Finite Group Databasebeta

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C22⋊C4×C3⋊D3×ΓL(1,𝔽8) #72247165

order6048
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24×C3

Aliases

direct productC22⋊C4×C3⋊D3×ΓL(1,𝔽8)

Splittings

FittingC32×C7×C22⋊C4 ⋊ C2×C3
HallC32×ΓL(1,𝔽8) ⋊ C2×C22⋊C4
SylowC7 ⋊ C3×C22⋊C4×C3⋊D3

Subgroups

subgroupG/H
center, FrattiniC22C22×C3⋊D3×ΓL(1,𝔽8)
commutatorC2×C32×C7C22×C3×C4
FittingC32×C7×C22⋊C4C2×C3
socle, cosocleC22×C32×C7C23×C3
prime radicalC23×C3⋊D3×ΓL(1,𝔽8)C2
hypercenterC22⋊C4C3⋊D3×ΓL(1,𝔽8)
hypocenterC32×C7C2×C3×C22⋊C4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C22⋊C4

1856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 504

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC22⋊C4×C3⋊D3×AGL(1,𝔽7), C3⋊D3×ΓL(1,𝔽8)×#120, C3⋊D3×#6445⋊C3
maximal normal subgroup ofC22⋊C4×C3⋊D3×AGL(1,𝔽7), C3⋊D3×ΓL(1,𝔽8)×#120, C3⋊D3×#6445⋊C3

GAP

Past the SmallGroups library coverage — no SmallGroups id.