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Cayley diagram

C5×#97 #3972

order160
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C4×#2021 (order 512) · Inn(G) = C22 · Out(G) = C4×#120
homologyAbelianization (H1) = C42×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#97

Subgroups

subgroupG/H
centerC2×C4×C5C22
commutatorC2C42×C5
Frattini, cosocleC2×C4C22×C5
socle, prime radicalC22×C5C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ G

40 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64684, #64685, #64686, …
maximal subgroup of#64684, #64685, #64686, …
maximal normal subgroup of#64684, #64685, #64686, …
maximal quotient ofC5×#322, C5×#336, C5×#347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 47);