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## G = C2×D39⋊C3order 468 = 22·32·13

### Direct product of C2 and D39⋊C3

Aliases: C2×D39⋊C3, D78⋊C3, C781C6, D392C6, C6⋊(C13⋊C6), C26⋊(C3×S3), C132(S3×C6), C13⋊C32D6, C392(C2×C6), (C2×C13⋊C3)⋊S3, C32(C2×C13⋊C6), (C6×C13⋊C3)⋊1C2, (C3×C13⋊C3)⋊2C22, SmallGroup(468,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — C2×D39⋊C3
 Chief series C1 — C13 — C39 — C3×C13⋊C3 — D39⋊C3 — C2×D39⋊C3
 Lower central C39 — C2×D39⋊C3
 Upper central C1 — C2

Generators and relations for C2×D39⋊C3
G = < a,b,c,d | a2=b39=c2=d3=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b22, dcd-1=b21c >

Character table of C2×D39⋊C3

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 13A 13B 26A 26B 39A 39B 39C 39D 78A 78B 78C 78D size 1 1 39 39 2 13 13 26 26 2 13 13 26 26 39 39 39 39 6 6 6 6 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 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 -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 -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 1 linear of order 2 ρ5 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 -1 1 -1 1 ζ32 ζ3 ζ32 ζ3 -1 ζ65 ζ6 ζ65 ζ6 ζ32 ζ65 ζ6 ζ3 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ7 1 1 -1 -1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ8 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ9 1 1 -1 -1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ10 1 -1 1 -1 1 ζ3 ζ32 ζ3 ζ32 -1 ζ6 ζ65 ζ6 ζ65 ζ3 ζ6 ζ65 ζ32 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ11 1 -1 -1 1 1 ζ3 ζ32 ζ3 ζ32 -1 ζ6 ζ65 ζ6 ζ65 ζ65 ζ32 ζ3 ζ6 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ12 1 -1 -1 1 1 ζ32 ζ3 ζ32 ζ3 -1 ζ65 ζ6 ζ65 ζ6 ζ6 ζ3 ζ32 ζ65 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ13 2 -2 0 0 -1 2 2 -1 -1 1 -2 -2 1 1 0 0 0 0 2 2 -2 -2 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ14 2 2 0 0 -1 2 2 -1 -1 -1 2 2 -1 -1 0 0 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 -2 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 1 1-√-3 1+√-3 ζ3 ζ32 0 0 0 0 2 2 -2 -2 -1 -1 -1 -1 1 1 1 1 complex lifted from S3×C6 ρ16 2 2 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ17 2 2 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ18 2 -2 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 1 1+√-3 1-√-3 ζ32 ζ3 0 0 0 0 2 2 -2 -2 -1 -1 -1 -1 1 1 1 1 complex lifted from S3×C6 ρ19 6 -6 0 0 6 0 0 0 0 -6 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 1+√13/2 1-√13/2 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 1-√13/2 1+√13/2 1+√13/2 1-√13/2 orthogonal lifted from C2×C13⋊C6 ρ20 6 6 0 0 6 0 0 0 0 6 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 orthogonal lifted from C13⋊C6 ρ21 6 -6 0 0 6 0 0 0 0 -6 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 1-√13/2 1+√13/2 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 1+√13/2 1-√13/2 1-√13/2 1+√13/2 orthogonal lifted from C2×C13⋊C6 ρ22 6 6 0 0 6 0 0 0 0 6 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 orthogonal lifted from C13⋊C6 ρ23 6 -6 0 0 -3 0 0 0 0 3 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 1+√13/2 1-√13/2 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13+ζ139+ζ133+ζ13 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132+ζ136+ζ135+ζ132 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132+ζ1311+ζ138+ζ137 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13+ζ1312+ζ1310+ζ134 orthogonal faithful ρ24 6 6 0 0 -3 0 0 0 0 -3 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 orthogonal lifted from D39⋊C3 ρ25 6 -6 0 0 -3 0 0 0 0 3 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 1-√13/2 1+√13/2 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132+ζ1311+ζ138+ζ137 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13+ζ139+ζ133+ζ13 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13+ζ1312+ζ1310+ζ134 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132+ζ136+ζ135+ζ132 orthogonal faithful ρ26 6 -6 0 0 -3 0 0 0 0 3 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 1+√13/2 1-√13/2 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13+ζ1312+ζ1310+ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132+ζ1311+ζ138+ζ137 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132+ζ136+ζ135+ζ132 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13+ζ139+ζ133+ζ13 orthogonal faithful ρ27 6 6 0 0 -3 0 0 0 0 -3 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 orthogonal lifted from D39⋊C3 ρ28 6 6 0 0 -3 0 0 0 0 -3 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 orthogonal lifted from D39⋊C3 ρ29 6 -6 0 0 -3 0 0 0 0 3 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 1-√13/2 1+√13/2 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132+ζ136+ζ135+ζ132 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13+ζ1312+ζ1310+ζ134 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13+ζ139+ζ133+ζ13 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132+ζ1311+ζ138+ζ137 orthogonal faithful ρ30 6 6 0 0 -3 0 0 0 0 -3 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 orthogonal lifted from D39⋊C3

Smallest permutation representation of C2×D39⋊C3
On 78 points
Generators in S78
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)
(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 66 67 68 69 70 71 72 73 74 75 76 77 78)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)
(2 17 23)(3 33 6)(4 10 28)(5 26 11)(7 19 16)(8 35 38)(9 12 21)(13 37 31)(15 30 36)(18 39 24)(20 32 29)(22 25 34)(40 61 46)(41 77 68)(42 54 51)(43 70 73)(44 47 56)(45 63 78)(48 72 66)(50 65 71)(52 58 76)(53 74 59)(55 67 64)(57 60 69)

G:=sub<Sym(78)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69), (2,17,23)(3,33,6)(4,10,28)(5,26,11)(7,19,16)(8,35,38)(9,12,21)(13,37,31)(15,30,36)(18,39,24)(20,32,29)(22,25,34)(40,61,46)(41,77,68)(42,54,51)(43,70,73)(44,47,56)(45,63,78)(48,72,66)(50,65,71)(52,58,76)(53,74,59)(55,67,64)(57,60,69)>;

G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69), (2,17,23)(3,33,6)(4,10,28)(5,26,11)(7,19,16)(8,35,38)(9,12,21)(13,37,31)(15,30,36)(18,39,24)(20,32,29)(22,25,34)(40,61,46)(41,77,68)(42,54,51)(43,70,73)(44,47,56)(45,63,78)(48,72,66)(50,65,71)(52,58,76)(53,74,59)(55,67,64)(57,60,69) );

G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48)], [(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),(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(40,57),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69)], [(2,17,23),(3,33,6),(4,10,28),(5,26,11),(7,19,16),(8,35,38),(9,12,21),(13,37,31),(15,30,36),(18,39,24),(20,32,29),(22,25,34),(40,61,46),(41,77,68),(42,54,51),(43,70,73),(44,47,56),(45,63,78),(48,72,66),(50,65,71),(52,58,76),(53,74,59),(55,67,64),(57,60,69)]])

Matrix representation of C2×D39⋊C3 in GL6(𝔽79)

 78 0 0 0 0 0 0 78 0 0 0 0 0 0 78 0 0 0 0 0 0 78 0 0 0 0 0 0 78 0 0 0 0 0 0 78
,
 3 73 34 33 18 25 34 42 37 58 6 63 76 15 26 41 10 34 74 68 17 12 61 78 73 3 10 63 16 22 67 75 7 73 73 27
,
 76 3 64 15 0 51 76 19 48 18 76 0 16 51 3 12 0 51 3 76 6 60 31 76 21 46 8 19 9 36 72 1 14 74 5 70
,
 15 64 14 2 78 63 0 0 0 0 1 0 1 0 0 0 0 0 78 2 14 64 15 2 0 0 0 0 0 1 0 1 0 0 0 0

G:=sub<GL(6,GF(79))| [78,0,0,0,0,0,0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,78],[3,34,76,74,73,67,73,42,15,68,3,75,34,37,26,17,10,7,33,58,41,12,63,73,18,6,10,61,16,73,25,63,34,78,22,27],[76,76,16,3,21,72,3,19,51,76,46,1,64,48,3,6,8,14,15,18,12,60,19,74,0,76,0,31,9,5,51,0,51,76,36,70],[15,0,1,78,0,0,64,0,0,2,0,1,14,0,0,14,0,0,2,0,0,64,0,0,78,1,0,15,0,0,63,0,0,2,1,0] >;

C2×D39⋊C3 in GAP, Magma, Sage, TeX

C_2\times D_{39}\rtimes C_3
% in TeX

G:=Group("C2xD39:C3");
// GroupNames label

G:=SmallGroup(468,35);
// by ID

G=gap.SmallGroup(468,35);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,483,10804,689]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^39=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^22,d*c*d^-1=b^21*c>;
// generators/relations

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