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C2×ΓL(1,𝔽8)×C3⋊Dic3 #13094754

order1512
isoclinismisoclinic class of C3⋊D3×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×ΓL(1,𝔽8)×C3⋊Dic3
unambiguous semidirect product(C2×C3)⋊(Dic3×ΓL(1,𝔽8))

Splittings

derivedC32×C7 ⋊ C2×C3×C4
HallC32×ΓL(1,𝔽8) ⋊ C2×C4
SylowC7 ⋊ C2×C3×C3⋊Dic3

Subgroups

subgroupG/H
center, hypercenterC22C3⋊D3×ΓL(1,𝔽8)
commutator, hypocenterC32×C7C2×C3×C4
FrattiniC2C2×C3⋊D3×ΓL(1,𝔽8)
Fitting, socleC22×C32×C7C2×C3
prime radicalC22×C32×ΓL(1,𝔽8)C2
cosocleC2×C32×C7C22×C3
LCSG ▷ C32×C7
UCSC1 ◁ C22

272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 504

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×C3⋊D3×ΓL(1,𝔽8), C2×C3⋊Dic3×AGL(1,𝔽7), C2×#13094713, …
maximal normal subgroup ofC2×C4×C3⋊D3×ΓL(1,𝔽8), C2×C3⋊Dic3×AGL(1,𝔽7), C2×#13094713, …
maximal quotient ofC4×ΓL(1,𝔽8)×C3⋊Dic3, ΓL(1,𝔽8)×#3757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 651);