Finite Group Databasebeta

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Cayley diagram

#4268⋊C3 #92663

order504
isoclinismisoclinic class of Dic42⋊C3
automorphismAut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = C2×C3⋊AGL(1,𝔽7) · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#4268⋊C3

Splittings

FittingC22×C3×C7 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ D4
SylowC7 ⋊ C3×#67

Subgroups

subgroupG/H
center, FrattiniC2C2×C3⋊AGL(1,𝔽7)
commutator, socle, cosocleC2×C3×C7C22×C3
FittingC22×C3×C7C2×C3
hypercenterC22C3⋊AGL(1,𝔽7)
hypocenterC3×C7C3×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C22

74 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)(4,5,6)(8,9,10)(12,14);(2,3)(4,5)(6,7)(9,10)(11,12)(13,14)

Polycyclic presentation:

2^2,3,2,3,7|b^a=bd,e^a=e^2,f^a=f^6,f^c=f^2

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66683⋊C3, #66686⋊C3, #67×AGL(1,𝔽7), …
maximal normal subgroup of#66683⋊C3, #66686⋊C3, #67×AGL(1,𝔽7), …
maximal quotient of#66629⋊C3, #66631⋊C3, #66632⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 89);