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C5×#66332 #13108147

order1620
isoclinismisoclinic class of #4185
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×#66332

Splittings

derived, Sylow#732 ⋊ C4×C5
HallC5×#732 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C5#4185
commutator, hypocenter#732C4×C5
FrattiniC2×C32C5×C3⋊D3
Fitting, prime radicalC2×C5×#732C2
socleC2×C3×C5#272
cosocleC2×#732C2×C5
LCSG ▷ #732
UCSC1 ◁ C2×C5

96 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)(6,7,8,9)(11,14)(12,18)(13,20)(15,23)(16,22)(17,27)(19,30)(21,25)(24,29)(26,35)(28,34)(31,32)(33,36);(10,12,18)(11,15,25)(13,19,31)(14,21,23)(16,28,36)(17,26,29)(20,32,30)(22,33,34)(24,35,27);(10,11,14)(12,15,24)(13,17,22)(16,27,20)(18,29,23)(19,26,21)(25,35,30)(28,33,32)(31,36,34)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 53);