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C5×#84100 #14412164

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

Aliases

direct productC5×#84100

Splittings

HallC3×C5 ⋊ #2888
SylowC3 ⋊ C5×#2888

Subgroups

subgroupG/H
centerC2×C5C22×#217
commutatorC22×C3C23×C4×C5
FrattiniC2×C4C23×C5×D3
Fitting, prime radicalC2×C3×C5×H(5,𝔽2)C2
socleC2×C3×C5C22×C22⋊C4
cosocleC2×C3×C4C24×C5
hypercenterC2×C5×H(5,𝔽2)D3
hypocenterC3C5×#2888
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C2×C5×#41 ◁ C2×C5×H(5,𝔽2)

1512 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)(11,17)(15,20)(18,23)(22,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(10,14)(12,19)(15,20)(18,23);(7,8)(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 137862);