Finite Group Databasebeta

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C32×#688 #10623847

order720
isoclinismisoclinic class of #686
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C3

Aliases

direct productC32×#688

Splittings

FittingC32×C5×Q8 ⋊ C2
HallC32×C5 ⋊ SD16
SylowC5 ⋊ C32×SD16

Subgroups

subgroupG/H
centerC2×C32C22⋊D5
commutator, cosocleC4×C5C22×C32
FrattiniC4C2×C32×D5
FittingC32×C5×Q8C2
socleC2×C32×C5D4
prime radicalC32×D20C2
hypercenterC32×Q8D5
hypocenterC5C32×SD16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×Q8

120 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)(7,8,10,11,9)(12,13,15,17)(14,19,18,16);(4,5,6)(8,9)(10,11)(13,16)(14,18)(17,19)

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×#688, C3×#91504, C3×#91505, …
maximal normal subgroup ofC3×D3×#688, C3×#91504, C3×#91505, …
maximal quotient ofC32×#3940, C32×#3941, C32×#3967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 316);