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Cayley diagram

C5×M32 #3985

order160
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C4×#41 (order 128) · Inn(G) = C22 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C1

Aliases

direct productC5×M32

Subgroups

subgroupG/H
centerC5×C8C22
commutatorC2C2×C5×C8
Frattini, cosocleC8C22×C5
socleC2×C5C2×C8
prime radicalC22×C5C8
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C5×C8 ◁ G

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21);(7,15)(9,17)(11,19)(13,21)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64731, #64732, #64733, …
maximal subgroup of#64731, #64732, #64733, …
maximal normal subgroup of#64731, #64732, #64733, …
maximal quotient ofC5×#346, C5×C22⋊C16, C5×C4⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 60);