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C5×Dic36 #10623580

order720
isoclinismisoclinic class of Dic36
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×Dic36

Splittings

HallC5×C9 ⋊ Q16
SylowC9 ⋊ C5×Q16

Subgroups

subgroupG/H
centerC2×C5D36
commutator, cosocleC4×C9C22×C5
FrattiniC3×C4C2×C5×D3
FittingC5×C8×C9C2
socle, prime radicalC2×C3×C5D12
hypercenterC5×C8D9
hypocenterC9C5×Q16
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083696, #13083697, #13083702, …
maximal normal subgroup of#13083696, #13083697, #13083702, …
maximal quotient ofC5×#63378, C5×#63376, C2×C5×Dic36

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 49);