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Cayley diagram

C5×#106 #3981

order160
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C4×#3003 (order 512) · Inn(G) = D4 · Out(G) = C2×C4×D4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#106

Subgroups

subgroupG/H
center, socle, prime radicalC22×C5D4
commutatorC4C2×C4×C5
Frattini, cosocleC2×C4C22×C5
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C4×C5 ◁ G

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64707, #64708, #65141, …
maximal subgroup of#64707, #64708, C5×#360, …
maximal normal subgroup of#64707, #64708, C5×#360, …
maximal quotient ofC5×#334, C5×#340, C5×#365, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 56);