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Cayley diagram

C3×ΓL(1,𝔽8)⋊Q8 #92649

order504
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)⋊Q8
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×Q8)
unambiguous semidirect productDic14⋊C32
unambiguous semidirect product(C3×Dic14)⋊C3

Splittings

HallC3×ΓL(1,𝔽8) ⋊ Q8
HallDic14 ⋊ C32
SylowC7 ⋊ C32×Q8

Subgroups

subgroupG/H
centerC2×C3C2×AGL(1,𝔽7)
commutator, cosocleC2×C7C22×C32
FrattiniC2C2×C3×AGL(1,𝔽7)
FittingC3×C4×C7C2×C3
socleC2×C3×C7C22×C3
prime radicalC2×C3×ΓL(1,𝔽8)C22
hypercenterC3×C4AGL(1,𝔽7)
hypocenterC7C32×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×C4

72 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:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66564⋊C3, #66566⋊C3, #66569⋊C3, …
maximal normal subgroup of#66564⋊C3, #66566⋊C3, #66569⋊C3, …
maximal quotient ofC3×(ΓL(1,𝔽8)⋊C4)⋊C4, C3×C4⋊(ΓL(1,𝔽8)⋊C4), C2×C3×ΓL(1,𝔽8)⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 75);