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C5×C3⋊GU(2,𝔽3) #13086215

order1440
isoclinismisoclinic class of C3⋊#226
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊GU(2,𝔽3)
unambiguous semidirect productC3⋊(C5×GU(2,𝔽3))

Splittings

derivedC3×SL(2,𝔽3) ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC4×C5C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C4×C5
FrattiniC4C5×C3⋊S4
FittingC3×C5×#41D3
socleC2×C3×C5SL(2,ℤ/4)
prime radicalC3×C5×#41⋊C3C2
cosocleC3×#41⋊C3C2×C5
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4×C5

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 360

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,4,5)(6,8,11,15)(7,13)(9,17,18,26)(10,21)(12,29,14,22)(16,19,28,20)(23,27,24,25)(31,32);(30,31,32);(6,9,17)(7,12,16)(8,14,22)(10,20,25)(11,19,28)(13,24,26)(15,23,27)(18,29,21)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 2532);