Finite Group Databasebeta

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C7×#6718 #13132351

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

Aliases

direct productC7×#6718

Splittings

HallC3×C5×C7 ⋊ Q16
HallC5×C7 ⋊ #216
HallC3×C7 ⋊ #689
HallC3×C5 ⋊ C7×Q16
SylowC5 ⋊ C7×#216
SylowC3 ⋊ C7×#689

Subgroups

subgroupG/H
centerC2×C7#1220
commutator, cosocleC3×C4×C5C22×C7
FrattiniC4C2×C7×D15
FittingC3×C5×C7×Q8C2
socle, prime radicalC2×C3×C5×C7D4
hypercenterC7×Q8D15
hypocenterC3×C5C7×Q16
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×Q8

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 1680

A minimal faithful permutation representation — 2 generators on 31 points:

(1,2,3,4,5,6,7)(8,9,11,12,10)(13,14,15)(16,18,20,26)(17,21,23,30)(19,28,29,25)(22,31,27,24);(9,10)(11,12)(14,15)(16,17,20,23)(18,24,26,31)(19,27,29,22)(21,28,30,25)

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, 371);