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## G = C22×C13⋊C3order 156 = 22·3·13

### Direct product of C22 and C13⋊C3

Aliases: C22×C13⋊C3, C262C6, (C2×C26)⋊3C3, C132(C2×C6), SmallGroup(156,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C22×C13⋊C3
 Chief series C1 — C13 — C13⋊C3 — C2×C13⋊C3 — C22×C13⋊C3
 Lower central C13 — C22×C13⋊C3
 Upper central C1 — C22

Generators and relations for C22×C13⋊C3
G = < a,b,c,d | a2=b2=c13=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

Character table of C22×C13⋊C3

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 13A 13B 13C 13D 26A 26B 26C 26D 26E 26F 26G 26H 26I 26J 26K 26L size 1 1 1 1 13 13 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 -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 linear of order 2 ρ5 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ32 ζ3 ζ6 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 6 ρ7 1 1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ3 ζ32 ζ65 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 6 ρ8 1 -1 1 -1 ζ3 ζ32 ζ65 ζ32 ζ3 ζ6 ζ65 ζ6 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 linear of order 6 ρ9 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ10 1 -1 1 -1 ζ32 ζ3 ζ6 ζ3 ζ32 ζ65 ζ6 ζ65 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 linear of order 6 ρ11 1 -1 -1 1 ζ3 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 ζ32 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 6 ρ12 1 -1 -1 1 ζ32 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 ζ3 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 6 ρ13 3 3 3 3 0 0 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 complex lifted from C13⋊C3 ρ14 3 -3 -3 3 0 0 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ1312-ζ1310-ζ134 -ζ139-ζ133-ζ13 complex lifted from C2×C13⋊C3 ρ15 3 3 3 3 0 0 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 complex lifted from C13⋊C3 ρ16 3 3 3 3 0 0 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 complex lifted from C13⋊C3 ρ17 3 3 -3 -3 0 0 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ139-ζ133-ζ13 ζ1312+ζ1310+ζ134 complex lifted from C2×C13⋊C3 ρ18 3 -3 3 -3 0 0 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 ζ136+ζ135+ζ132 -ζ1311-ζ138-ζ137 complex lifted from C2×C13⋊C3 ρ19 3 -3 -3 3 0 0 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ1311-ζ138-ζ137 -ζ136-ζ135-ζ132 complex lifted from C2×C13⋊C3 ρ20 3 -3 3 -3 0 0 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 ζ139+ζ133+ζ13 -ζ1312-ζ1310-ζ134 complex lifted from C2×C13⋊C3 ρ21 3 -3 3 -3 0 0 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 ζ1311+ζ138+ζ137 -ζ136-ζ135-ζ132 complex lifted from C2×C13⋊C3 ρ22 3 3 3 3 0 0 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 complex lifted from C13⋊C3 ρ23 3 3 -3 -3 0 0 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ136-ζ135-ζ132 ζ1311+ζ138+ζ137 complex lifted from C2×C13⋊C3 ρ24 3 -3 -3 3 0 0 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ136-ζ135-ζ132 -ζ1311-ζ138-ζ137 complex lifted from C2×C13⋊C3 ρ25 3 3 -3 -3 0 0 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ1311-ζ138-ζ137 ζ136+ζ135+ζ132 complex lifted from C2×C13⋊C3 ρ26 3 3 -3 -3 0 0 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ1312-ζ1310-ζ134 ζ139+ζ133+ζ13 complex lifted from C2×C13⋊C3 ρ27 3 -3 -3 3 0 0 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ139-ζ133-ζ13 -ζ1312-ζ1310-ζ134 complex lifted from C2×C13⋊C3 ρ28 3 -3 3 -3 0 0 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 ζ1312+ζ1310+ζ134 -ζ139-ζ133-ζ13 complex lifted from C2×C13⋊C3

Smallest permutation representation of C22×C13⋊C3
On 52 points
Generators in S52
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)
(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 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 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)
(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)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52), (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)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (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), (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( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52), (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)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (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), (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([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52)], [(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),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52)], [(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)], [(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)]])

C22×C13⋊C3 is a maximal subgroup of   D26⋊C6

Matrix representation of C22×C13⋊C3 in GL4(𝔽79) generated by

 78 0 0 0 0 78 0 0 0 0 78 0 0 0 0 78
,
 78 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 50 66 1 0 1 0 0 0 0 1 0
,
 23 0 0 0 0 1 0 0 0 12 49 66 0 40 67 29
G:=sub<GL(4,GF(79))| [78,0,0,0,0,78,0,0,0,0,78,0,0,0,0,78],[78,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,50,1,0,0,66,0,1,0,1,0,0],[23,0,0,0,0,1,12,40,0,0,49,67,0,0,66,29] >;

C22×C13⋊C3 in GAP, Magma, Sage, TeX

C_2^2\times C_{13}\rtimes C_3
% in TeX

G:=Group("C2^2xC13:C3");
// GroupNames label

G:=SmallGroup(156,12);
// by ID

G=gap.SmallGroup(156,12);
# by ID

G:=PCGroup([4,-2,-2,-3,-13,155]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^13=d^3=1,a*b=b*a,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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