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C5×#66410 #13108340

order1620
isoclinismisoclinic class of #272
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#66410

Splittings

HallC5×#737 ⋊ C4
Sylow#737 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5×C9C3⋊D3
commutator, hypocenterH(3,𝔽3)C3×C4×C5
FrattiniC2×C3C3×C5×C3⋊D3
FittingC2×C5×#737C2
socleC2×C3×C5C3×C3⋊D3
prime radicalC2×C5×H(3,𝔽3)C2×C3
cosocleC2×H(3,𝔽3)C2×C3×C5
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C5×C9

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 540

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4,5)(6,22,18,9,7,28,16,12,10)(8,30,27,23,11,32,14,29,17,13,20,31,15,19,26,24,21,25)(33,34,35,36);(8,14,15)(11,20,21)(13,23,24)(17,26,27)(19,29,30)(25,31,32);(6,8,13)(7,11,19)(9,15,23)(10,17,25)(12,21,29)(14,24,16)(18,27,31)(20,30,22)(26,32,28)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 246);