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D32×AGL(1,𝔽7) #13094895

order1512
isoclinismisoclinic class of D32×AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productD32×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×D32)
unambiguous semidirect product(D3×D7)⋊(C3×D3)
unambiguous semidirect product(D32×D7)⋊C3
unambiguous semidirect product(C3×D3×AGL(1,𝔽7))⋊C2
unambiguous semidirect product(C3⋊D3×AGL(1,𝔽7))⋊C2
unambiguous semidirect product(D3×C3⋊AGL(1,𝔽7))⋊C2

Splittings

Fitting, socle, derivedC32×C7 ⋊ C23×C3
HallC32×ΓL(1,𝔽8) ⋊ C23
SylowC7 ⋊ C2×C3×D32

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C7C23×C3
LCSG ▷ C32×C7
UCSC1

324 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 42

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72213636, C2×D32×AGL(1,𝔽7), AGL(1,𝔽7)×GO+(4,𝔽2), …
maximal normal subgroup ofC2×D32×AGL(1,𝔽7), AGL(1,𝔽7)×GO+(4,𝔽2), C5×D32×AGL(1,𝔽7), …
maximal quotient of#72213636, C2×D32×AGL(1,𝔽7), D32×ΓL(1,𝔽8)⋊C4, …
automorphism group ofD3×D21, C22×C3×ΓL(1,𝔽8), C3×D3×ΓL(1,𝔽8), …
holomorph ofD3×C7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 792);