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C4×D3×AGL(1,𝔽7) #11759366

order1008
isoclinismisoclinic class of D3×AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC4×D3×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C4×D3)
unambiguous semidirect product(D3×D7)⋊(C3×C4)
unambiguous semidirect product(C4×D3×D7)⋊C3

Splittings

FittingC3×C4×C7 ⋊ C22×C3
derivedC3×C7 ⋊ C22×C3×C4
HallC3×ΓL(1,𝔽8) ⋊ C22×C4
SylowC7 ⋊ C2×C3×C4×D3

Subgroups

subgroupG/H
center, hypercenterC4D3×AGL(1,𝔽7)
commutator, hypocenterC3×C7C22×C3×C4
FrattiniC2C2×D3×AGL(1,𝔽7)
FittingC3×C4×C7C22×C3
socle, cosocleC2×C3×C7C23×C3
prime radicalC2×D3×AGL(1,𝔽7)C2
LCSG ▷ C3×C7
UCSC1 ◁ C4

232 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 84

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽5)×AGL(1,𝔽7), C4×C3⋊D3×AGL(1,𝔽7), C2×C4×D3×AGL(1,𝔽7), …
maximal normal subgroup ofC2×C4×D3×AGL(1,𝔽7), C3×C4×D3×AGL(1,𝔽7), D3×D4×AGL(1,𝔽7), …
maximal quotient ofD3×AGL(1,𝔽5)×AGL(1,𝔽7), C4×C3⋊D3×AGL(1,𝔽7), C2×C4×D3×AGL(1,𝔽7), …
automorphism group ofC5×D21, C3×C5×ΓL(1,𝔽8), C5×D3×D7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1008, 537);