Finite Group Databasebeta

← Prev (#72244170) Next (#72244172) →

C22⋊C4×C3⋊AGL(1,𝔽7) #72244171

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

Aliases

direct productC22⋊C4×C3⋊AGL(1,𝔽7)

Splittings

FittingC3×C7×C22⋊C4 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×C22⋊C4
SylowC7 ⋊ C3×D3×C22⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C22×C3⋊AGL(1,𝔽7)
commutatorC2×C3×C7C22×C3×C4
FittingC3×C7×C22⋊C4C2×C3
socle, cosocleC22×C3×C7C23×C3
prime radicalC23×C3⋊AGL(1,𝔽7)C2
hypercenterC22⋊C4C3⋊AGL(1,𝔽7)
hypocenterC3×C7C2×C3×C22⋊C4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊C4

562 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 120 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 of#120×C3⋊AGL(1,𝔽7), D3×C22⋊C4×AGL(1,𝔽7), #72244221, …
maximal normal subgroup of#120×C3⋊AGL(1,𝔽7), D3×C22⋊C4×AGL(1,𝔽7), #72244221, …
maximal quotient of#100×C3⋊AGL(1,𝔽7), C22⋊C4×C3⋊(C3⋊AGL(1,𝔽7)), #104×C3⋊AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.