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C3×ΓL(1,𝔽8)×B(4,𝔽2) #72246162

order4032
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productC3×ΓL(1,𝔽8)×B(4,𝔽2)

Splittings

FittingC3×C7×B(4,𝔽2) ⋊ C3
HallC7×B(4,𝔽2) ⋊ C32
SylowC7 ⋊ C32×B(4,𝔽2)

Subgroups

subgroupG/H
centerC2×C3ΓL(1,𝔽8)×#120
commutator, cosocleC23×C7C23×C32
FrattiniC23C23×C3×ΓL(1,𝔽8)
FittingC3×C7×B(4,𝔽2)C3
socleC2×C3×C7C3×#120
hypercenterC3×B(4,𝔽2)ΓL(1,𝔽8)
hypocenterC7C32×B(4,𝔽2)
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×B(4,𝔽2)

1188 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×B(4,𝔽2), ΓL(1,𝔽8)2×B(4,𝔽2), D3×ΓL(1,𝔽8)×B(4,𝔽2), …
maximal normal subgroup ofC3×AGL(1,𝔽7)×B(4,𝔽2), D3×ΓL(1,𝔽8)×B(4,𝔽2), B(4,𝔽2)×C3⋊AGL(1,𝔽7)
maximal quotient ofΓL(1,𝔽8)2×B(4,𝔽2)

GAP

Past the SmallGroups library coverage — no SmallGroups id.