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C5×#6403 #11764500

order1120
isoclinismisoclinic class of #6400
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#6403

Splittings

HallC5×C7 ⋊ Q32
SylowC7 ⋊ C5×Q32

Subgroups

subgroupG/H
centerC2×C5#1141
commutator, cosocleC7×C8C22×C5
FrattiniC8C2×C5×D7
FittingC5×C7×Q16C2
socle, prime radicalC2×C5×C7D8
hypercenterC5×Q16D7
hypocenterC7C5×Q32
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8 ◁ C5×Q16

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 44
transitive degreeμt(G) = 1120

A minimal faithful permutation representation — 2 generators on 44 points:

(1,2,3,4,5)(6,7,9,11,12,10,8)(13,15,18,26)(14,19,22,36)(16,28,31,44)(17,33,29,25)(20,38,39,43)(21,40,32,35)(23,41,42,27)(24,37,34,30);(7,8)(9,10)(11,12)(13,14,18,22)(15,23,26,42)(16,27,31,41)(17,32,29,21)(19,34,36,24)(20,37,39,30)(25,43,33,38)(28,40,44,35)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 106);