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Cayley diagram

ΓL(1,𝔽8)×C3⋊C8 #92617

order504
isoclinismisoclinic class of D3×ΓL(1,𝔽8)
automorphismAut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = D3×ΓL(1,𝔽8) · Out(G) = C23
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×C3⋊C8
unambiguous semidirect productC3⋊(C8×ΓL(1,𝔽8))
unambiguous semidirect product(C7×C3⋊C8)⋊C3

Splittings

derivedC3×C7 ⋊ C3×C8
HallC3×ΓL(1,𝔽8) ⋊ C8
SylowC7 ⋊ C3×C3⋊C8

Subgroups

subgroupG/H
center, Frattini, hypercenterC4D3×ΓL(1,𝔽8)
commutator, hypocenterC3×C7C3×C8
Fitting, cosocleC3×C4×C7C2×C3
socleC2×C3×C7C3×C4
prime radicalC2×C3×ΓL(1,𝔽8)C4
LCSG ▷ C3×C7
UCSC1 ◁ C4

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3)(5,6,7)(8,10,9)(11,15,17,12,14,18,13,16);(1,2,3)(4,5,6,8,7,9,10)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊C8×AGL(1,𝔽7), (C3⋊AGL(1,𝔽7))⋊C8, #66557⋊C3, …
maximal normal subgroup ofC3⋊C8×AGL(1,𝔽7), (C3⋊AGL(1,𝔽7))⋊C8, #66557⋊C3, …
maximal quotient ofΓL(1,𝔽8)×C3⋊C16, C2×ΓL(1,𝔽8)×C3⋊C8, ΓL(1,𝔽8)×C9⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 43);