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C5×D11×D147 #44568388

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

Aliases

direct productC5×D11×D147

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 ⋊ C2×D147
SylowC49 ⋊ C5×D3×D11
HallC3×C11 ⋊ C2×C5×D49
HallC3×C5 ⋊ D11×D49
SylowC11 ⋊ C2×C5×D147
SylowC3 ⋊ C5×D11×D49

Subgroups

subgroupG/H
center, hypercenterC5D11×D147
commutator, cosocle, hypocenterC3×C11×C49C22×C5
FrattiniC7C5×D11×D21
FittingC3×C5×C11×C49C22
socleC3×C5×C7×C11C2×D7
LCSG ▷ C3×C11×C49
UCSC1 ◁ C5

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

Conjugacy classes & character table

2625 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 90);