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C5×#6554 #11765433

order1120
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C25

Aliases

direct productC5×#6554

Splittings

FittingC5×C7×#41 ⋊ C2
HallC5×C7 ⋊ 2−1+4
SylowC7 ⋊ C5×2−1+4

Subgroups

subgroupG/H
centerC2×C5C23×D7
commutator, cosocleC2×C7C24×C5
FrattiniC2C23×C5×D7
FittingC5×C7×#41C2
socleC2×C5×C7C24
hypercenterC5×#41D7
hypocenterC7C5×2−1+4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C5×#41

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

235 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 1039);