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Cayley diagram

C23×C3×ΓL(1,𝔽8) #92768

order504
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC23×C3×ΓL(1,𝔽8)

Splittings

Fitting, socleC23×C3×C7 ⋊ C3
derived, SylowC7 ⋊ C23×C32
HallC23×C7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC23×C3ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C23×C32
FrattiniC1G
Fitting, socleC23×C3×C7C3
LCSG ▷ C7
UCSC1 ◁ C23×C3

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×(C22⋊Dic7)⋊C3, ΓL(1,𝔽8)×#217, #66636⋊C3, …
maximal normal subgroup ofC3×(C22⋊Dic7)⋊C3, ΓL(1,𝔽8)×#217, #66636⋊C3, …
maximal quotient ofC22×C3×C4×ΓL(1,𝔽8), C2×C3×D4×ΓL(1,𝔽8), C2×C3×Q8×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 194);