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C5×D33⋊D49 #44568392

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

Aliases

direct productC5×D33⋊D49
unambiguous semidirect productD147⋊(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 ⋊ D3×D11
HallC3×C5×C11 ⋊ C2×D49
HallC3×C49 ⋊ C2×C5×D11
HallC5×C11 ⋊ D3×D49
SylowC49 ⋊ C5×D3×D11
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×D21⋊D11
FittingC3×C5×C11×C49C22
socleC3×C5×C7×C11C2×D7
LCSG ▷ C3×C11×C49
UCSC1 ◁ C5

120 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 40 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:

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

Conjugacy classes & character table

2265 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 94);