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C5×D4×AGL(1,𝔽7) #13132559

order1680
automorphism?
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C23

Aliases

direct productC5×D4×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C5×D4)
unambiguous semidirect product(D4×D7)⋊(C3×C5)
unambiguous semidirect product(C5×D4×D7)⋊C3

Splittings

FittingC5×C7×D4 ⋊ C2×C3
HallC5×D4×D7 ⋊ C3
HallD4×D7 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ C2×D4
HallC5×C7 ⋊ C2×C3×D4
HallΓL(1,𝔽8) ⋊ C2×C5×D4
SylowC7 ⋊ C2×C3×C5×D4

Subgroups

subgroupG/H
centerC2×C5C22×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3×C5
FrattiniC2C22×C5×AGL(1,𝔽7)
FittingC5×C7×D4C2×C3
socleC2×C5×C7C23×C3
hypercenterC5×D4AGL(1,𝔽7)
hypocenterC7C2×C3×C5×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C5×D4

216 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 140

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4×D5×AGL(1,𝔽7), C2×C5×D4×AGL(1,𝔽7), C5×D8×AGL(1,𝔽7), …
maximal normal subgroup ofD4×D5×AGL(1,𝔽7), C2×C5×D4×AGL(1,𝔽7), C5×D8×AGL(1,𝔽7), …
maximal quotient ofC2×C5×D4×AGL(1,𝔽7), C5×D8×AGL(1,𝔽7), C5×#6474⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 579);