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Cayley diagram

C5×#108 #3983

order160
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×C4×AGL(1,ℤ/8) (order 256) · Inn(G) = D4 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#108

Subgroups

subgroupG/H
centerC4×C5D4
commutatorC4C2×C4×C5
Frattini, cosocleC2×C4C22×C5
socleC2×C5C4⋊C4
prime radicalC22×C5Q8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ G

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18,19,20,21);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64713, #64714, #64715, …
maximal subgroup of#64713, #64714, #64715, …
maximal normal subgroup of#64713, #64714, #64715, …
maximal quotient ofC5×#334, C5×#335, C5×#341, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 58);