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C5×#71819 #14313411

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

Aliases

direct productC5×#71819

Splittings

HallC3×C5 ⋊ #2126
SylowC3 ⋊ C5×#2126

Subgroups

subgroupG/H
centerC4×C5#834
commutatorC2×C3×C4C22×C4×C5
FrattiniC2×D4C22×C5×D3
Fitting, prime radicalC3×C5×#532C2
socleC2×C3×C5C2×#99
cosocleC2×C3×D4C23×C5
hypercenterC5×#532D3
hypocenterC3C5×#2126
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ C5×#532

516 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)(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22);(10,14)(12,20)(15,21)(16,17)(18,24)(22,23);(7,8)(9,10)(11,15)(12,16,20,14)(13,17)(18,22,24,21)(19,23)

Conjugacy classes & character table

270 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 39109);