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C3×#64929 #11746706

order960
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#64929

Splittings

HallC3×C5 ⋊ #356
SylowC5 ⋊ C3×#356

Subgroups

subgroupG/H
centerC2×C3#4011
commutatorC2×C4×C5C2×C3×C4
FrattiniC2×C4C2×C3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C4
socleC2×C3×C5#99
prime radicalC22×C3×C5C22⋊C4
cosocleC2×Dic10C22×C3
hypercenterC2×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×#356
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#10598836, C3×#10598576, #14279382, …
maximal normal subgroup ofC3×#10598836, C3×#10598576, #14279382, …
maximal quotient ofC3×#10598827, C3×#10598711, C3×#10598718, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 901);