Finite Group Databasebeta

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Cayley diagram

C5×#131 #4117

order160
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = C22 · Out(G) = C22×C4×D3
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#131

Subgroups

subgroupG/H
centerC5×C8C22
commutatorC2C22×C4×C5
Frattini, cosocleC4C23×C5
socleC2×C5C22×C4
prime radicalC5×#41C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C5×C8 ◁ 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10613745⋊C3, #10613746⋊C3, #10613744⋊C3, …
Fitting subgroup of#64781, #65429, #65430, …
maximal subgroup of#64781, C5×#350, #65429, …
maximal normal subgroup of#64781, C5×#350, #65429, …
maximal quotient ofC5×#405, C5×#408, C5×#424, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 192);