Finite Group Databasebeta

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Cayley diagram

C5×#135 #4121

order160
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×C4×AGL(1,ℤ/8) (order 256) · Inn(G) = D4 · Out(G) = C23×C4
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#135

Subgroups

subgroupG/H
centerC4×C5D4
commutator, Frattini, cosocleC4C23×C5
socleC2×C5C2×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ G

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64786, #64787, #65482, …
maximal subgroup of#64786, #64787, C5×#359, …
maximal normal subgroup of#64786, #64787, C5×#359, …
maximal quotient ofC5×#416, C5×#427, C4×C5×D8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 196);