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C5×#78688 #14411965

order1920
automorphism?
homologyAbelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26

Aliases

direct productC5×#78688

Splittings

FittingC2×C3×C5×#99 ⋊ C2
HallC3×C5 ⋊ #2883
SylowC3 ⋊ C5×#2883

Subgroups

subgroupG/H
centerC2×C5C2×D3×C22⋊C4
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC2×C3×C5×#99C2
socleC2×C3×C5C22×C22⋊C4
prime radicalC2×C5×#1004C2
cosocleC23×C3C24×C5
hypercenterC2×C5×#99D3
hypocenterC3C5×#2883
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C24×C5 ◁ C2×C5×#99

1600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 137663);