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C5×C7×#67 #11719813

order840
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C5×C7 · Schur multiplier (H2) = C2

Aliases

direct productC5×C7×#67

Splittings

FittingC22×C3×C5×C7 ⋊ C2
HallC3×C5×C7 ⋊ D4
HallC3×C7 ⋊ C5×D4
HallC3×C5 ⋊ C7×D4
SylowC3 ⋊ C5×C7×D4

Subgroups

subgroupG/H
centerC2×C5×C7C2×D3
commutator, cosocleC2×C3C22×C5×C7
FrattiniC2C2×C5×D3×C7
FittingC22×C3×C5×C7C2
socleC2×C3×C5×C7C22
hypercenterC22×C5×C7D3
hypocenterC3C5×C7×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5×C7 ◁ C22×C5×C7

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)(6,7,8,9,10,11,12)(13,14,15)(17,19);(14,15)(16,17)(18,19)

Conjugacy classes & character table

315 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×#66684, C5×#66686, C5×D7×#67, …
maximal normal subgroup ofC5×#66684, C5×#66686, C5×D7×#67, …
maximal quotient ofC5×C7×Dic3⋊C4, C5×C7×(C2×D3)⋊C4, C5×C7×#213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 115);