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AGL(1,𝔽7)×(C2×D3)⋊C4 #72244153

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

Aliases

direct productAGL(1,𝔽7)×(C2×D3)⋊C4

Splittings

FittingC2×C3×C4×C7 ⋊ C22×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×C22⋊C4
SylowC7 ⋊ C2×C3×(C2×D3)⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C2×D3×AGL(1,𝔽7)
commutatorC2×C3×C7C22×C3×C4
FittingC2×C3×C4×C7C22×C3
socle, cosocleC22×C3×C7C23×C3
prime radicalC22×D3×AGL(1,𝔽7)C2
hypercenterC2×C4D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×C22⋊C4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4

582 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 131 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

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#937, D3×C22⋊C4×AGL(1,𝔽7), AGL(1,𝔽7)×#882, …
maximal normal subgroup ofAGL(1,𝔽7)×#937, D3×C22⋊C4×AGL(1,𝔽7), AGL(1,𝔽7)×#882
maximal quotient ofAGL(1,𝔽7)×#823, AGL(1,𝔽7)×#3752, AGL(1,𝔽7)×#805, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.