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C3×#66108 #11755785

order960
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#66108

Splittings

FittingC3×C5×AGL(1,ℤ/8) ⋊ C2
HallC3×C5 ⋊ #576
SylowC5 ⋊ C3×#576

Subgroups

subgroupG/H
centerC2×C3C2×D4×D5
commutator, cosocleC4×C5C24×C3
FrattiniC4C23×C3×D5
FittingC3×C5×AGL(1,ℤ/8)C2
socleC2×C3×C5C22×D4
hypercenterC3×AGL(1,ℤ/8)D5
hypocenterC5C3×#576
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

536 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)(4,5,7,8,6)(9,11)(10,15)(12,20)(13,17)(14,21)(16,24)(18,19)(22,23);(11,18)(12,19)(15,23)(16,22)(17,20)(21,24);(10,14)(11,17)(16,22)(18,20);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,22)(20,24)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#10612011, C3×#10612670, C3×#10612663, …
maximal normal subgroup ofC3×#10612011, C3×#10612670, C3×#10612663, …
maximal quotient ofC3×#10610613, C3×#10610656, C3×#10610941, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 9980);