Finite Group Databasebeta

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C32×D40 #10623837

order720
isoclinismisoclinic class of #677
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC32×D40

Splittings

FittingC32×C5×C8 ⋊ C2
HallC32×C5 ⋊ D8
SylowC5 ⋊ C32×D8

Subgroups

subgroupG/H
centerC2×C32D20
commutator, cosocleC4×C5C22×C32
FrattiniC4C2×C32×D5
FittingC32×C5×C8C2
socleC2×C32×C5D4
hypercenterC32×C8D5
hypocenterC5C32×D8
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×C8

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 360

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#90937, C3×#90938, C3×#90940, …
maximal normal subgroup ofC3×#90937, C3×#90938, C3×#90940, …
maximal quotient ofC32×#3932, C32×D80, C32×Dic40, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 306);