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## G = D52order 104 = 23·13

### Dihedral group

Aliases: D52, C4⋊D13, C131D4, C521C2, D261C2, C2.4D26, C26.3C22, sometimes denoted D104 or Dih52 or Dih104, SmallGroup(104,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D52
 Chief series C1 — C13 — C26 — D26 — D52
 Lower central C13 — C26 — D52
 Upper central C1 — C2 — C4

Generators and relations for D52
G = < a,b | a52=b2=1, bab=a-1 >

26C2
26C2
13C22
13C22
2D13
2D13
13D4

Character table of D52

 class 1 2A 2B 2C 4 13A 13B 13C 13D 13E 13F 26A 26B 26C 26D 26E 26F 52A 52B 52C 52D 52E 52F 52G 52H 52I 52J 52K 52L size 1 1 26 26 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 -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 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 -1 linear of order 2 ρ5 2 -2 0 0 0 2 2 2 2 2 2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 0 2 ζ1311+ζ132 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ138+ζ135 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ1310+ζ133 ζ139+ζ134 ζ1311+ζ132 ζ138+ζ135 ζ1312+ζ13 ζ137+ζ136 ζ137+ζ136 ζ1312+ζ13 orthogonal lifted from D13 ρ7 2 2 0 0 -2 ζ1312+ζ13 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 -ζ139-ζ134 -ζ1312-ζ13 -ζ1311-ζ132 -ζ138-ζ135 -ζ138-ζ135 -ζ1311-ζ132 -ζ1312-ζ13 -ζ139-ζ134 -ζ137-ζ136 -ζ1310-ζ133 -ζ1310-ζ133 -ζ137-ζ136 orthogonal lifted from D26 ρ8 2 2 0 0 2 ζ138+ζ135 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ137+ζ136 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1312+ζ13 ζ1310+ζ133 ζ138+ζ135 ζ137+ζ136 ζ139+ζ134 ζ1311+ζ132 ζ1311+ζ132 ζ139+ζ134 orthogonal lifted from D13 ρ9 2 2 0 0 -2 ζ1311+ζ132 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 -ζ138-ζ135 -ζ1311-ζ132 -ζ139-ζ134 -ζ1310-ζ133 -ζ1310-ζ133 -ζ139-ζ134 -ζ1311-ζ132 -ζ138-ζ135 -ζ1312-ζ13 -ζ137-ζ136 -ζ137-ζ136 -ζ1312-ζ13 orthogonal lifted from D26 ρ10 2 2 0 0 -2 ζ139+ζ134 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 -ζ1310-ζ133 -ζ139-ζ134 -ζ138-ζ135 -ζ137-ζ136 -ζ137-ζ136 -ζ138-ζ135 -ζ139-ζ134 -ζ1310-ζ133 -ζ1311-ζ132 -ζ1312-ζ13 -ζ1312-ζ13 -ζ1311-ζ132 orthogonal lifted from D26 ρ11 2 2 0 0 2 ζ1312+ζ13 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ139+ζ134 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ138+ζ135 ζ1311+ζ132 ζ1312+ζ13 ζ139+ζ134 ζ137+ζ136 ζ1310+ζ133 ζ1310+ζ133 ζ137+ζ136 orthogonal lifted from D13 ρ12 2 2 0 0 2 ζ1310+ζ133 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1312+ζ13 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ1311+ζ132 ζ137+ζ136 ζ1310+ζ133 ζ1312+ζ13 ζ138+ζ135 ζ139+ζ134 ζ139+ζ134 ζ138+ζ135 orthogonal lifted from D13 ρ13 2 2 0 0 -2 ζ138+ζ135 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 -ζ137-ζ136 -ζ138-ζ135 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1312-ζ13 -ζ1310-ζ133 -ζ138-ζ135 -ζ137-ζ136 -ζ139-ζ134 -ζ1311-ζ132 -ζ1311-ζ132 -ζ139-ζ134 orthogonal lifted from D26 ρ14 2 2 0 0 2 ζ137+ζ136 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ1311+ζ132 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ139+ζ134 ζ1312+ζ13 ζ137+ζ136 ζ1311+ζ132 ζ1310+ζ133 ζ138+ζ135 ζ138+ζ135 ζ1310+ζ133 orthogonal lifted from D13 ρ15 2 2 0 0 -2 ζ1310+ζ133 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 -ζ1312-ζ13 -ζ1310-ζ133 -ζ137-ζ136 -ζ1311-ζ132 -ζ1311-ζ132 -ζ137-ζ136 -ζ1310-ζ133 -ζ1312-ζ13 -ζ138-ζ135 -ζ139-ζ134 -ζ139-ζ134 -ζ138-ζ135 orthogonal lifted from D26 ρ16 2 2 0 0 -2 ζ137+ζ136 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 -ζ1311-ζ132 -ζ137-ζ136 -ζ1312-ζ13 -ζ139-ζ134 -ζ139-ζ134 -ζ1312-ζ13 -ζ137-ζ136 -ζ1311-ζ132 -ζ1310-ζ133 -ζ138-ζ135 -ζ138-ζ135 -ζ1310-ζ133 orthogonal lifted from D26 ρ17 2 2 0 0 2 ζ139+ζ134 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ1310+ζ133 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ137+ζ136 ζ138+ζ135 ζ139+ζ134 ζ1310+ζ133 ζ1311+ζ132 ζ1312+ζ13 ζ1312+ζ13 ζ1311+ζ132 orthogonal lifted from D13 ρ18 2 -2 0 0 0 ζ138+ζ135 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ137+ζ136 -ζ138-ζ135 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1311-ζ132 -ζ139-ζ134 -ζ137-ζ136 -ζ4ζ137+ζ4ζ136 -ζ43ζ138+ζ43ζ135 -ζ43ζ1310+ζ43ζ133 -ζ43ζ1312+ζ43ζ13 ζ43ζ1312-ζ43ζ13 ζ43ζ1310-ζ43ζ133 ζ43ζ138-ζ43ζ135 ζ4ζ137-ζ4ζ136 ζ4ζ139-ζ4ζ134 ζ4ζ1311-ζ4ζ132 -ζ4ζ1311+ζ4ζ132 -ζ4ζ139+ζ4ζ134 orthogonal faithful ρ19 2 -2 0 0 0 ζ138+ζ135 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ137+ζ136 -ζ138-ζ135 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1311-ζ132 -ζ139-ζ134 -ζ137-ζ136 ζ4ζ137-ζ4ζ136 ζ43ζ138-ζ43ζ135 ζ43ζ1310-ζ43ζ133 ζ43ζ1312-ζ43ζ13 -ζ43ζ1312+ζ43ζ13 -ζ43ζ1310+ζ43ζ133 -ζ43ζ138+ζ43ζ135 -ζ4ζ137+ζ4ζ136 -ζ4ζ139+ζ4ζ134 -ζ4ζ1311+ζ4ζ132 ζ4ζ1311-ζ4ζ132 ζ4ζ139-ζ4ζ134 orthogonal faithful ρ20 2 -2 0 0 0 ζ1310+ζ133 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1312+ζ13 -ζ1310-ζ133 -ζ137-ζ136 -ζ1311-ζ132 -ζ139-ζ134 -ζ138-ζ135 -ζ1312-ζ13 ζ43ζ1312-ζ43ζ13 -ζ43ζ1310+ζ43ζ133 -ζ4ζ137+ζ4ζ136 -ζ4ζ1311+ζ4ζ132 ζ4ζ1311-ζ4ζ132 ζ4ζ137-ζ4ζ136 ζ43ζ1310-ζ43ζ133 -ζ43ζ1312+ζ43ζ13 -ζ43ζ138+ζ43ζ135 -ζ4ζ139+ζ4ζ134 ζ4ζ139-ζ4ζ134 ζ43ζ138-ζ43ζ135 orthogonal faithful ρ21 2 -2 0 0 0 ζ1310+ζ133 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1312+ζ13 -ζ1310-ζ133 -ζ137-ζ136 -ζ1311-ζ132 -ζ139-ζ134 -ζ138-ζ135 -ζ1312-ζ13 -ζ43ζ1312+ζ43ζ13 ζ43ζ1310-ζ43ζ133 ζ4ζ137-ζ4ζ136 ζ4ζ1311-ζ4ζ132 -ζ4ζ1311+ζ4ζ132 -ζ4ζ137+ζ4ζ136 -ζ43ζ1310+ζ43ζ133 ζ43ζ1312-ζ43ζ13 ζ43ζ138-ζ43ζ135 ζ4ζ139-ζ4ζ134 -ζ4ζ139+ζ4ζ134 -ζ43ζ138+ζ43ζ135 orthogonal faithful ρ22 2 -2 0 0 0 ζ139+ζ134 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ1310+ζ133 -ζ139-ζ134 -ζ138-ζ135 -ζ137-ζ136 -ζ1312-ζ13 -ζ1311-ζ132 -ζ1310-ζ133 ζ43ζ1310-ζ43ζ133 -ζ4ζ139+ζ4ζ134 ζ43ζ138-ζ43ζ135 -ζ4ζ137+ζ4ζ136 ζ4ζ137-ζ4ζ136 -ζ43ζ138+ζ43ζ135 ζ4ζ139-ζ4ζ134 -ζ43ζ1310+ζ43ζ133 ζ4ζ1311-ζ4ζ132 -ζ43ζ1312+ζ43ζ13 ζ43ζ1312-ζ43ζ13 -ζ4ζ1311+ζ4ζ132 orthogonal faithful ρ23 2 -2 0 0 0 ζ1312+ζ13 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ139+ζ134 -ζ1312-ζ13 -ζ1311-ζ132 -ζ138-ζ135 -ζ1310-ζ133 -ζ137-ζ136 -ζ139-ζ134 -ζ4ζ139+ζ4ζ134 ζ43ζ1312-ζ43ζ13 ζ4ζ1311-ζ4ζ132 -ζ43ζ138+ζ43ζ135 ζ43ζ138-ζ43ζ135 -ζ4ζ1311+ζ4ζ132 -ζ43ζ1312+ζ43ζ13 ζ4ζ139-ζ4ζ134 -ζ4ζ137+ζ4ζ136 ζ43ζ1310-ζ43ζ133 -ζ43ζ1310+ζ43ζ133 ζ4ζ137-ζ4ζ136 orthogonal faithful ρ24 2 -2 0 0 0 ζ1311+ζ132 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ138+ζ135 -ζ1311-ζ132 -ζ139-ζ134 -ζ1310-ζ133 -ζ137-ζ136 -ζ1312-ζ13 -ζ138-ζ135 ζ43ζ138-ζ43ζ135 ζ4ζ1311-ζ4ζ132 -ζ4ζ139+ζ4ζ134 -ζ43ζ1310+ζ43ζ133 ζ43ζ1310-ζ43ζ133 ζ4ζ139-ζ4ζ134 -ζ4ζ1311+ζ4ζ132 -ζ43ζ138+ζ43ζ135 ζ43ζ1312-ζ43ζ13 ζ4ζ137-ζ4ζ136 -ζ4ζ137+ζ4ζ136 -ζ43ζ1312+ζ43ζ13 orthogonal faithful ρ25 2 -2 0 0 0 ζ1311+ζ132 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ138+ζ135 -ζ1311-ζ132 -ζ139-ζ134 -ζ1310-ζ133 -ζ137-ζ136 -ζ1312-ζ13 -ζ138-ζ135 -ζ43ζ138+ζ43ζ135 -ζ4ζ1311+ζ4ζ132 ζ4ζ139-ζ4ζ134 ζ43ζ1310-ζ43ζ133 -ζ43ζ1310+ζ43ζ133 -ζ4ζ139+ζ4ζ134 ζ4ζ1311-ζ4ζ132 ζ43ζ138-ζ43ζ135 -ζ43ζ1312+ζ43ζ13 -ζ4ζ137+ζ4ζ136 ζ4ζ137-ζ4ζ136 ζ43ζ1312-ζ43ζ13 orthogonal faithful ρ26 2 -2 0 0 0 ζ1312+ζ13 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ139+ζ134 -ζ1312-ζ13 -ζ1311-ζ132 -ζ138-ζ135 -ζ1310-ζ133 -ζ137-ζ136 -ζ139-ζ134 ζ4ζ139-ζ4ζ134 -ζ43ζ1312+ζ43ζ13 -ζ4ζ1311+ζ4ζ132 ζ43ζ138-ζ43ζ135 -ζ43ζ138+ζ43ζ135 ζ4ζ1311-ζ4ζ132 ζ43ζ1312-ζ43ζ13 -ζ4ζ139+ζ4ζ134 ζ4ζ137-ζ4ζ136 -ζ43ζ1310+ζ43ζ133 ζ43ζ1310-ζ43ζ133 -ζ4ζ137+ζ4ζ136 orthogonal faithful ρ27 2 -2 0 0 0 ζ137+ζ136 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ1311+ζ132 -ζ137-ζ136 -ζ1312-ζ13 -ζ139-ζ134 -ζ138-ζ135 -ζ1310-ζ133 -ζ1311-ζ132 -ζ4ζ1311+ζ4ζ132 ζ4ζ137-ζ4ζ136 -ζ43ζ1312+ζ43ζ13 -ζ4ζ139+ζ4ζ134 ζ4ζ139-ζ4ζ134 ζ43ζ1312-ζ43ζ13 -ζ4ζ137+ζ4ζ136 ζ4ζ1311-ζ4ζ132 ζ43ζ1310-ζ43ζ133 -ζ43ζ138+ζ43ζ135 ζ43ζ138-ζ43ζ135 -ζ43ζ1310+ζ43ζ133 orthogonal faithful ρ28 2 -2 0 0 0 ζ137+ζ136 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ1311+ζ132 -ζ137-ζ136 -ζ1312-ζ13 -ζ139-ζ134 -ζ138-ζ135 -ζ1310-ζ133 -ζ1311-ζ132 ζ4ζ1311-ζ4ζ132 -ζ4ζ137+ζ4ζ136 ζ43ζ1312-ζ43ζ13 ζ4ζ139-ζ4ζ134 -ζ4ζ139+ζ4ζ134 -ζ43ζ1312+ζ43ζ13 ζ4ζ137-ζ4ζ136 -ζ4ζ1311+ζ4ζ132 -ζ43ζ1310+ζ43ζ133 ζ43ζ138-ζ43ζ135 -ζ43ζ138+ζ43ζ135 ζ43ζ1310-ζ43ζ133 orthogonal faithful ρ29 2 -2 0 0 0 ζ139+ζ134 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ1310+ζ133 -ζ139-ζ134 -ζ138-ζ135 -ζ137-ζ136 -ζ1312-ζ13 -ζ1311-ζ132 -ζ1310-ζ133 -ζ43ζ1310+ζ43ζ133 ζ4ζ139-ζ4ζ134 -ζ43ζ138+ζ43ζ135 ζ4ζ137-ζ4ζ136 -ζ4ζ137+ζ4ζ136 ζ43ζ138-ζ43ζ135 -ζ4ζ139+ζ4ζ134 ζ43ζ1310-ζ43ζ133 -ζ4ζ1311+ζ4ζ132 ζ43ζ1312-ζ43ζ13 -ζ43ζ1312+ζ43ζ13 ζ4ζ1311-ζ4ζ132 orthogonal faithful

Smallest permutation representation of D52
On 52 points
Generators in S52
```(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)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)```

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

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

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

D52 is a maximal subgroup of
C104⋊C2  D104  D4⋊D13  Q8⋊D13  D525C2  D4×D13  D52⋊C2  D52⋊C3  C3⋊D52  D156
D52 is a maximal quotient of
C104⋊C2  D104  Dic52  C523C4  D26⋊C4  C3⋊D52  D156

Matrix representation of D52 in GL2(𝔽53) generated by

 21 38 15 7
,
 21 38 47 32
`G:=sub<GL(2,GF(53))| [21,15,38,7],[21,47,38,32] >;`

D52 in GAP, Magma, Sage, TeX

`D_{52}`
`% in TeX`

`G:=Group("D52");`
`// GroupNames label`

`G:=SmallGroup(104,6);`
`// by ID`

`G=gap.SmallGroup(104,6);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-13,49,21,1539]);`
`// Polycyclic`

`G:=Group<a,b|a^52=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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