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Cayley diagram

C3×Q8×ΓL(1,𝔽8) #92699

order504
isoclinismisoclinic class of D4×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×Q8×ΓL(1,𝔽8)

Splittings

FittingC3×C7×Q8 ⋊ C3
HallC7×Q8 ⋊ C32
SylowC7 ⋊ C32×Q8

Subgroups

subgroupG/H
centerC2×C3C22×ΓL(1,𝔽8)
commutator, cosocleC2×C7C22×C32
FrattiniC2C22×C3×ΓL(1,𝔽8)
FittingC3×C7×Q8C3
socleC2×C3×C7C22×C3
prime radicalC2×C3×ΓL(1,𝔽8)C22
hypercenterC3×Q8ΓL(1,𝔽8)
hypocenterC7C32×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×Q8

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:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#1143⋊C3, C3×#1144⋊C3, ΓL(1,𝔽8)×#215, …
maximal normal subgroup ofC3×#1143⋊C3, C3×#1144⋊C3, ΓL(1,𝔽8)×#215, …
maximal quotient ofC3×C4⋊C4×ΓL(1,𝔽8), C2×C3×Q8×ΓL(1,𝔽8), Q8×C9×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 125);