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C32×#683 #10623842

order720
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC32×#683

Splittings

HallC32×C5 ⋊ C4⋊C4
SylowC5 ⋊ C32×C4⋊C4

Subgroups

subgroupG/H
centerC22×C32C2×D5
commutatorC2×C5C2×C32×C4
FrattiniC22C2×C32×D5
FittingC2×C32×C4×C5C2
socle, prime radicalC22×C32×C5C22
cosocleC22×C5C22×C32
hypercenterC2×C32×C4D5
hypocenterC5C32×C4⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C32 ◁ C2×C32×C4

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 720

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

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

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#91334, C3×#91335, C3×#91336, …
maximal normal subgroup ofC3×#91334, C3×#91335, C3×#91336, …
maximal quotient ofC32×#3939, C32×#3940, C32×#3946, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 311);