Finite Group Databasebeta

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C5×#63252 #13062821

order1360
isoclinismisoclinic class of Dic34
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#63252

Splittings

FittingC2×C4×C5×C17 ⋊ C2
HallC5×C17 ⋊ #41
SylowC17 ⋊ C5×#41

Subgroups

subgroupG/H
centerC4×C5C2×D17
commutator, cosocleC2×C17C23×C5
FrattiniC2C22×C5×D17
FittingC2×C4×C5×C17C2
socleC2×C5×C17C23
hypercenterC2×C4×C5D17
hypocenterC17C5×#41
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C4×C5 ◁ C2×C4×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 680

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3,4,5)(6,7,9,11,13,15,17,19,21,22,20,18,16,14,12,10,8)(24,28)(27,30);(23,25,26,29)(24,27,28,30);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28)(29,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1360, 189);