Finite Group Databasebeta

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C5×#6437 #11765032

order1120
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22×C4

Aliases

direct productC5×#6437

Splittings

FittingC42×C5×C7 ⋊ C2
HallC5×C7 ⋊ #127
SylowC7 ⋊ C5×#127

Subgroups

subgroupG/H
centerC22×C5C22×D7
commutator, cosocleC22×C7C23×C5
FrattiniC22C22×C5×D7
FittingC42×C5×C7C2
socleC22×C5×C7C23
hypercenterC42×C5D7
hypocenterC7C5×#127
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C5 ◁ C42×C5

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 560

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 638);