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C5×#6439 #11765034

order1120
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#6439

Splittings

FittingC42×C5×C7 ⋊ C2
HallC5×C7 ⋊ #126
SylowC7 ⋊ C5×#126

Subgroups

subgroupG/H
centerC22×C5C22×D7
commutator, cosocleC22×C7C23×C5
FrattiniC22C22×C5×D7
FittingC42×C5×C7C2
socleC22×C5×C7C23
prime radicalC22×C5×D7C22
hypercenterC42×C5D7
hypocenterC7C5×#126
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C5 ◁ C42×C5

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 560

A minimal faithful permutation representation — 3 generators on 28 points:

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

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, 640);