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C5×#4268 #11719818

order840
isoclinismisoclinic class of Dic42
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#4268

Splittings

FittingC22×C3×C5×C7 ⋊ C2
HallC3×C5×C7 ⋊ D4
HallC5×C7 ⋊ #67
HallC3×C7 ⋊ C5×D4
HallC3×C5 ⋊ C22⋊D7
SylowC7 ⋊ C5×#67
SylowC3 ⋊ C5×C22⋊D7

Subgroups

subgroupG/H
centerC2×C5C2×D21
commutator, cosocleC2×C3×C7C22×C5
FrattiniC2C2×C5×D21
FittingC22×C3×C5×C7C2
socleC2×C3×C5×C7C22
hypercenterC22×C5D21
hypocenterC3×C7C5×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C5 ◁ C22×C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 420

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,4,5)(6,7,9,11,12,10,8)(13,14,15)(17,19);(7,8)(9,10)(11,12)(14,15)(16,17)(18,19)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#66683, C5×#66686, C5×D7×#67, …
maximal normal subgroup ofC5×#66683, C5×#66686, C5×D7×#67, …
maximal quotient ofC5×#66629, C5×#66631, C5×#66632, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 120);