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C5×#3676 #10623595

order720
isoclinismisoclinic class of Dic18
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#3676

Splittings

HallC5×C9 ⋊ C22⋊C4
SylowC9 ⋊ C5×C22⋊C4

Subgroups

subgroupG/H
centerC22×C5C2×D9
commutatorC2×C9C2×C4×C5
FrattiniC22×C3C2×C5×D3
FittingC23×C5×C9C2
socleC22×C3×C5C2×D3
prime radicalC23×C3×C5D3
cosocleC22×C9C22×C5
hypercenterC23×C5D9
hypocenterC9C5×C22⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C5 ◁ C23×C5

102 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)(20,22);(7,9)(8,12)(10,11)(13,14)(15,16,17,18)(19,20)(21,22)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083754, #13083756, C5×#63365, …
maximal normal subgroup of#13083754, #13083756, C5×#63365, …
maximal quotient ofC5×#63389, C5×C9⋊#95, C5×#63391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 64);