Finite Group Databasebeta

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C7×#6717 #13132350

order1680
isoclinismisoclinic class of #6715
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×#6717

Splittings

FittingC3×C5×C7×Q8 ⋊ C2
HallC3×C5×C7 ⋊ SD16
HallC5×C7 ⋊ #215
HallC3×C7 ⋊ #688
HallC3×C5 ⋊ C7×SD16
SylowC5 ⋊ C7×#215
SylowC3 ⋊ C7×#688

Subgroups

subgroupG/H
centerC2×C7#1220
commutator, cosocleC3×C4×C5C22×C7
FrattiniC4C2×C7×D15
FittingC3×C5×C7×Q8C2
socleC2×C3×C5×C7D4
prime radicalC7×D60C2
hypercenterC7×Q8D15
hypocenterC3×C5C7×SD16
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×Q8

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 840

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5,6,7)(8,9,11,12,10)(13,14,15)(16,17,19,21)(18,23,22,20);(9,10)(11,12)(14,15)(17,20)(18,22)(21,23)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 370);