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## G = D5×C13⋊C3order 390 = 2·3·5·13

### Direct product of D5 and C13⋊C3

Aliases: D5×C13⋊C3, C653C6, (D5×C13)⋊C3, C132(C3×D5), C5⋊(C2×C13⋊C3), (C5×C13⋊C3)⋊3C2, SmallGroup(390,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C65 — D5×C13⋊C3
 Chief series C1 — C13 — C65 — C5×C13⋊C3 — D5×C13⋊C3
 Lower central C65 — D5×C13⋊C3
 Upper central C1

Generators and relations for D5×C13⋊C3
G = < a,b,c,d | a5=b2=c13=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

Character table of D5×C13⋊C3

 class 1 2 3A 3B 5A 5B 6A 6B 13A 13B 13C 13D 15A 15B 15C 15D 26A 26B 26C 26D 65A 65B 65C 65D 65E 65F 65G 65H size 1 5 13 13 2 2 65 65 3 3 3 3 26 26 26 26 15 15 15 15 6 6 6 6 6 6 6 6 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 ζ32 ζ3 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 1 1 ζ3 ζ32 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 -1 ζ32 ζ3 1 1 ζ65 ζ6 1 1 1 1 ζ3 ζ32 ζ3 ζ32 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ6 1 -1 ζ3 ζ32 1 1 ζ6 ζ65 1 1 1 1 ζ32 ζ3 ζ32 ζ3 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ7 2 0 2 2 -1-√5/2 -1+√5/2 0 0 2 2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 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 orthogonal lifted from D5 ρ8 2 0 2 2 -1+√5/2 -1-√5/2 0 0 2 2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 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 orthogonal lifted from D5 ρ9 2 0 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 0 0 2 2 2 2 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 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 complex lifted from C3×D5 ρ10 2 0 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 0 0 2 2 2 2 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 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 complex lifted from C3×D5 ρ11 2 0 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 0 0 2 2 2 2 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 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 complex lifted from C3×D5 ρ12 2 0 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 0 0 2 2 2 2 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 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 complex lifted from C3×D5 ρ13 3 3 0 0 3 3 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 0 0 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 complex lifted from C13⋊C3 ρ14 3 3 0 0 3 3 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 0 0 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 complex lifted from C13⋊C3 ρ15 3 -3 0 0 3 3 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 0 0 0 0 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 complex lifted from C2×C13⋊C3 ρ16 3 -3 0 0 3 3 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 0 0 0 0 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 complex lifted from C2×C13⋊C3 ρ17 3 3 0 0 3 3 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 0 0 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 complex lifted from C13⋊C3 ρ18 3 -3 0 0 3 3 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 0 0 0 0 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 complex lifted from C2×C13⋊C3 ρ19 3 3 0 0 3 3 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 0 0 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 complex lifted from C13⋊C3 ρ20 3 -3 0 0 3 3 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 0 0 0 0 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 complex lifted from C2×C13⋊C3 ρ21 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 0 0 0 0 0 0 0 0 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 complex faithful ρ22 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 0 0 0 0 0 0 0 0 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 complex faithful ρ23 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 0 0 0 0 0 0 0 0 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 complex faithful ρ24 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 0 0 0 0 0 0 0 0 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 complex faithful ρ25 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 0 0 0 0 0 0 0 0 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 complex faithful ρ26 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 0 0 0 0 0 0 0 0 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 complex faithful ρ27 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 0 0 0 0 0 0 0 0 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 complex faithful ρ28 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 0 0 0 0 0 0 0 0 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ1311+ζ5ζ138+ζ5ζ137 complex faithful

Smallest permutation representation of D5×C13⋊C3
On 65 points
Generators in S65
(1 53 40 27 14)(2 54 41 28 15)(3 55 42 29 16)(4 56 43 30 17)(5 57 44 31 18)(6 58 45 32 19)(7 59 46 33 20)(8 60 47 34 21)(9 61 48 35 22)(10 62 49 36 23)(11 63 50 37 24)(12 64 51 38 25)(13 65 52 39 26)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)
(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 56 57 58 59 60 61 62 63 64 65)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)

G:=sub<Sym(65)| (1,53,40,27,14)(2,54,41,28,15)(3,55,42,29,16)(4,56,43,30,17)(5,57,44,31,18)(6,58,45,32,19)(7,59,46,33,20)(8,60,47,34,21)(9,61,48,35,22)(10,62,49,36,23)(11,63,50,37,24)(12,64,51,38,25)(13,65,52,39,26), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65), (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)(53,54,55,56,57,58,59,60,61,62,63,64,65), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)>;

G:=Group( (1,53,40,27,14)(2,54,41,28,15)(3,55,42,29,16)(4,56,43,30,17)(5,57,44,31,18)(6,58,45,32,19)(7,59,46,33,20)(8,60,47,34,21)(9,61,48,35,22)(10,62,49,36,23)(11,63,50,37,24)(12,64,51,38,25)(13,65,52,39,26), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65), (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)(53,54,55,56,57,58,59,60,61,62,63,64,65), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64) );

G=PermutationGroup([[(1,53,40,27,14),(2,54,41,28,15),(3,55,42,29,16),(4,56,43,30,17),(5,57,44,31,18),(6,58,45,32,19),(7,59,46,33,20),(8,60,47,34,21),(9,61,48,35,22),(10,62,49,36,23),(11,63,50,37,24),(12,64,51,38,25),(13,65,52,39,26)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65)], [(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),(53,54,55,56,57,58,59,60,61,62,63,64,65)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51),(54,56,62),(55,59,58),(57,65,63),(60,61,64)]])

Matrix representation of D5×C13⋊C3 in GL5(𝔽1171)

 1 247 0 0 0 1015 112 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 247 0 0 0 0 1170 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 762 1006 0 0 1 0 339 0 0 0 1 20
,
 750 0 0 0 0 0 750 0 0 0 0 0 520 482 890 0 0 256 876 1072 0 0 514 630 946

G:=sub<GL(5,GF(1171))| [1,1015,0,0,0,247,112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,247,1170,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,72,1,0,0,0,762,0,1,0,0,1006,339,20],[750,0,0,0,0,0,750,0,0,0,0,0,520,256,514,0,0,482,876,630,0,0,890,1072,946] >;

D5×C13⋊C3 in GAP, Magma, Sage, TeX

D_5\times C_{13}\rtimes C_3
% in TeX

G:=Group("D5xC13:C3");
// GroupNames label

G:=SmallGroup(390,2);
// by ID

G=gap.SmallGroup(390,2);
# by ID

G:=PCGroup([4,-2,-3,-5,-13,290,727]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^13=d^3=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^-1=c^9>;
// generators/relations

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