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C11×C3⋊ΓL(1,𝔽16) #14520452

order1980
isoclinismisoclinic class of C3⋊ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C4×C11 · Schur multiplier (H2) = C3

Aliases

direct productC11×C3⋊ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C11×ΓL(1,𝔽16))

Splittings

Fitting, socle, HallC32×C5×C11 ⋊ C4
derived, HallC32×C5 ⋊ C4×C11
HallC32×C11 ⋊ AGL(1,𝔽5)
HallC5×C11 ⋊ C3⋊Dic3
SylowC32 ⋊ C11×AGL(1,𝔽5)
SylowC5 ⋊ C11×C3⋊Dic3

Subgroups

subgroupG/H
center, hypercenterC11C3⋊ΓL(1,𝔽16)
commutator, hypocenterC32×C5C4×C11
FrattiniC1G
Fitting, socleC32×C5×C11C4
prime radicalC32×D5×C11C2
cosocleC32×D5C2×C11
LCSG ▷ C32×C5
UCSC1 ◁ C11

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 495

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,4,5,6,7,8,9,10,11)(13,14,15,16)(18,19)(21,22);(12,13,16,14,15)(20,21,22);(17,18,19)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×C11×C3⋊ΓL(1,𝔽16), C11×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), C121×C3⋊ΓL(1,𝔽16), …
maximal normal subgroup ofC7×C11×C3⋊ΓL(1,𝔽16), C121×C3⋊ΓL(1,𝔽16), C112×C3⋊ΓL(1,𝔽16), …
maximal quotient ofC7×C11×C3⋊ΓL(1,𝔽16), C11×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), C121×C3⋊ΓL(1,𝔽16), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1980, 77);