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C3×AGL(1,𝔽67) #27210706

order13266
automorphism?
homologyAbelianization (H1) = C2×C32×C11 · Schur multiplier (H2) = C3

Aliases

direct productC3×AGL(1,𝔽67)
unambiguous semidirect product(D67⋊C3)⋊(C3×C11)
unambiguous semidirect product(C3×D67⋊C3)⋊C11
unambiguous semidirect product(D67⋊C11)⋊C32
unambiguous semidirect product((C67⋊C3)⋊C11)⋊(C2×C3)
unambiguous semidirect product(C3×D67⋊C11)⋊C3

Splittings

Fitting, socleC3×C67 ⋊ C2×C3×C11
derived, SylowC67 ⋊ C2×C32×C11
HallC3×(C67⋊C3)⋊C11 ⋊ C2
HallD67⋊C11 ⋊ C32
HallC3×D67⋊C3 ⋊ C11
HallC67⋊C11 ⋊ C2×C32
HallC3×C67⋊C3 ⋊ C2×C11
HallD67 ⋊ C32×C11

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽67)
commutator, cosocle, hypocenterC67C2×C32×C11
FrattiniC1G
Fitting, socleC3×C67C2×C3×C11
LCSG ▷ C67
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 70
transitive degreeμt(G) = 201

A minimal faithful permutation representation — 2 generators on 70 points:

(2,16,26,36,51,61,37,66,62,64,25,11,5,6,46,60,67,17,21,63,59,32,4,22,12,15,19,58,23,34,31,42,49,3,10,14,24,30,40,29,44,50,55,48,9,8,13,28,39,52,33,43,54,65,56,7,20,18,27,38,45,47,57,41,53,35);(1,2,6,8,19,41,27,14,29,12,25,50,61,52,39,55,54,44,30,53,22,46,56,10,21,45,57,36,47,9,7,17,35,59,65,49,33,4,11,23,24,34,58,43,16,32,28,20,42,51,66,63,64,60,67,40,62,48,18,37,26,15,31,38,5,13,3)(68,69,70)

Conjugacy classes & character table

201 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽67), C2×C3×AGL(1,𝔽67)
maximal normal subgroup ofD3×AGL(1,𝔽67), C2×C3×AGL(1,𝔽67)
maximal quotient ofC3×((C67⋊C3)⋊C4)⋊C11, C2×C3×AGL(1,𝔽67)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13266, 20);