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A4⋊AGL(1,𝔽11) #13050436

order1320
isoclinismisoclinic class of A4⋊AGL(1,𝔽11)
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productA4⋊AGL(1,𝔽11)
unambiguous semidirect product(C11⋊C5)⋊S4
unambiguous semidirect product(C11⋊S4)⋊C5

Splittings

Fitting, socleC22×C11 ⋊ C5×D3
derivedC11×A4 ⋊ C2×C5
HallC11⋊S4 ⋊ C5
HallC11⋊C5 ⋊ S4
SylowC11 ⋊ C5×S4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC11×A4C2×C5
Fitting, socleC22×C11C5×D3
LCSG ▷ C11×A4
UCSC1

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 44

Polycyclic presentation:

2,5,3,11,2^2|c^a=c^2,d^a=d^10,e^a=ef,d^b=d^4,e^c=f,f^c=ef

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×AGL(1,𝔽11), C2×A4⋊AGL(1,𝔽11), #72222662, …
maximal normal subgroup ofS4×AGL(1,𝔽11), C2×A4⋊AGL(1,𝔽11), C3×A4⋊AGL(1,𝔽11)
maximal quotient ofC2×A4⋊AGL(1,𝔽11), #72222644, #72222662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1320, 140);