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C2×C3⋊(C49⋊AGL(1,𝔽11)) #44568366

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

Aliases

direct productC2×C3⋊(C49⋊AGL(1,𝔽11))
unambiguous semidirect product(C2×C3)⋊(C49⋊AGL(1,𝔽11))
unambiguous semidirect product(C2×C49)⋊(C3⋊AGL(1,𝔽11))
unambiguous semidirect product(C2×C11⋊C5)⋊D147
unambiguous semidirect product(C2×D1617)⋊C5

Splittings

FittingC2×C3×C11×C49 ⋊ C2×C5
derived, HallC3×C11×C49 ⋊ C22×C5
HallC3×C49×C11⋊C5 ⋊ C22
HallC2×D1617 ⋊ C5
HallC49×C11⋊C5 ⋊ C2×D3
HallC11×C49 ⋊ C2×C5×D3
HallC3×C11⋊C5 ⋊ C2×D49
HallC3×C49 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×D147
SylowC49 ⋊ C2×C3⋊AGL(1,𝔽11)
HallC3×C11 ⋊ C2×C5×D49
SylowC11 ⋊ C2×C5×D147
SylowC3 ⋊ C2×C49⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C49⋊AGL(1,𝔽11))
commutator, cosocle, hypocenterC3×C11×C49C22×C5
FrattiniC7C2×C3⋊(C7⋊AGL(1,𝔽11))
FittingC2×C3×C11×C49C2×C5
socleC2×C3×C7×C11C5×D7
LCSG ▷ C3×C11×C49
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 65
transitive degreeμt(G) = 3234

A minimal faithful permutation representation — 2 generators on 65 points:

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

Conjugacy classes & character table

1044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 68);