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C5×#64313 #13089326

order1440
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C25

Aliases

direct productC5×#64313

Splittings

FittingC32×C5×D4 ⋊ C22
HallC32×C5 ⋊ H(5,𝔽2)
SylowC32 ⋊ C5×H(5,𝔽2)

Subgroups

subgroupG/H
centerC2×C5C22×D32
commutator, cosocleC2×C32C24×C5
FrattiniC2C22×C5×D32
FittingC32×C5×D4C22
socleC2×C32×C5C24
hypercenterC5×D4D32
hypocenterC32C5×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C5×D4

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,3,4,5)(7,11)(8,12)(14,15,16)(17,18,19);(6,8)(7,10)(9,12)(11,13);(8,12)(10,13)(15,16);(6,7)(8,10)(9,11)(12,13)(18,19)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 5643);