Finite Group Databasebeta

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C7×#6758 #13132618

order1680
isoclinismisoclinic class of #6758
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×#6758

Splittings

HallC3×C5×C7 ⋊ M16
HallC5×C7 ⋊ #208
HallC3×C7 ⋊ #700
HallC3×C5 ⋊ C7×M16
SylowC5 ⋊ C7×#208
SylowC3 ⋊ C7×#700

Subgroups

subgroupG/H
centerC2×C7C2×ΓL(1,𝔽16)
commutatorC2×C3×C5C2×C4×C7
FrattiniC2C2×C7×ΓL(1,𝔽16)
FittingC3×C4×C5×C7C4
socleC2×C3×C5×C7C2×C4
prime radicalC2×C3×C7×D5C4
cosocleC3×Dic5C22×C7
hypercenterC4×C7ΓL(1,𝔽16)
hypocenterC3×C5C7×M16
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C7 ◁ C4×C7

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)(8,9,12,10,11)(13,14,15)(16,18,20,19)(17,21,22,23);(9,10,11,12)(14,15)(16,17,19,21,20,22,18,23)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 638);