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Cayley diagram

ΓL(1,𝔽8)×Dic6 #92654

order504
isoclinismisoclinic class of ΓL(1,𝔽8)×Dic6
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×Dic6
unambiguous semidirect productC3⋊(Q8×ΓL(1,𝔽8))
unambiguous semidirect product(C7×Dic6)⋊C3

Splittings

HallC3×ΓL(1,𝔽8) ⋊ Q8
SylowC7 ⋊ C3×Dic6

Subgroups

subgroupG/H
center, FrattiniC2C2×D3×ΓL(1,𝔽8)
commutator, socle, cosocleC2×C3×C7C22×C3
FittingC3×C4×C7C2×C3
prime radicalC2×C3×ΓL(1,𝔽8)C22
hypercenterC4D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×Q8
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4

54 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:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66563⋊C3, #66568⋊C3, #66569⋊C3, …
maximal normal subgroup of#66563⋊C3, #66568⋊C3, #66569⋊C3, …
maximal quotient ofΓL(1,𝔽8)×Dic3⋊C4, ΓL(1,𝔽8)×C4⋊Dic3, C2×ΓL(1,𝔽8)×Dic6, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 80);