Finite Group Databasebeta

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Cayley diagram

C3×C4×D3×C7 #92708

order504
isoclinismisoclinic class of D3
automorphismAut(G) = C24×C3×D3 (order 288) · Inn(G) = D3 · Out(G) = C24×C3
homologyAbelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC3×C4×D3×C7

Splittings

FittingC32×C4×C7 ⋊ C2
derivedC3 ⋊ C2×C3×C4×C7
HallC32×C7 ⋊ C2×C4
SylowC32 ⋊ C2×C4×C7

Subgroups

subgroupG/H
center, hypercenterC3×C4×C7D3
commutator, hypocenterC3C2×C3×C4×C7
FrattiniC2C2×C3×D3×C7
FittingC32×C4×C7C2
socleC2×C32×C7C22
prime radicalC2×C3×D3×C7C2
cosocleC2×C3C22×C3×C7
LCSG ▷ C3
UCSC1 ◁ C3×C4×C7

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 168

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

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×C7⋊C8, C3×#66558, D3×C7×C3⋊C8, …
maximal normal subgroup ofC3×D3×C7⋊C8, C3×#66558, D3×C7×C3⋊C8, …
maximal quotient ofC3×D3×C7×C8, C3×C7×ΓL(1,𝔽25), C3×C4×C7×Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 134);