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C3×C22⋊C4×AGL(1,𝔽7) #72244137

order2016
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C24×C3

Aliases

direct productC3×C22⋊C4×AGL(1,𝔽7)

Splittings

FittingC3×C7×C22⋊C4 ⋊ C2×C3
HallD7×C22⋊C4 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C2×C22⋊C4
SylowC7 ⋊ C2×C32×C22⋊C4

Subgroups

subgroupG/H
centerC22×C3C22×AGL(1,𝔽7)
commutatorC2×C7C22×C32×C4
FrattiniC22C22×C3×AGL(1,𝔽7)
FittingC3×C7×C22⋊C4C2×C3
socleC22×C3×C7C23×C3
prime radicalC23×C3×AGL(1,𝔽7)C2
cosocleC22×C7C23×C32
hypercenterC3×C22⋊C4AGL(1,𝔽7)
hypocenterC7C2×C32×C22⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#120×AGL(1,𝔽7), D3×C22⋊C4×AGL(1,𝔽7), AGL(1,𝔽7)×#882, …
maximal normal subgroup ofC3×#120×AGL(1,𝔽7), D3×C22⋊C4×AGL(1,𝔽7), AGL(1,𝔽7)×#882, …
maximal quotient ofC3×#100×AGL(1,𝔽7), C3×#104×AGL(1,𝔽7), C22⋊C4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.