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C2×D4×C3⋊D3×ΓL(1,𝔽8) #72245578

order6048
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C3

Aliases

direct productC2×D4×C3⋊D3×ΓL(1,𝔽8)

Splittings

FittingC2×C32×C7×D4 ⋊ C2×C3
HallC32×ΓL(1,𝔽8) ⋊ C22×D4
SylowC7 ⋊ C2×C3×D4×C3⋊D3

Subgroups

subgroupG/H
centerC22C22×C3⋊D3×ΓL(1,𝔽8)
commutator, cosocleC2×C32×C7C24×C3
FrattiniC2C23×C3⋊D3×ΓL(1,𝔽8)
FittingC2×C32×C7×D4C2×C3
socleC22×C32×C7C23×C3
hypercenterC2×D4C3⋊D3×ΓL(1,𝔽8)
hypocenterC32×C7C22×C3×D4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C2×D4

3264 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 ofC2×D4×C3⋊D3×AGL(1,𝔽7), C3⋊D3×ΓL(1,𝔽8)×#127, C3⋊D3×#6503⋊C3, …
maximal normal subgroup ofC2×D4×C3⋊D3×AGL(1,𝔽7), C3⋊D3×ΓL(1,𝔽8)×#127, C3⋊D3×#6503⋊C3, …
maximal quotient ofC3⋊D3×ΓL(1,𝔽8)×#127, C3⋊D3×ΓL(1,𝔽8)×#120

GAP

Past the SmallGroups library coverage — no SmallGroups id.