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C2×C5×(C22⋊D7)⋊C3 #13132564

order1680
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C23

Aliases

direct productC2×C5×(C22⋊D7)⋊C3
unambiguous semidirect product(C2×C22⋊D7)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C22⋊D7)⋊C3

Splittings

FittingC23×C5×C7 ⋊ C2×C3
HallC2×C5×C22⋊D7 ⋊ C3
HallC2×C22⋊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
centerC22×C5C2×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3×C5
FrattiniC2C22×C5×AGL(1,𝔽7)
FittingC23×C5×C7C2×C3
socleC22×C5×C7C22×C3
hypercenterC23×C5AGL(1,𝔽7)
hypocenterC7C2×C3×C5×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C5 ◁ C23×C5

216 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 92 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

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D5×(C22⋊D7)⋊C3, C2×C5×D4×AGL(1,𝔽7), C2×#10587963⋊C3, …
maximal normal subgroup ofC2×D5×(C22⋊D7)⋊C3, C2×C5×D4×AGL(1,𝔽7), C2×#10587963⋊C3, …
maximal quotient ofC5×#6495⋊C3, C5×#6512⋊C3, C5×#10620789, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 584);