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Cayley diagram

C32×C22⋊D5 #67164

order360
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC32×C22⋊D5

Splittings

FittingC22×C32×C5 ⋊ C2
HallC32×C5 ⋊ D4
SylowC5 ⋊ C32×D4

Subgroups

subgroupG/H
centerC2×C32C2×D5
commutator, cosocleC2×C5C22×C32
FrattiniC2C2×C32×D5
FittingC22×C32×C5C2
socleC2×C32×C5C22
hypercenterC22×C32D5
hypocenterC5C32×D4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C32 ◁ C22×C32

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#6779, C3×#6780, C3×D3×C22⋊D5, …
maximal normal subgroup ofC3×#6779, C3×#6780, C3×D3×C22⋊D5, …
maximal quotient ofC32×#683, C32×#685, C32×#686, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 94);