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Cayley diagram

D5×Q32 #65206

order320
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productD5×Q32

Splittings

FittingC5×Q32 ⋊ C2
SylowC5 ⋊ C2×Q32

Subgroups

subgroupG/H
centerC2D5×D8
commutator, cosocleC5×C8C23
FrattiniC8C22×D5
FittingC5×Q32C2
socleC2×C5C2×D8
prime radicalC2×D5D8
hypercenterQ32D5
hypocenterC5C2×Q32
LCSG ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,3,6,14)(2,7,10,24)(4,16,19,32)(5,21,17,13)(8,26,27,31)(9,28,20,23)(11,29,30,15)(12,25,22,18)(33,34,36,37,35);(1,2,6,10)(3,11,14,30)(4,15,19,29)(5,20,17,9)(7,22,24,12)(8,25,27,18)(13,31,21,26)(16,28,32,23);(34,35)(36,37)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10598682, #10598685, D5×SD64, …
maximal normal subgroup of#10598682, #10598685, D5×SD64, …
maximal quotient of#10600000, #10600004, #10600008, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 544);