Finite Group Databasebeta

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Cayley diagram

C3×#4268 #92716

order504
isoclinismisoclinic class of Dic42
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×#4268

Splittings

FittingC22×C32×C7 ⋊ C2
HallC32×C7 ⋊ D4
SylowC32 ⋊ C22⋊D7
SylowC7 ⋊ C3×#67

Subgroups

subgroupG/H
centerC2×C3C2×D21
commutator, cosocleC2×C3×C7C22×C3
FrattiniC2C2×C3×D21
FittingC22×C32×C7C2
socleC2×C32×C7C22
hypercenterC22×C3D21
hypocenterC3×C7C3×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3 ◁ C22×C3

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,4,6,7,5,3)(8,9,10)(11,12,13)(15,17);(2,3)(4,5)(6,7)(12,13)(14,15)(16,17)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#66683, C3×#66686, C3×D7×#67, …
maximal normal subgroup ofC3×#66683, C3×#66686, C3×D7×#67, …
maximal quotient ofC3×#66629, C3×#66631, C3×#66632, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 142);