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C7×D5×D261 #49141290

order36540
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×D5×D261
unambiguous semidirect product(C9×D5)⋊(C7×D29)

Splittings

Fitting, HallC5×C7×C9×C29 ⋊ C22
derived, HallC5×C9×C29 ⋊ C22×C7
HallC7×C9×C29 ⋊ C2×D5
HallC5×C7×C29 ⋊ C2×D9
HallC5×C7×C9 ⋊ C2×D29
HallC9×C29 ⋊ C2×C7×D5
HallC7×C29 ⋊ D5×D9
HallC5×C29 ⋊ C2×C7×D9
HallC7×C9 ⋊ D5×D29
HallC5×C9 ⋊ C2×C7×D29
HallC5×C7 ⋊ C2×D261
SylowC29 ⋊ C7×D5×D9
SylowC9 ⋊ C7×D5×D29
SylowC5 ⋊ C2×C7×D261

Subgroups

subgroupG/H
center, hypercenterC7D5×D261
commutator, cosocle, hypocenterC5×C9×C29C22×C7
FrattiniC3C7×D5×D87
FittingC5×C7×C9×C29C22
socleC3×C5×C7×C29C2×D3
LCSG ▷ C5×C9×C29
UCSC1 ◁ C7

120 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 50
transitive degreeμt(G) = 9135

A minimal faithful permutation representation — 2 generators on 50 points:

(1,2,4,5,3)(6,7,8,9,10,11,12)(14,16)(15,19)(17,18)(20,21)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50);(2,3)(4,5)(13,21,16,19,18,17,15,14,20)(22,23,25,27,29,31,33,35,37,39,41,43,45,47,49,50,48,46,44,42,40,38,36,34,32,30,28,26,24)

Conjugacy classes & character table

3696 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 132);