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C4×C3⋊D3×ΓL(1,𝔽8) #13094752

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

Aliases

direct productC4×C3⋊D3×ΓL(1,𝔽8)

Splittings

FittingC32×C4×C7 ⋊ C2×C3
derivedC32×C7 ⋊ C2×C3×C4
HallC32×ΓL(1,𝔽8) ⋊ C2×C4
SylowC7 ⋊ C3×C4×C3⋊D3

Subgroups

subgroupG/H
center, hypercenterC4C3⋊D3×ΓL(1,𝔽8)
commutator, hypocenterC32×C7C2×C3×C4
FrattiniC2C2×C3⋊D3×ΓL(1,𝔽8)
FittingC32×C4×C7C2×C3
socle, cosocleC2×C32×C7C22×C3
prime radicalC2×C3⋊D3×ΓL(1,𝔽8)C2
LCSG ▷ C32×C7
UCSC1 ◁ C4

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 252

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊D3×AGL(1,𝔽5)×ΓL(1,𝔽8), C4×ΓL(1,𝔽8)×#157, C4×C3⋊D3×AGL(1,𝔽7), …
maximal normal subgroup ofC4×ΓL(1,𝔽8)×#157, C4×C3⋊D3×AGL(1,𝔽7), C4×ΓL(1,𝔽8)⋊#157, …
maximal quotient ofC3⋊D3×AGL(1,𝔽5)×ΓL(1,𝔽8), C2×C4×C3⋊D3×ΓL(1,𝔽8), C4×ΓL(1,𝔽8)×C3⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 649);