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C5×#6551 #11765430

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

Aliases

direct productC5×#6551

Splittings

FittingC2×C5×C7×Q8 ⋊ C2
HallC5×C7 ⋊ 2−1+4
SylowC7 ⋊ C5×2−1+4

Subgroups

subgroupG/H
centerC2×C5C23×D7
commutator, cosocleC2×C7C24×C5
FrattiniC2C23×C5×D7
FittingC2×C5×C7×Q8C2
socleC2×C5×C7C24
hypercenterC2×C5×Q8D7
hypocenterC7C5×2−1+4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C2×C5×Q8

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

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, 1036);