Finite Group Databasebeta

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D3×M16×ΓL(1,𝔽8) #72244536

order2016
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productD3×M16×ΓL(1,𝔽8)

Splittings

FittingC3×C7×M16 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×M16
SylowC7 ⋊ C3×D3×M16

Subgroups

subgroupG/H
center, FrattiniC4C22×D3×ΓL(1,𝔽8)
commutator, socleC2×C3×C7C22×C3×C4
FittingC3×C7×M16C2×C3
prime radicalC22×D3×ΓL(1,𝔽8)C4
cosocleC3×C4×C7C23×C3
hypercenterM16D3×ΓL(1,𝔽8)
hypocenterC3×C7C2×C3×M16
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ M16

292 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 117 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

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽8)×#100, D3×M16×AGL(1,𝔽7), D3×#6397⋊C3, …
maximal normal subgroup ofD3×ΓL(1,𝔽8)×#100, D3×M16×AGL(1,𝔽7), D3×#6397⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#63818

GAP

Past the SmallGroups library coverage — no SmallGroups id.