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## G = D5×D11order 220 = 22·5·11

### Direct product of D5 and D11

Aliases: D5×D11, D55⋊C2, C51D22, C55⋊C22, C111D10, (D5×C11)⋊C2, (C5×D11)⋊C2, SmallGroup(220,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — D5×D11
 Chief series C1 — C11 — C55 — C5×D11 — D5×D11
 Lower central C55 — D5×D11
 Upper central C1

Generators and relations for D5×D11
G = < a,b,c,d | a5=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
11C2
55C2
55C22
11C10
11D5
5C22
5D11
11D10
5D22

Character table of D5×D11

 class 1 2A 2B 2C 5A 5B 10A 10B 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E 55A 55B 55C 55D 55E 55F 55G 55H 55I 55J size 1 5 11 55 2 2 22 22 2 2 2 2 2 10 10 10 10 10 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 0 2 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 2 2 2 2 0 0 0 0 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ6 2 0 2 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 2 2 2 2 0 0 0 0 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ7 2 0 -2 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 2 2 2 2 2 0 0 0 0 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D10 ρ8 2 0 -2 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 2 2 2 2 2 0 0 0 0 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D10 ρ9 2 -2 0 0 2 2 0 0 ζ116+ζ115 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 -ζ119-ζ112 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 orthogonal lifted from D22 ρ10 2 -2 0 0 2 2 0 0 ζ117+ζ114 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 -ζ116-ζ115 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 orthogonal lifted from D22 ρ11 2 2 0 0 2 2 0 0 ζ118+ζ113 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 orthogonal lifted from D11 ρ12 2 2 0 0 2 2 0 0 ζ119+ζ112 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 orthogonal lifted from D11 ρ13 2 -2 0 0 2 2 0 0 ζ1110+ζ11 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 -ζ117-ζ114 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 orthogonal lifted from D22 ρ14 2 2 0 0 2 2 0 0 ζ116+ζ115 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 orthogonal lifted from D11 ρ15 2 -2 0 0 2 2 0 0 ζ118+ζ113 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 -ζ1110-ζ11 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 orthogonal lifted from D22 ρ16 2 2 0 0 2 2 0 0 ζ117+ζ114 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 orthogonal lifted from D11 ρ17 2 2 0 0 2 2 0 0 ζ1110+ζ11 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 orthogonal lifted from D11 ρ18 2 -2 0 0 2 2 0 0 ζ119+ζ112 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 -ζ118-ζ113 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 orthogonal lifted from D22 ρ19 4 0 0 0 -1+√5 -1-√5 0 0 2ζ119+2ζ112 2ζ1110+2ζ11 2ζ118+2ζ113 2ζ116+2ζ115 2ζ117+2ζ114 0 0 0 0 0 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 orthogonal faithful ρ20 4 0 0 0 -1+√5 -1-√5 0 0 2ζ117+2ζ114 2ζ119+2ζ112 2ζ116+2ζ115 2ζ1110+2ζ11 2ζ118+2ζ113 0 0 0 0 0 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 orthogonal faithful ρ21 4 0 0 0 -1-√5 -1+√5 0 0 2ζ118+2ζ113 2ζ117+2ζ114 2ζ1110+2ζ11 2ζ119+2ζ112 2ζ116+2ζ115 0 0 0 0 0 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 orthogonal faithful ρ22 4 0 0 0 -1-√5 -1+√5 0 0 2ζ1110+2ζ11 2ζ116+2ζ115 2ζ117+2ζ114 2ζ118+2ζ113 2ζ119+2ζ112 0 0 0 0 0 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 orthogonal faithful ρ23 4 0 0 0 -1+√5 -1-√5 0 0 2ζ1110+2ζ11 2ζ116+2ζ115 2ζ117+2ζ114 2ζ118+2ζ113 2ζ119+2ζ112 0 0 0 0 0 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 orthogonal faithful ρ24 4 0 0 0 -1-√5 -1+√5 0 0 2ζ117+2ζ114 2ζ119+2ζ112 2ζ116+2ζ115 2ζ1110+2ζ11 2ζ118+2ζ113 0 0 0 0 0 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 orthogonal faithful ρ25 4 0 0 0 -1-√5 -1+√5 0 0 2ζ119+2ζ112 2ζ1110+2ζ11 2ζ118+2ζ113 2ζ116+2ζ115 2ζ117+2ζ114 0 0 0 0 0 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 orthogonal faithful ρ26 4 0 0 0 -1-√5 -1+√5 0 0 2ζ116+2ζ115 2ζ118+2ζ113 2ζ119+2ζ112 2ζ117+2ζ114 2ζ1110+2ζ11 0 0 0 0 0 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 orthogonal faithful ρ27 4 0 0 0 -1+√5 -1-√5 0 0 2ζ118+2ζ113 2ζ117+2ζ114 2ζ1110+2ζ11 2ζ119+2ζ112 2ζ116+2ζ115 0 0 0 0 0 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 orthogonal faithful ρ28 4 0 0 0 -1+√5 -1-√5 0 0 2ζ116+2ζ115 2ζ118+2ζ113 2ζ119+2ζ112 2ζ117+2ζ114 2ζ1110+2ζ11 0 0 0 0 0 ζ53ζ117+ζ53ζ114+ζ52ζ117+ζ52ζ114 ζ53ζ118+ζ53ζ113+ζ52ζ118+ζ52ζ113 ζ53ζ1110+ζ53ζ11+ζ52ζ1110+ζ52ζ11 ζ53ζ116+ζ53ζ115+ζ52ζ116+ζ52ζ115 ζ53ζ119+ζ53ζ112+ζ52ζ119+ζ52ζ112 ζ54ζ118+ζ54ζ113+ζ5ζ118+ζ5ζ113 ζ54ζ1110+ζ54ζ11+ζ5ζ1110+ζ5ζ11 ζ54ζ116+ζ54ζ115+ζ5ζ116+ζ5ζ115 ζ54ζ119+ζ54ζ112+ζ5ζ119+ζ5ζ112 ζ54ζ117+ζ54ζ114+ζ5ζ117+ζ5ζ114 orthogonal faithful

Smallest permutation representation of D5×D11
On 55 points
Generators in S55
(1 54 43 32 21)(2 55 44 33 22)(3 45 34 23 12)(4 46 35 24 13)(5 47 36 25 14)(6 48 37 26 15)(7 49 38 27 16)(8 50 39 28 17)(9 51 40 29 18)(10 52 41 30 19)(11 53 42 31 20)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)(23 24 25 26 27 28 29 30 31 32 33)(34 35 36 37 38 39 40 41 42 43 44)(45 46 47 48 49 50 51 52 53 54 55)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)

G:=sub<Sym(55)| (1,54,43,32,21)(2,55,44,33,22)(3,45,34,23,12)(4,46,35,24,13)(5,47,36,25,14)(6,48,37,26,15)(7,49,38,27,16)(8,50,39,28,17)(9,51,40,29,18)(10,52,41,30,19)(11,53,42,31,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)>;

G:=Group( (1,54,43,32,21)(2,55,44,33,22)(3,45,34,23,12)(4,46,35,24,13)(5,47,36,25,14)(6,48,37,26,15)(7,49,38,27,16)(8,50,39,28,17)(9,51,40,29,18)(10,52,41,30,19)(11,53,42,31,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54) );

G=PermutationGroup([[(1,54,43,32,21),(2,55,44,33,22),(3,45,34,23,12),(4,46,35,24,13),(5,47,36,25,14),(6,48,37,26,15),(7,49,38,27,16),(8,50,39,28,17),(9,51,40,29,18),(10,52,41,30,19),(11,53,42,31,20)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55)], [(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33),(34,35,36,37,38,39,40,41,42,43,44),(45,46,47,48,49,50,51,52,53,54,55)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54)]])

D5×D11 is a maximal quotient of   D552C4  C55⋊D4  C5⋊D44  C11⋊D20  C55⋊Q8

Matrix representation of D5×D11 in GL4(𝔽331) generated by

 1 0 0 0 0 1 0 0 0 0 330 1 0 0 213 117
,
 330 0 0 0 0 330 0 0 0 0 330 0 0 0 213 1
,
 0 1 0 0 330 159 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(331))| [1,0,0,0,0,1,0,0,0,0,330,213,0,0,1,117],[330,0,0,0,0,330,0,0,0,0,330,213,0,0,0,1],[0,330,0,0,1,159,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

D5×D11 in GAP, Magma, Sage, TeX

D_5\times D_{11}
% in TeX

G:=Group("D5xD11");
// GroupNames label

G:=SmallGroup(220,11);
// by ID

G=gap.SmallGroup(220,11);
# by ID

G:=PCGroup([4,-2,-2,-5,-11,102,3203]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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