,22,24,25,23,21,19,17)(26,27,29,31,33,35,37,39,41,43,44,42,40,38,36,34,32,30,28)

Conjugacy classes & character table

3180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 93);