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C5×D33×D49 #44568390

order32340
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D33×D49
unambiguous semidirect product(C3×D49)⋊(C5×D11)

Splittings

Fitting, HallC3×C5×C11×C49 ⋊ C22
derived, HallC3×C11×C49 ⋊ C22×C5
HallC5×C11×C49 ⋊ C2×D3
HallC3×C5×C49 ⋊ C2×D11
HallC11×C49 ⋊ C2×C5×D3
HallC5×C49 ⋊ C2×D33
HallC3×C5×C11 ⋊ C2×D49
HallC3×C49 ⋊ C2×C5×D11
HallC5×C11 ⋊ D3×D49
SylowC49 ⋊ C2×C5×D33
HallC3×C11 ⋊ C2×C5×D49
HallC3×C5 ⋊ D11×D49
SylowC11 ⋊ C5×D3×D49
SylowC3 ⋊ C5×D11×D49

Subgroups

subgroupG/H
center, hypercenterC5D33×D49
commutator, cosocle, hypocenterC3×C11×C49C22×C5
FrattiniC7C5×D7×D33
FittingC3×C5×C11×C49C22
socleC3×C5×C7×C11C2×D7
LCSG ▷ C3×C11×C49
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) = 68
transitive degreeμt(G) = 8085

A minimal faithful permutation representation — 2 generators on 68 points:

(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,20)(12,23)(14,25)(17,29)(18,28)(19,27)(21,32)(22,34)(24,36)(26,38)(30,39)(31,40)(33,42)(35,44)(37,46)(41,47)(43,48)(45,49)(50,51,53,55,57,59,60,58,56,54,52)(61,62,63)(64,65,66,67,68);(1,15,40,48,49,41,18,3,2,27,42,44,46,30,8,6,5,32,34,36,38,17,14,12,11,20,23,25,29,26,24,22,21,10,13,16,39,37,35,33,19,4,7,28,47,45,43,31,9)(51,52)(53,54)(55,56)(57,58)(59,60)(62,63)

Conjugacy classes & character table

2340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 92);