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C5×#72090 #14302330

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

Aliases

direct productC5×#72090

Splittings

HallC3×C5 ⋊ #1903
SylowC3 ⋊ C5×#1903

Subgroups

subgroupG/H
centerC2×C5#5220
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC2×C3×C5×AGL(1,ℤ/8)C2
socleC2×C3×C5#386
cosocleC22×C3×C4C23×C5
hypercenterC2×C5×AGL(1,ℤ/8)D3
hypocenterC3C5×#1903
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×AGL(1,ℤ/8)

648 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)(10,16)(11,18)(13,22)(14,21)(15,23)(17,24)(19,20);(7,8)(9,11,14,10,15,19,13,17)(12,18,21,16,23,20,22,24);(7,8)(9,10)(11,14)(12,20,23,18)(13,19)(15,17)(16,22,24,21)

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, 28028);