direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C31⋊C6, C62⋊C6, D62⋊C3, D31⋊C6, C31⋊(C2×C6), C31⋊C3⋊C22, (C2×C31⋊C3)⋊C2, SmallGroup(372,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C2×C31⋊C6 |
Generators and relations for C2×C31⋊C6
G = < a,b,c | a2=b31=c6=1, ab=ba, ac=ca, cbc-1=b6 >
Character table of C2×C31⋊C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 31A | 31B | 31C | 31D | 31E | 62A | 62B | 62C | 62D | 62E | |
| size | 1 | 1 | 31 | 31 | 31 | 31 | 31 | 31 | 31 | 31 | 31 | 31 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ10 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ11 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ13 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 | -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 | -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 | -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 | -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 | orthogonal faithful |
| ρ14 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 | -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 | -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 | -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 | -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 | orthogonal faithful |
| ρ15 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 | -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 | -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 | -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 | -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 | orthogonal faithful |
| ρ16 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | orthogonal lifted from C31⋊C6 |
| ρ17 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | orthogonal lifted from C31⋊C6 |
| ρ18 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | orthogonal lifted from C31⋊C6 |
| ρ19 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | orthogonal lifted from C31⋊C6 |
| ρ20 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 | -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 | -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 | -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 | -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 | orthogonal faithful |
| ρ21 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 | -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 | -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 | -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 | -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 | orthogonal faithful |
| ρ22 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | orthogonal lifted from C31⋊C6 |
►On 62 points
Generators in S62
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)
(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)
(2 27 26 31 6 7)(3 22 20 30 11 13)(4 17 14 29 16 19)(5 12 8 28 21 25)(9 23 15 24 10 18)(33 58 57 62 37 38)(34 53 51 61 42 44)(35 48 45 60 47 50)(36 43 39 59 52 56)(40 54 46 55 41 49)
G:=sub<Sym(62)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62), (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), (2,27,26,31,6,7)(3,22,20,30,11,13)(4,17,14,29,16,19)(5,12,8,28,21,25)(9,23,15,24,10,18)(33,58,57,62,37,38)(34,53,51,61,42,44)(35,48,45,60,47,50)(36,43,39,59,52,56)(40,54,46,55,41,49)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62), (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), (2,27,26,31,6,7)(3,22,20,30,11,13)(4,17,14,29,16,19)(5,12,8,28,21,25)(9,23,15,24,10,18)(33,58,57,62,37,38)(34,53,51,61,42,44)(35,48,45,60,47,50)(36,43,39,59,52,56)(40,54,46,55,41,49) );
G=PermutationGroup([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62)], [(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)], [(2,27,26,31,6,7),(3,22,20,30,11,13),(4,17,14,29,16,19),(5,12,8,28,21,25),(9,23,15,24,10,18),(33,58,57,62,37,38),(34,53,51,61,42,44),(35,48,45,60,47,50),(36,43,39,59,52,56),(40,54,46,55,41,49)]])
Matrix representation of C2×C31⋊C6 ►in GL6(𝔽373)
| 372 | 0 | 0 | 0 | 0 | 0 |
| 0 | 372 | 0 | 0 | 0 | 0 |
| 0 | 0 | 372 | 0 | 0 | 0 |
| 0 | 0 | 0 | 372 | 0 | 0 |
| 0 | 0 | 0 | 0 | 372 | 0 |
| 0 | 0 | 0 | 0 | 0 | 372 |
| 323 | 363 | 144 | 363 | 323 | 372 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 372 | 323 | 363 | 144 | 363 | 323 |
| 100 | 50 | 204 | 366 | 271 | 109 |
| 160 | 17 | 169 | 169 | 17 | 160 |
| 109 | 271 | 366 | 204 | 50 | 100 |
| 323 | 363 | 144 | 363 | 323 | 372 |
G:=sub<GL(6,GF(373))| [372,0,0,0,0,0,0,372,0,0,0,0,0,0,372,0,0,0,0,0,0,372,0,0,0,0,0,0,372,0,0,0,0,0,0,372],[323,1,0,0,0,0,363,0,1,0,0,0,144,0,0,1,0,0,363,0,0,0,1,0,323,0,0,0,0,1,372,0,0,0,0,0],[1,372,100,160,109,323,0,323,50,17,271,363,0,363,204,169,366,144,0,144,366,169,204,363,0,363,271,17,50,323,0,323,109,160,100,372] >;
C2×C31⋊C6 in GAP, Magma, Sage, TeX
C_2\times C_{31}\rtimes C_6 % in TeX
G:=Group("C2xC31:C6"); // GroupNames label
G:=SmallGroup(372,7);
// by ID
G=gap.SmallGroup(372,7);
# by ID
G:=PCGroup([4,-2,-2,-3,-31,5763,1211]);
// Polycyclic
G:=Group<a,b,c|a^2=b^31=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations
Export
Subgroup lattice of C2×C31⋊C6 in TeX
Character table of C2×C31⋊C6 in TeX