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C3⋊AGL(1,𝔽67) #27210707

order13266
automorphism?
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊AGL(1,𝔽67)
unambiguous semidirect product(C3⋊(D67⋊C3))⋊C11
unambiguous semidirect product((C67⋊C3)⋊C11)⋊D3
unambiguous semidirect product(C3⋊(D67⋊C11))⋊C3

Splittings

Fitting, socle, derivedC3×C67 ⋊ C2×C3×C11
HallC3×(C67⋊C3)⋊C11 ⋊ C2
HallC3⋊(D67⋊C3) ⋊ C11
HallC67⋊C11 ⋊ C3×D3
HallC3×C67⋊C3 ⋊ C2×C11
SylowC67 ⋊ C3×D3×C11

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C67C2×C3×C11
LCSG ▷ C3×C67
UCSC1

36 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:

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

Polycyclic presentation:

2,3,11,3,67|d^a=d^2,e^a=e^66,e^b=e^29,e^c=e^64

Conjugacy classes & character table

102 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, 21);