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Dic5×#733 #13108115

order1620
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productDic5×#733

Splittings

HallC5×#733 ⋊ C4
SylowC5 ⋊ C4×#733

Subgroups

subgroupG/H
centerC2×C3D5×H(3,𝔽3)
commutatorC32×C5C32×C4
FrattiniC2×C32C32×D5
FittingC2×C5×#733C2
socleC2×C3×C5C2×H(3,𝔽3)
prime radical, cosocleC2×C32×C5C2×C32
hypercenterC2×#733D5
hypocenterC5C4×#733
LCSG ▷ C32×C5 ▷ C3×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C32 ◁ C2×#733

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 36 points:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 21);