Finite Group Databasebeta

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C5×#66629 #13132329

order1680
isoclinismisoclinic class of Dic42
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#66629

Splittings

HallC3×C5×C7 ⋊ C4⋊C4
HallC5×C7 ⋊ Dic3⋊C4
HallC3×C7 ⋊ C5×C4⋊C4
HallC3×C5 ⋊ Dic7⋊C4
SylowC7 ⋊ C5×Dic3⋊C4
SylowC3 ⋊ C5×Dic7⋊C4

Subgroups

subgroupG/H
centerC22×C5C2×D21
commutatorC2×C3×C7C2×C4×C5
FrattiniC22C2×C5×D21
FittingC2×C3×C4×C5×C7C2
socle, prime radicalC22×C3×C5×C7C22
cosocleC22×C3×C7C22×C5
hypercenterC2×C4×C5D21
hypocenterC3×C7C5×C4⋊C4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22×C5 ◁ C2×C4×C5

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 1680

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 349);