Finite Group Databasebeta

← Prev (#11765105) Next (#11765107) →

C5×#6511 #11765106

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

Aliases

direct productC5×#6511

Splittings

HallC5×C7 ⋊ #131
SylowC7 ⋊ C5×#131

Subgroups

subgroupG/H
centerC4×C5C22×D7
commutatorC2×C7C22×C4×C5
FrattiniC4C22×C5×D7
Fitting, prime radicalC5×C7×#41C2
socleC2×C5×C7C22×C4
cosocleC4×C7C23×C5
hypercenterC5×#41D7
hypocenterC7C5×#131
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C4×C5 ◁ C5×#41

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

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, 712);