Finite Group Databasebeta

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C3×ΓL(1,𝔽8)×#1004 #72269443

order6048
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3

Aliases

direct productC3×ΓL(1,𝔽8)×#1004

Splittings

FittingC2×C32×C7×D4 ⋊ C2×C3
HallC32×ΓL(1,𝔽8) ⋊ H(5,𝔽2)
SylowC7 ⋊ C32×#1004

Subgroups

subgroupG/H
centerC2×C3C23×D3×ΓL(1,𝔽8)
commutator, cosocleC2×C3×C7C24×C32
FrattiniC2C23×C3×D3×ΓL(1,𝔽8)
FittingC2×C32×C7×D4C2×C3
socleC2×C32×C7C24×C3
hypercenterC2×C3×D4D3×ΓL(1,𝔽8)
hypocenterC3×C7C32×H(5,𝔽2)
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3 ◁ C2×C3×D4

2424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

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

Relationships

maximal subgroup ofΓL(1,𝔽8)×#10596675
maximal normal subgroup ofΓL(1,𝔽8)×#10596675

GAP

Past the SmallGroups library coverage — no SmallGroups id.