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C5×D72 #10623584

order720
isoclinismisoclinic class of Dic36
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D72

Splittings

FittingC5×C8×C9 ⋊ C2
HallC5×C9 ⋊ D8
SylowC9 ⋊ C5×D8

Subgroups

subgroupG/H
centerC2×C5D36
commutator, cosocleC4×C9C22×C5
FrattiniC3×C4C2×C5×D3
FittingC5×C8×C9C2
socleC2×C3×C5D12
hypercenterC5×C8D9
hypocenterC9C5×D8
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8

66 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)(6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22);(7,13)(8,12)(9,11)(15,22)(16,21)(17,20)(18,19)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083694, #13083695, #13083700, …
maximal normal subgroup of#13083694, #13083695, #13083700, …
maximal quotient ofC5×#63359, C5×D144, C5×Dic72, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 53);