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C32×#709 #10624243

order720
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3

Aliases

direct productC32×#709

Splittings

FittingC2×C32×C4×C5 ⋊ C2
HallC32×C5 ⋊ #41
SylowC5 ⋊ C32×#41

Subgroups

subgroupG/H
centerC32×C4C2×D5
commutator, cosocleC2×C5C23×C32
FrattiniC2C22×C32×D5
FittingC2×C32×C4×C5C2
socleC2×C32×C5C23
hypercenterC2×C32×C4D5
hypocenterC5C32×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C32×C4 ◁ C2×C32×C4

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#90976, C3×#90977, #13084744, …
maximal normal subgroup ofC3×#90976, C3×#90977, #13084744, …
maximal quotient ofC32×C4×Dic10, C32×#4016, C32×#4018, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 712);