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Cayley diagram

C5×AGL(1,ℤ/8) #4122

order160
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) = C4×D42 (order 256) · Inn(G) = C2×D4 · Out(G) = C22×C4
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC2×C5C2×D4
commutator, Frattini, cosocleC4C23×C5
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ G

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#65487, #65488, #65489, …
maximal subgroup of#65487, #65488, #65489, …
maximal normal subgroup of#65487, #65488, #65489, …
maximal quotient ofC5×#417, C5×#418, C5×#428, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 197);