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C17×C3⋊ΓL(1,𝔽16) #15690366

order3060
isoclinismisoclinic class of C3⋊ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C4×C17 · Schur multiplier (H2) = C3

Aliases

direct productC17×C3⋊ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C17×ΓL(1,𝔽16))

Splittings

Fitting, socle, HallC32×C5×C17 ⋊ C4
derived, HallC32×C5 ⋊ C4×C17
HallC32×C17 ⋊ AGL(1,𝔽5)
HallC5×C17 ⋊ C3⋊Dic3
SylowC32 ⋊ C17×AGL(1,𝔽5)
SylowC5 ⋊ C17×C3⋊Dic3

Subgroups

subgroupG/H
center, hypercenterC17C3⋊ΓL(1,𝔽16)
commutator, hypocenterC32×C5C4×C17
FrattiniC1G
Fitting, socleC32×C5×C17C4
prime radicalC32×D5×C17C2
cosocleC32×D5C2×C17
LCSG ▷ C32×C5
UCSC1 ◁ C17

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 765

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,20,21,22)(24,25)(27,28);(18,19,22,20,21)(26,27,28);(23,24,25)

Conjugacy classes & character table

357 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC17×C3⋊(C3⋊#1035), C5×C17×C3⋊ΓL(1,𝔽16), C17×C3⋊(C3⋊#1043), …
maximal normal subgroup ofC5×C17×C3⋊ΓL(1,𝔽16), C7×C17×C3⋊ΓL(1,𝔽16), C11×C17×C3⋊ΓL(1,𝔽16)
maximal quotient ofC17×C3⋊(C3⋊#1035), C5×C17×C3⋊ΓL(1,𝔽16), C17×C3⋊(C3⋊#1043), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3060, 71);