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Cayley diagram

Dic3×AGL(1,𝔽7) #92641

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

Aliases

direct productDic3×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×Dic3)
unambiguous semidirect product(C3×AGL(1,𝔽7))⋊C4
unambiguous semidirect product(Dic3×D7)⋊C3

Splittings

derivedC3×C7 ⋊ C2×C3×C4
HallC3×ΓL(1,𝔽8) ⋊ C2×C4
SylowC7 ⋊ C2×C3×Dic3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D3×AGL(1,𝔽7)
commutator, hypocenterC3×C7C2×C3×C4
Fitting, socle, cosocleC2×C3×C7C22×C3
prime radicalC2×C3×AGL(1,𝔽7)C2
LCSG ▷ C3×C7
UCSC1 ◁ C2

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:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic6×AGL(1,𝔽7), #66669⋊C3, #66670⋊C3, …
maximal normal subgroup ofDic6×AGL(1,𝔽7), #66669⋊C3, #66670⋊C3, …
maximal quotient ofC3⋊C8×AGL(1,𝔽7), #66557⋊C3, Dic3×ΓL(1,𝔽8)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 67);