Finite Group Databasebeta

← Prev (#10623847) Next (#10623849) →

C32×#689 #10623848

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

Aliases

direct productC32×#689

Splittings

HallC32×C5 ⋊ Q16
SylowC5 ⋊ C32×Q16

Subgroups

subgroupG/H
centerC2×C32C22⋊D5
commutator, cosocleC4×C5C22×C32
FrattiniC4C2×C32×D5
FittingC32×C5×Q8C2
socle, prime radicalC2×C32×C5D4
hypercenterC32×Q8D5
hypocenterC5C32×Q16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×Q8

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3)(7,8,10,11,9)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(4,5,6)(8,9)(10,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×#689, C3×#91510, C3×#91511, …
maximal normal subgroup ofC3×D3×#689, C3×#91510, C3×#91511, …
maximal quotient ofC32×#3939, C32×#3942, C32×#3967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 317);