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C5×#6470 #11765065

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

Aliases

direct productC5×#6470

Splittings

FittingC5×C7×M16 ⋊ C2
HallC5×C7 ⋊ #131
SylowC7 ⋊ C5×#131

Subgroups

subgroupG/H
centerC4×C5C22×D7
commutatorC2×C7C22×C4×C5
FrattiniC4C22×C5×D7
FittingC5×C7×M16C2
socleC2×C5×C7C22×C4
prime radicalC5×#1157C2
cosocleC4×C7C23×C5
hypercenterC5×M16D7
hypocenterC7C5×#131
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C4×C5 ◁ C5×M16

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 671);