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Cayley diagram

C5×#104 #3979

order160
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C22×C4×D4 (order 128) · Inn(G) = D4 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#104

Subgroups

subgroupG/H
centerC4×C5D4
commutatorC4C2×C4×C5
Frattini, cosocleC2×C4C22×C5
socleC2×C5C22⋊C4
prime radicalC5×#41C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ G

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD5×#104, #65110, #65111, …
maximal subgroup ofD5×#104, #65110, #65111, …
maximal normal subgroup ofD5×#104, #65110, #65111, …
maximal quotient ofC5×D4⋊C8, C5×Q8⋊C8, C5×#327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 54);