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Cayley diagram

D3×C22⋊ΓL(1,𝔽8) #92759

order504
isoclinismisoclinic class of D3×C22⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

direct productD3×C22⋊ΓL(1,𝔽8)
unambiguous semidirect productC3⋊(C2×C22⋊ΓL(1,𝔽8))
unambiguous semidirect product(C22×D3)⋊ΓL(1,𝔽8)
unambiguous semidirect product(D3×C7)⋊A4

Splittings

Fitting, socle, derivedC22×C3×C7 ⋊ C2×C3
SylowC7 ⋊ D3×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C3×C7C2×C3
LCSG ▷ C22×C3×C7
UCSC1

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D7⋊A4, C2×D3×C22⋊ΓL(1,𝔽8), D9×C22⋊ΓL(1,𝔽8), …
maximal normal subgroup ofD3×D7⋊A4, C2×D3×C22⋊ΓL(1,𝔽8), D3×A4×ΓL(1,𝔽8), …
maximal quotient ofC3⋊(#41⋊ΓL(1,𝔽8)), D3×C7⋊SL(2,𝔽3), Dic3×C22⋊ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 185);