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C5×#63256 #13062825

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

Aliases

direct productC5×#63256

Splittings

FittingC5×Q8×C17 ⋊ C2
HallC5×C17 ⋊ #41
SylowC17 ⋊ C5×#41

Subgroups

subgroupG/H
centerC2×C5C22×D17
commutator, cosocleC2×C17C23×C5
FrattiniC2C22×C5×D17
FittingC5×Q8×C17C2
socleC2×C5×C17C23
hypercenterC5×Q8D17
hypocenterC17C5×#41
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C5 ◁ C5×Q8

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1360, 193);