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Cayley diagram

C4×C3⋊AGL(1,𝔽7) #92660

order504
isoclinismisoclinic class of C3⋊AGL(1,𝔽7)
automorphismAut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = C3⋊AGL(1,𝔽7) · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×C3⋊AGL(1,𝔽7)
unambiguous semidirect product(C4×D21)⋊C3

Splittings

FittingC3×C4×C7 ⋊ C2×C3
derivedC3×C7 ⋊ C2×C3×C4
HallC3×ΓL(1,𝔽8) ⋊ C2×C4
SylowC7 ⋊ C3×C4×D3

Subgroups

subgroupG/H
center, hypercenterC4C3⋊AGL(1,𝔽7)
commutator, hypocenterC3×C7C2×C3×C4
FrattiniC2C2×C3⋊AGL(1,𝔽7)
FittingC3×C4×C7C2×C3
socle, cosocleC2×C3×C7C22×C3
prime radicalC2×C3⋊AGL(1,𝔽7)C2
LCSG ▷ C3×C7
UCSC1 ◁ C4

70 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C3⋊AGL(1,𝔽7))⋊C8, #66559⋊C3, C8×C3⋊AGL(1,𝔽7), …
maximal normal subgroup of(C3⋊AGL(1,𝔽7))⋊C8, #66559⋊C3, C8×C3⋊AGL(1,𝔽7), …
maximal quotient ofC8×C3⋊AGL(1,𝔽7), #66622⋊C3, C4×C3⋊(ΓL(1,𝔽8)⋊C4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 86);