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Cayley diagram

C3×(C22⋊D7)⋊C3 #92653

order504
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphismAut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C2×D3
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×(C22⋊D7)⋊C3
unambiguous semidirect product(C22⋊D7)⋊C32
unambiguous semidirect product(C3×C22⋊D7)⋊C3

Splittings

FittingC22×C3×C7 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ D4
HallC22⋊D7 ⋊ C32
SylowC7 ⋊ C32×D4

Subgroups

subgroupG/H
centerC2×C3C2×AGL(1,𝔽7)
commutator, cosocleC2×C7C22×C32
FrattiniC2C2×C3×AGL(1,𝔽7)
FittingC22×C3×C7C2×C3
socleC2×C3×C7C22×C3
hypercenterC22×C3AGL(1,𝔽7)
hypocenterC7C32×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C22×C3

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66683⋊C3, #66684⋊C3, D3×(C22⋊D7)⋊C3, …
maximal normal subgroup of#66683⋊C3, #66684⋊C3, D3×(C22⋊D7)⋊C3, …
maximal quotient ofC3×(ΓL(1,𝔽8)⋊C4)⋊C4, C3×(C2×AGL(1,𝔽7))⋊C4, C3×#1141⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 79);