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C5×#64315 #13089328

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

Aliases

direct productC5×#64315

Splittings

HallC32×C5 ⋊ 2−1+4
SylowC32 ⋊ C5×2−1+4

Subgroups

subgroupG/H
centerC2×C5C22×D32
commutator, cosocleC2×C32C24×C5
FrattiniC2C22×C5×D32
FittingC32×C5×Q8C22
socleC2×C32×C5C24
hypercenterC5×Q8D32
hypocenterC32C5×2−1+4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C5×Q8

622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 4 generators on 27 points:

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

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, 5645);