Finite Group Databasebeta

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C5×#71854 #14302836

order1920
automorphism?
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23×C4

Aliases

direct productC5×#71854

Splittings

HallC3×C5 ⋊ #1910
SylowC3 ⋊ C5×#1910

Subgroups

subgroupG/H
centerC2×C5#5220
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#535C2
socleC2×C3×C5#386
cosocleC22×C3×C4C23×C5
hypercenterC5×#535D3
hypocenterC3C5×#1910
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#535

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 28534);