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PGL(2,𝔽11) #13050429

order1320
isoclinismisoclinic class of PGL(2,𝔽11)
automorphismAut(G) = PGL(2,𝔽11) (order 1320) · Inn(G) = G · Out(G) = C1
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

projective general linearPGL(2,𝔽11)
unambiguous semidirect productPSL(2,𝔽11)⋊C2

Splittings

socle, derivedPSL(2,𝔽11) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, socle, cosocle, hypocenterPSL(2,𝔽11)C2
LCSG ▷ PSL(2,𝔽11)
UCSC1

29 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 12 points:

(1,4,3,9,5,7,10,11,6,8,12);(1,10,3,12)(2,5,8,9)(4,11,6,7)

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×PGL(2,𝔽11), C3×PGL(2,𝔽11)
maximal normal subgroup ofC5×PGL(2,𝔽11), C3×PGL(2,𝔽11)
maximal quotient of#72210865, C5×PGL(2,𝔽11), #72223872, …
automorphism group ofPSL(2,𝔽11), SL(2,𝔽11), PGL(2,𝔽11)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1320, 133);