Finite Group Databasebeta

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C5×#71855 #14303040

order1920
automorphism?
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24

Aliases

direct productC5×#71855

Splittings

HallC3×C5 ⋊ #1914
SylowC3 ⋊ C5×#1914

Subgroups

subgroupG/H
centerC2×C5#5220
commutatorC23×C3C22×C4×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#535C2
socleC2×C3×C5#386
cosocleC22×C3×C4C23×C5
hypercenterC5×#535D3
hypocenterC3C5×#1914
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#535

680 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)(14,20)(16,19)(17,24)(21,23);(7,8)(9,11,13,16,14,17,23,10)(12,18,22,19,20,24,21,15);(7,8)(9,10,13,17,14,16,23,11)(12,19,21,24,20,15,22,18)

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, 28738);