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C2×C4×C5⋊AGL(1,𝔽7) #13132576

order1680
isoclinismisoclinic class of C5⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4×C5⋊AGL(1,𝔽7)
unambiguous semidirect product(C2×C4×D35)⋊C3

Splittings

FittingC2×C4×C5×C7 ⋊ C2×C3
derived, HallC5×C7 ⋊ C22×C3×C4
HallC2×C4×D35 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C22×C4
HallΓL(1,𝔽8) ⋊ C2×C4×D5
SylowC7 ⋊ C2×C3×C4×D5
SylowC5 ⋊ C2×C4×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2×C4C5⋊AGL(1,𝔽7)
commutator, hypocenterC5×C7C22×C3×C4
FrattiniC2C22×C5⋊AGL(1,𝔽7)
FittingC2×C4×C5×C7C2×C3
socleC22×C5×C7C22×C3
prime radicalC22×C5⋊AGL(1,𝔽7)C2
cosocleC2×C5×C7C23×C3
LCSG ▷ C5×C7
UCSC1 ◁ C2×C4

216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 280

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×D5×AGL(1,𝔽7), C2×D4×C5⋊AGL(1,𝔽7), C2×#10587950⋊C3, …
maximal normal subgroup ofC2×C4×D5×AGL(1,𝔽7), C2×D4×C5⋊AGL(1,𝔽7), C2×#10587950⋊C3, …
maximal quotient ofM16×C5⋊AGL(1,𝔽7), C22⋊C4×C5⋊AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 596);