Finite Group Databasebeta

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C5×#66636 #13132336

order1680
isoclinismisoclinic class of Dic42
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#66636

Splittings

HallC3×C5×C7 ⋊ C22⋊C4
HallC5×C7 ⋊ #217
HallC3×C7 ⋊ C5×C22⋊C4
HallC3×C5 ⋊ C22⋊Dic7
SylowC7 ⋊ C5×#217
SylowC3 ⋊ C5×C22⋊Dic7

Subgroups

subgroupG/H
centerC22×C5C2×D21
commutatorC2×C3×C7C2×C4×C5
FrattiniC22C2×C5×D21
Fitting, prime radicalC23×C3×C5×C7C2
socleC22×C3×C5×C7C22
cosocleC22×C3×C7C22×C5
hypercenterC23×C5D21
hypocenterC3×C7C5×C22⋊C4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22×C5 ◁ C23×C5

136 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)(6,7,9,11,12,10,8)(13,14,15)(21,23);(7,8)(9,10)(11,12)(14,15)(16,17,18,19)(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, 356);