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C5×#6467 #11765062

order1120
isoclinismisoclinic class of #1132
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#6467

Splittings

FittingC2×C5×C7×C8 ⋊ C2
HallC5×C7 ⋊ #135
SylowC7 ⋊ C5×#135

Subgroups

subgroupG/H
centerC4×C5D28
commutator, cosocleC4×C7C23×C5
FrattiniC4C22×C5×D7
FittingC2×C5×C7×C8C2
socleC2×C5×C7C2×D4
hypercenterC2×C5×C8D7
hypocenterC7C5×#135
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×C8

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

Conjugacy classes & character table

310 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 668);