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C7×#1202 #11719765

order840
isoclinismisoclinic class of #1201
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#1202

Splittings

HallC3×C5×C7 ⋊ D4
HallC5×C7 ⋊ #67
HallC3×C7 ⋊ D20
HallC3×C5 ⋊ C7×D4
SylowC5 ⋊ C7×#67
SylowC3 ⋊ C7×D20

Subgroups

subgroupG/H
center, hypercenterC2×C7D3×D5
commutator, cosocleC2×C3×C5C22×C7
FrattiniC2D3×C7×D5
Fitting, socleC2×C3×C5×C7C22
hypocenterC3×C5C7×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×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)(8,9,11,12,10)(14,15)(16,17,18,19);(9,10)(11,12)(13,14,15)(17,19)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×#1202, #13132499, #13132501, …
maximal normal subgroup ofD7×#1202, #13132499, #13132501, …
maximal quotient ofC7×#6653, C7×#6657, C7×#6658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 67);