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Cayley diagram

C2×C3×C22⋊ΓL(1,𝔽8) #92770

order504
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C22⋊ΓL(1,𝔽8)
unambiguous semidirect product(C2×C3×C7)⋊A4

Splittings

Fitting, socleC23×C3×C7 ⋊ C3
derivedC22×C7 ⋊ C2×C32
HallC23×C7 ⋊ C32
SylowC23 ⋊ C3×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×A4

Subgroups

subgroupG/H
center, hypercenterC2×C3C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C2×C32
FrattiniC1G
Fitting, socleC23×C3×C7C3
LCSG ▷ C22×C7
UCSC1 ◁ C2×C3

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 126

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×Dic7⋊A4, Dic3×C22⋊ΓL(1,𝔽8), C3⋊(Dic7⋊A4), …
maximal normal subgroup ofC3×Dic7⋊A4, Dic3×C22⋊ΓL(1,𝔽8), C3⋊(Dic7⋊A4), …
maximal quotient ofC3×C4×C22⋊ΓL(1,𝔽8), C2×C3×C7⋊SL(2,𝔽3), C3×#41⋊ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 196);