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C5×#6475 #11765070

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

Aliases

direct productC5×#6475

Splittings

FittingC5×C7×D8 ⋊ C2
HallC5×C7 ⋊ #135
SylowC7 ⋊ C5×#135

Subgroups

subgroupG/H
centerC2×C5D4×D7
commutator, cosocleC4×C7C23×C5
FrattiniC4C22×C5×D7
FittingC5×C7×D8C2
socleC2×C5×C7C2×D4
hypercenterC5×D8D7
hypocenterC7C5×#135
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×D8

124 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)(14,19)(15,22)(16,25)(18,26)(21,28)(23,24);(13,15)(14,18)(16,22)(17,24)(19,21)(20,28)(23,25)(26,27);(7,8)(9,10)(11,12)(13,14)(15,21)(16,20)(17,19)(18,24)(22,26)(23,28)(25,27)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 676);