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C5×#3667 #10623586

order720
isoclinismisoclinic class of Dic18
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#3667

Splittings

HallC5×C9 ⋊ M16
SylowC9 ⋊ C5×M16

Subgroups

subgroupG/H
centerC4×C5C2×D9
commutatorC2×C9C2×C4×C5
FrattiniC3×C4C2×C5×D3
FittingC2×C4×C5×C9C2
socleC2×C3×C5C2×Dic3
prime radicalC22×C3×C5Dic3
cosocleC4×C9C22×C5
hypercenterC2×C4×C5D9
hypocenterC9C5×M16
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C4×C5 ◁ C2×C4×C5

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 360

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4,5)(6,11,13,12,7,9,8,14,10)(16,19)(20,22);(7,9)(8,12)(10,11)(13,14)(15,16,17,20,18,19,21,22)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083712, #13083731, #13083736, …
maximal normal subgroup of#13083712, #13083731, #13083736, …
maximal quotient ofC5×#63362, C5×C4⋊(C9⋊C8), C5×#63389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 55);