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C32×#676 #10623835

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

Aliases

direct productC32×#676

Splittings

FittingC32×C5×C8 ⋊ C2
HallC32×C5 ⋊ M16
SylowC5 ⋊ C32×M16

Subgroups

subgroupG/H
centerC32×C4C2×D5
commutatorC2×C5C2×C32×C4
FrattiniC4C2×C32×D5
FittingC32×C5×C8C2
socleC2×C32×C5C2×C4
prime radicalC2×C32×D5C4
cosocleC4×C5C22×C32
hypercenterC32×C8D5
hypocenterC5C32×M16
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C32×C4 ◁ C32×C8

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

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×#676, C3×#91246, C3×#91249, …
maximal normal subgroup ofC3×D3×#676, C3×#91246, C3×#91249, …
maximal quotient ofC32×#3946, C32×#3947, C32×#3952, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 304);