Finite Group Databasebeta

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C2×D4×C3⋊AGL(1,𝔽7) #72244111

order2016
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC2×D4×C3⋊AGL(1,𝔽7)

Splittings

FittingC2×C3×C7×D4 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ C22×D4
SylowC7 ⋊ C2×C3×D3×D4

Subgroups

subgroupG/H
centerC22C22×C3⋊AGL(1,𝔽7)
commutator, cosocleC2×C3×C7C24×C3
FrattiniC2C23×C3⋊AGL(1,𝔽7)
FittingC2×C3×C7×D4C2×C3
socleC22×C3×C7C23×C3
hypercenterC2×D4C3⋊AGL(1,𝔽7)
hypocenterC3×C7C22×C3×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

998 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 232 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

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×D4×AGL(1,𝔽7), #120×C3⋊AGL(1,𝔽7), #72244193, …
maximal normal subgroup ofC2×D3×D4×AGL(1,𝔽7), #120×C3⋊AGL(1,𝔽7), #72244193, …
maximal quotient of#120×C3⋊AGL(1,𝔽7), #127×C3⋊AGL(1,𝔽7), AGL(1,ℤ/8)×C3⋊AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.