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## G = C13⋊S4order 312 = 23·3·13

### The semidirect product of C13 and S4 acting via S4/A4=C2

Aliases: C13⋊S4, A4⋊D13, C22⋊D39, (C2×C26)⋊2S3, (A4×C13)⋊1C2, SmallGroup(312,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C13 — C13⋊S4
 Chief series C1 — C22 — C2×C26 — A4×C13 — C13⋊S4
 Lower central A4×C13 — C13⋊S4
 Upper central C1

Generators and relations for C13⋊S4
G = < a,b,c,d,e | a13=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

3C2
78C2
4C3
39C22
39C4
52S3
3C26
6D13
4C39
39D4
3D26
4D39
13S4

Character table of C13⋊S4

 class 1 2A 2B 3 4 13A 13B 13C 13D 13E 13F 26A 26B 26C 26D 26E 26F 39A 39B 39C 39D 39E 39F 39G 39H 39I 39J 39K 39L size 1 3 78 8 78 2 2 2 2 2 2 6 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 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 2 2 0 -1 0 2 2 2 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 0 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1310+ζ133 ζ138+ζ135 ζ1311+ζ132 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 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 ρ5 2 2 0 2 0 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ1312+ζ13 ζ137+ζ136 ζ138+ζ135 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 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 ρ6 2 2 0 2 0 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ137+ζ136 ζ1310+ζ133 ζ139+ζ134 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 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 ρ7 2 2 0 2 0 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1311+ζ132 ζ1312+ζ13 ζ1310+ζ133 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 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 ρ8 2 2 0 2 0 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ139+ζ134 ζ1311+ζ132 ζ137+ζ136 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 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 ρ9 2 2 0 2 0 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ138+ζ135 ζ139+ζ134 ζ1312+ζ13 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 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 ρ10 2 2 0 -1 0 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ1312+ζ13 ζ137+ζ136 ζ138+ζ135 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ1312+ζ13 ζ3ζ1312-ζ3ζ13-ζ13 ζ32ζ1310-ζ32ζ133-ζ133 ζ3ζ137-ζ3ζ136-ζ136 ζ3ζ1311-ζ3ζ132-ζ132 ζ32ζ1311-ζ32ζ132-ζ132 -ζ3ζ137+ζ3ζ136-ζ137 -ζ32ζ1310+ζ32ζ133-ζ1310 -ζ3ζ1312+ζ3ζ13-ζ1312 -ζ3ζ138+ζ3ζ135-ζ138 -ζ32ζ139+ζ32ζ134-ζ139 ζ32ζ139-ζ32ζ134-ζ134 -ζ32ζ138+ζ32ζ135-ζ138 orthogonal lifted from D39 ρ11 2 2 0 -1 0 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ137+ζ136 ζ1310+ζ133 ζ139+ζ134 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ137+ζ136 -ζ3ζ137+ζ3ζ136-ζ137 -ζ32ζ138+ζ32ζ135-ζ138 -ζ32ζ1310+ζ32ζ133-ζ1310 ζ3ζ1312-ζ3ζ13-ζ13 -ζ3ζ1312+ζ3ζ13-ζ1312 ζ32ζ1310-ζ32ζ133-ζ133 -ζ3ζ138+ζ3ζ135-ζ138 ζ3ζ137-ζ3ζ136-ζ136 -ζ32ζ139+ζ32ζ134-ζ139 ζ3ζ1311-ζ3ζ132-ζ132 ζ32ζ1311-ζ32ζ132-ζ132 ζ32ζ139-ζ32ζ134-ζ134 orthogonal lifted from D39 ρ12 2 2 0 -1 0 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ138+ζ135 ζ139+ζ134 ζ1312+ζ13 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ138+ζ135 -ζ32ζ138+ζ32ζ135-ζ138 ζ32ζ1311-ζ32ζ132-ζ132 -ζ32ζ139+ζ32ζ134-ζ139 ζ32ζ1310-ζ32ζ133-ζ133 -ζ32ζ1310+ζ32ζ133-ζ1310 ζ32ζ139-ζ32ζ134-ζ134 ζ3ζ1311-ζ3ζ132-ζ132 -ζ3ζ138+ζ3ζ135-ζ138 ζ3ζ1312-ζ3ζ13-ζ13 -ζ3ζ137+ζ3ζ136-ζ137 ζ3ζ137-ζ3ζ136-ζ136 -ζ3ζ1312+ζ3ζ13-ζ1312 orthogonal lifted from D39 ρ13 2 2 0 -1 0 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ138+ζ135 ζ139+ζ134 ζ1312+ζ13 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ138+ζ135 -ζ3ζ138+ζ3ζ135-ζ138 ζ3ζ1311-ζ3ζ132-ζ132 ζ32ζ139-ζ32ζ134-ζ134 -ζ32ζ1310+ζ32ζ133-ζ1310 ζ32ζ1310-ζ32ζ133-ζ133 -ζ32ζ139+ζ32ζ134-ζ139 ζ32ζ1311-ζ32ζ132-ζ132 -ζ32ζ138+ζ32ζ135-ζ138 -ζ3ζ1312+ζ3ζ13-ζ1312 ζ3ζ137-ζ3ζ136-ζ136 -ζ3ζ137+ζ3ζ136-ζ137 ζ3ζ1312-ζ3ζ13-ζ13 orthogonal lifted from D39 ρ14 2 2 0 -1 0 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ139+ζ134 ζ1311+ζ132 ζ137+ζ136 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ139+ζ134 ζ32ζ139-ζ32ζ134-ζ134 -ζ3ζ1312+ζ3ζ13-ζ1312 ζ3ζ1311-ζ3ζ132-ζ132 -ζ32ζ138+ζ32ζ135-ζ138 -ζ3ζ138+ζ3ζ135-ζ138 ζ32ζ1311-ζ32ζ132-ζ132 ζ3ζ1312-ζ3ζ13-ζ13 -ζ32ζ139+ζ32ζ134-ζ139 -ζ3ζ137+ζ3ζ136-ζ137 ζ32ζ1310-ζ32ζ133-ζ133 -ζ32ζ1310+ζ32ζ133-ζ1310 ζ3ζ137-ζ3ζ136-ζ136 orthogonal lifted from D39 ρ15 2 2 0 -1 0 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1310+ζ133 ζ138+ζ135 ζ1311+ζ132 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ1310+ζ133 ζ32ζ1310-ζ32ζ133-ζ133 ζ32ζ139-ζ32ζ134-ζ134 -ζ3ζ138+ζ3ζ135-ζ138 -ζ3ζ137+ζ3ζ136-ζ137 ζ3ζ137-ζ3ζ136-ζ136 -ζ32ζ138+ζ32ζ135-ζ138 -ζ32ζ139+ζ32ζ134-ζ139 -ζ32ζ1310+ζ32ζ133-ζ1310 ζ3ζ1311-ζ3ζ132-ζ132 ζ3ζ1312-ζ3ζ13-ζ13 -ζ3ζ1312+ζ3ζ13-ζ1312 ζ32ζ1311-ζ32ζ132-ζ132 orthogonal lifted from D39 ρ16 2 2 0 -1 0 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ137+ζ136 ζ1310+ζ133 ζ139+ζ134 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ137+ζ136 ζ3ζ137-ζ3ζ136-ζ136 -ζ3ζ138+ζ3ζ135-ζ138 ζ32ζ1310-ζ32ζ133-ζ133 -ζ3ζ1312+ζ3ζ13-ζ1312 ζ3ζ1312-ζ3ζ13-ζ13 -ζ32ζ1310+ζ32ζ133-ζ1310 -ζ32ζ138+ζ32ζ135-ζ138 -ζ3ζ137+ζ3ζ136-ζ137 ζ32ζ139-ζ32ζ134-ζ134 ζ32ζ1311-ζ32ζ132-ζ132 ζ3ζ1311-ζ3ζ132-ζ132 -ζ32ζ139+ζ32ζ134-ζ139 orthogonal lifted from D39 ρ17 2 2 0 -1 0 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1310+ζ133 ζ138+ζ135 ζ1311+ζ132 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ1310+ζ133 -ζ32ζ1310+ζ32ζ133-ζ1310 -ζ32ζ139+ζ32ζ134-ζ139 -ζ32ζ138+ζ32ζ135-ζ138 ζ3ζ137-ζ3ζ136-ζ136 -ζ3ζ137+ζ3ζ136-ζ137 -ζ3ζ138+ζ3ζ135-ζ138 ζ32ζ139-ζ32ζ134-ζ134 ζ32ζ1310-ζ32ζ133-ζ133 ζ32ζ1311-ζ32ζ132-ζ132 -ζ3ζ1312+ζ3ζ13-ζ1312 ζ3ζ1312-ζ3ζ13-ζ13 ζ3ζ1311-ζ3ζ132-ζ132 orthogonal lifted from D39 ρ18 2 2 0 -1 0 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1311+ζ132 ζ1312+ζ13 ζ1310+ζ133 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1311+ζ132 ζ3ζ1311-ζ3ζ132-ζ132 -ζ3ζ137+ζ3ζ136-ζ137 -ζ3ζ1312+ζ3ζ13-ζ1312 -ζ32ζ139+ζ32ζ134-ζ139 ζ32ζ139-ζ32ζ134-ζ134 ζ3ζ1312-ζ3ζ13-ζ13 ζ3ζ137-ζ3ζ136-ζ136 ζ32ζ1311-ζ32ζ132-ζ132 -ζ32ζ1310+ζ32ζ133-ζ1310 -ζ3ζ138+ζ3ζ135-ζ138 -ζ32ζ138+ζ32ζ135-ζ138 ζ32ζ1310-ζ32ζ133-ζ133 orthogonal lifted from D39 ρ19 2 2 0 -1 0 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ139+ζ134 ζ1311+ζ132 ζ137+ζ136 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ139+ζ134 -ζ32ζ139+ζ32ζ134-ζ139 ζ3ζ1312-ζ3ζ13-ζ13 ζ32ζ1311-ζ32ζ132-ζ132 -ζ3ζ138+ζ3ζ135-ζ138 -ζ32ζ138+ζ32ζ135-ζ138 ζ3ζ1311-ζ3ζ132-ζ132 -ζ3ζ1312+ζ3ζ13-ζ1312 ζ32ζ139-ζ32ζ134-ζ134 ζ3ζ137-ζ3ζ136-ζ136 -ζ32ζ1310+ζ32ζ133-ζ1310 ζ32ζ1310-ζ32ζ133-ζ133 -ζ3ζ137+ζ3ζ136-ζ137 orthogonal lifted from D39 ρ20 2 2 0 -1 0 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ1312+ζ13 ζ137+ζ136 ζ138+ζ135 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ1312+ζ13 -ζ3ζ1312+ζ3ζ13-ζ1312 -ζ32ζ1310+ζ32ζ133-ζ1310 -ζ3ζ137+ζ3ζ136-ζ137 ζ32ζ1311-ζ32ζ132-ζ132 ζ3ζ1311-ζ3ζ132-ζ132 ζ3ζ137-ζ3ζ136-ζ136 ζ32ζ1310-ζ32ζ133-ζ133 ζ3ζ1312-ζ3ζ13-ζ13 -ζ32ζ138+ζ32ζ135-ζ138 ζ32ζ139-ζ32ζ134-ζ134 -ζ32ζ139+ζ32ζ134-ζ139 -ζ3ζ138+ζ3ζ135-ζ138 orthogonal lifted from D39 ρ21 2 2 0 -1 0 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1311+ζ132 ζ1312+ζ13 ζ1310+ζ133 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1311+ζ132 ζ32ζ1311-ζ32ζ132-ζ132 ζ3ζ137-ζ3ζ136-ζ136 ζ3ζ1312-ζ3ζ13-ζ13 ζ32ζ139-ζ32ζ134-ζ134 -ζ32ζ139+ζ32ζ134-ζ139 -ζ3ζ1312+ζ3ζ13-ζ1312 -ζ3ζ137+ζ3ζ136-ζ137 ζ3ζ1311-ζ3ζ132-ζ132 ζ32ζ1310-ζ32ζ133-ζ133 -ζ32ζ138+ζ32ζ135-ζ138 -ζ3ζ138+ζ3ζ135-ζ138 -ζ32ζ1310+ζ32ζ133-ζ1310 orthogonal lifted from D39 ρ22 3 -1 1 0 -1 3 3 3 3 3 3 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ23 3 -1 -1 0 1 3 3 3 3 3 3 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ24 6 -2 0 0 0 3ζ138+3ζ135 3ζ1310+3ζ133 3ζ137+3ζ136 3ζ1311+3ζ132 3ζ139+3ζ134 3ζ1312+3ζ13 -ζ137-ζ136 -ζ138-ζ135 -ζ1311-ζ132 -ζ1310-ζ133 -ζ139-ζ134 -ζ1312-ζ13 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 6 -2 0 0 0 3ζ1310+3ζ133 3ζ137+3ζ136 3ζ1312+3ζ13 3ζ139+3ζ134 3ζ138+3ζ135 3ζ1311+3ζ132 -ζ1312-ζ13 -ζ1310-ζ133 -ζ139-ζ134 -ζ137-ζ136 -ζ138-ζ135 -ζ1311-ζ132 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ26 6 -2 0 0 0 3ζ139+3ζ134 3ζ138+3ζ135 3ζ1310+3ζ133 3ζ1312+3ζ13 3ζ1311+3ζ132 3ζ137+3ζ136 -ζ1310-ζ133 -ζ139-ζ134 -ζ1312-ζ13 -ζ138-ζ135 -ζ1311-ζ132 -ζ137-ζ136 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ27 6 -2 0 0 0 3ζ137+3ζ136 3ζ1312+3ζ13 3ζ1311+3ζ132 3ζ138+3ζ135 3ζ1310+3ζ133 3ζ139+3ζ134 -ζ1311-ζ132 -ζ137-ζ136 -ζ138-ζ135 -ζ1312-ζ13 -ζ1310-ζ133 -ζ139-ζ134 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ28 6 -2 0 0 0 3ζ1312+3ζ13 3ζ1311+3ζ132 3ζ139+3ζ134 3ζ1310+3ζ133 3ζ137+3ζ136 3ζ138+3ζ135 -ζ139-ζ134 -ζ1312-ζ13 -ζ1310-ζ133 -ζ1311-ζ132 -ζ137-ζ136 -ζ138-ζ135 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ29 6 -2 0 0 0 3ζ1311+3ζ132 3ζ139+3ζ134 3ζ138+3ζ135 3ζ137+3ζ136 3ζ1312+3ζ13 3ζ1310+3ζ133 -ζ138-ζ135 -ζ1311-ζ132 -ζ137-ζ136 -ζ139-ζ134 -ζ1312-ζ13 -ζ1310-ζ133 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of C13⋊S4
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 22)(2 23)(3 24)(4 25)(5 26)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 40)(37 41)(38 42)(39 43)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 27)(14 50)(15 51)(16 52)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)
(14 33 50)(15 34 51)(16 35 52)(17 36 40)(18 37 41)(19 38 42)(20 39 43)(21 27 44)(22 28 45)(23 29 46)(24 30 47)(25 31 48)(26 32 49)
(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(14 40)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 29)(30 39)(31 38)(32 37)(33 36)(34 35)```

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

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

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

Matrix representation of C13⋊S4 in GL5(𝔽157)

 4 58 0 0 0 99 62 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 0 156 1 0 0 0 156 0 0 0 1 156 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 156 0 0 1 0 156 0 0 0 0 156
,
 156 1 0 0 0 156 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 95 58 0 0 0 153 62 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(5,GF(157))| [4,99,0,0,0,58,62,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,156,156,156,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,156,156,156],[156,156,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[95,153,0,0,0,58,62,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;`

C13⋊S4 in GAP, Magma, Sage, TeX

`C_{13}\rtimes S_4`
`% in TeX`

`G:=Group("C13:S4");`
`// GroupNames label`

`G:=SmallGroup(312,48);`
`// by ID`

`G=gap.SmallGroup(312,48);`
`# by ID`

`G:=PCGroup([5,-2,-3,-13,-2,2,41,1082,3123,1568,1954,2934]);`
`// Polycyclic`

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

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