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Cayley diagram

C5×#102 #3977

order160
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C23×C4×D4 (order 256) · Inn(G) = D4 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#102

Subgroups

subgroupG/H
center, socleC22×C5D4
commutatorC4C2×C4×C5
Frattini, cosocleC2×C4C22×C5
prime radicalC2×C5×D4C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C4×C5 ◁ G

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 80

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#65045, #65046, #65047, …
maximal subgroup of#65045, #65046, #65047, …
maximal normal subgroup of#65045, #65046, #65047, …
maximal quotient ofC5×D4⋊C8, C5×#327, C5×#331, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 52);