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## G = C4×C13⋊C3order 156 = 22·3·13

### Direct product of C4 and C13⋊C3

Aliases: C4×C13⋊C3, C52⋊C3, C134C12, C26.2C6, C2.(C2×C13⋊C3), (C2×C13⋊C3).2C2, SmallGroup(156,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C4×C13⋊C3
 Chief series C1 — C13 — C26 — C2×C13⋊C3 — C4×C13⋊C3
 Lower central C13 — C4×C13⋊C3
 Upper central C1 — C4

Generators and relations for C4×C13⋊C3
G = < a,b,c | a4=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

Character table of C4×C13⋊C3

 class 1 2 3A 3B 4A 4B 6A 6B 12A 12B 12C 12D 13A 13B 13C 13D 26A 26B 26C 26D 52A 52B 52C 52D 52E 52F 52G 52H size 1 1 13 13 1 1 13 13 13 13 13 13 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 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 ζ3 ζ32 ζ32 ζ3 1 1 1 1 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 ζ6 ζ65 ζ65 ζ6 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ5 1 1 ζ32 ζ3 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ6 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ7 1 -1 1 1 i -i -1 -1 -i i -i i 1 1 1 1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 4 ρ8 1 -1 1 1 -i i -1 -1 i -i i -i 1 1 1 1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ9 1 -1 ζ32 ζ3 i -i ζ65 ζ6 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 1 1 1 1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 12 ρ10 1 -1 ζ32 ζ3 -i i ζ65 ζ6 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 1 1 1 1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 12 ρ11 1 -1 ζ3 ζ32 -i i ζ6 ζ65 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 1 1 1 1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 12 ρ12 1 -1 ζ3 ζ32 i -i ζ6 ζ65 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 1 1 1 1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 12 ρ13 3 3 0 0 -3 -3 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 complex lifted from C2×C13⋊C3 ρ14 3 3 0 0 -3 -3 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 complex lifted from C2×C13⋊C3 ρ15 3 3 0 0 3 3 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 complex lifted from C13⋊C3 ρ16 3 3 0 0 -3 -3 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 complex lifted from C2×C13⋊C3 ρ17 3 3 0 0 -3 -3 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 complex lifted from C2×C13⋊C3 ρ18 3 3 0 0 3 3 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 complex lifted from C13⋊C3 ρ19 3 3 0 0 3 3 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 complex lifted from C13⋊C3 ρ20 3 3 0 0 3 3 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 complex lifted from C13⋊C3 ρ21 3 -3 0 0 3i -3i 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 -ζ1311-ζ138-ζ137 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ139-ζ133-ζ13 ζ43ζ1311+ζ43ζ138+ζ43ζ137 ζ43ζ139+ζ43ζ133+ζ43ζ13 ζ43ζ136+ζ43ζ135+ζ43ζ132 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 ζ4ζ139+ζ4ζ133+ζ4ζ13 ζ4ζ136+ζ4ζ135+ζ4ζ132 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 ζ4ζ1311+ζ4ζ138+ζ4ζ137 complex faithful ρ22 3 -3 0 0 3i -3i 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 -ζ1312-ζ1310-ζ134 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1311-ζ138-ζ137 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 ζ43ζ1311+ζ43ζ138+ζ43ζ137 ζ43ζ139+ζ43ζ133+ζ43ζ13 ζ43ζ136+ζ43ζ135+ζ43ζ132 ζ4ζ1311+ζ4ζ138+ζ4ζ137 ζ4ζ139+ζ4ζ133+ζ4ζ13 ζ4ζ136+ζ4ζ135+ζ4ζ132 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 complex faithful ρ23 3 -3 0 0 -3i 3i 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 -ζ136-ζ135-ζ132 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ1312-ζ1310-ζ134 ζ4ζ136+ζ4ζ135+ζ4ζ132 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 ζ4ζ1311+ζ4ζ138+ζ4ζ137 ζ4ζ139+ζ4ζ133+ζ4ζ13 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 ζ43ζ1311+ζ43ζ138+ζ43ζ137 ζ43ζ139+ζ43ζ133+ζ43ζ13 ζ43ζ136+ζ43ζ135+ζ43ζ132 complex faithful ρ24 3 -3 0 0 -3i 3i 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 -ζ1312-ζ1310-ζ134 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1311-ζ138-ζ137 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 ζ4ζ1311+ζ4ζ138+ζ4ζ137 ζ4ζ139+ζ4ζ133+ζ4ζ13 ζ4ζ136+ζ4ζ135+ζ4ζ132 ζ43ζ1311+ζ43ζ138+ζ43ζ137 ζ43ζ139+ζ43ζ133+ζ43ζ13 ζ43ζ136+ζ43ζ135+ζ43ζ132 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 complex faithful ρ25 3 -3 0 0 3i -3i 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 -ζ136-ζ135-ζ132 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ1312-ζ1310-ζ134 ζ43ζ136+ζ43ζ135+ζ43ζ132 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 ζ43ζ1311+ζ43ζ138+ζ43ζ137 ζ43ζ139+ζ43ζ133+ζ43ζ13 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 ζ4ζ1311+ζ4ζ138+ζ4ζ137 ζ4ζ139+ζ4ζ133+ζ4ζ13 ζ4ζ136+ζ4ζ135+ζ4ζ132 complex faithful ρ26 3 -3 0 0 -3i 3i 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 -ζ139-ζ133-ζ13 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ136-ζ135-ζ132 ζ4ζ139+ζ4ζ133+ζ4ζ13 ζ4ζ136+ζ4ζ135+ζ4ζ132 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 ζ4ζ1311+ζ4ζ138+ζ4ζ137 ζ43ζ136+ζ43ζ135+ζ43ζ132 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 ζ43ζ1311+ζ43ζ138+ζ43ζ137 ζ43ζ139+ζ43ζ133+ζ43ζ13 complex faithful ρ27 3 -3 0 0 3i -3i 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 -ζ139-ζ133-ζ13 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ136-ζ135-ζ132 ζ43ζ139+ζ43ζ133+ζ43ζ13 ζ43ζ136+ζ43ζ135+ζ43ζ132 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 ζ43ζ1311+ζ43ζ138+ζ43ζ137 ζ4ζ136+ζ4ζ135+ζ4ζ132 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 ζ4ζ1311+ζ4ζ138+ζ4ζ137 ζ4ζ139+ζ4ζ133+ζ4ζ13 complex faithful ρ28 3 -3 0 0 -3i 3i 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 -ζ1311-ζ138-ζ137 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ139-ζ133-ζ13 ζ4ζ1311+ζ4ζ138+ζ4ζ137 ζ4ζ139+ζ4ζ133+ζ4ζ13 ζ4ζ136+ζ4ζ135+ζ4ζ132 ζ4ζ1312+ζ4ζ1310+ζ4ζ134 ζ43ζ139+ζ43ζ133+ζ43ζ13 ζ43ζ136+ζ43ζ135+ζ43ζ132 ζ43ζ1312+ζ43ζ1310+ζ43ζ134 ζ43ζ1311+ζ43ζ138+ζ43ζ137 complex faithful

Smallest permutation representation of C4×C13⋊C3
On 52 points
Generators in S52
(1 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)
(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)
(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)

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

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

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

C4×C13⋊C3 is a maximal subgroup of   C132C24  Dic26⋊C3  D52⋊C3

Matrix representation of C4×C13⋊C3 in GL4(𝔽157) generated by

 28 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 119 52 1 0 1 0 0 0 0 1 0
,
 144 0 0 0 0 1 0 0 0 104 118 52 0 73 53 38
G:=sub<GL(4,GF(157))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,119,1,0,0,52,0,1,0,1,0,0],[144,0,0,0,0,1,104,73,0,0,118,53,0,0,52,38] >;

C4×C13⋊C3 in GAP, Magma, Sage, TeX

C_4\times C_{13}\rtimes C_3
% in TeX

G:=Group("C4xC13:C3");
// GroupNames label

G:=SmallGroup(156,2);
// by ID

G=gap.SmallGroup(156,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-13,24,295]);
// Polycyclic

G:=Group<a,b,c|a^4=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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