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Cayley diagram

C5×Q32 #3988

order160
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C4×AGL(1,ℤ/16) (order 512) · Inn(G) = D8 · Out(G) = C2×C42
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×Q32

Subgroups

subgroupG/H
center, socle, prime radicalC2×C5D8
commutator, Frattini, cosocleC8C22×C5
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8 ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64741, #64742, D5×Q32, …
maximal subgroup of#64741, #64742, C5×SD64, …
maximal normal subgroup of#64741, #64742, C5×SD64, …
maximal quotient ofC5×#358, C5×#366, C2×C5×Q32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 63);