direct product, metacyclic, supersoluble, monomial, Z-group, 7-hyperelementary
Aliases: C2×C29⋊C7, C58⋊C7, C29⋊2C14, SmallGroup(406,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C2×C29⋊C7 |
Generators and relations for C2×C29⋊C7
G = < a,b,c | a2=b29=c7=1, ab=ba, ac=ca, cbc-1=b20 >
Character table of C2×C29⋊C7
| class | 1 | 2 | 7A | 7B | 7C | 7D | 7E | 7F | 14A | 14B | 14C | 14D | 14E | 14F | 29A | 29B | 29C | 29D | 58A | 58B | 58C | 58D | |
| size | 1 | 1 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | |
| 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 | ζ74 | ζ73 | ζ76 | ζ72 | ζ75 | ζ7 | -ζ74 | -ζ73 | -ζ76 | -ζ72 | -ζ75 | -ζ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 14 |
| ρ4 | 1 | -1 | ζ7 | ζ76 | ζ75 | ζ74 | ζ73 | ζ72 | -ζ7 | -ζ76 | -ζ75 | -ζ74 | -ζ73 | -ζ72 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 14 |
| ρ5 | 1 | -1 | ζ75 | ζ72 | ζ74 | ζ76 | ζ7 | ζ73 | -ζ75 | -ζ72 | -ζ74 | -ζ76 | -ζ7 | -ζ73 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 14 |
| ρ6 | 1 | -1 | ζ72 | ζ75 | ζ73 | ζ7 | ζ76 | ζ74 | -ζ72 | -ζ75 | -ζ73 | -ζ7 | -ζ76 | -ζ74 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 14 |
| ρ7 | 1 | 1 | ζ74 | ζ73 | ζ76 | ζ72 | ζ75 | ζ7 | ζ74 | ζ73 | ζ76 | ζ72 | ζ75 | ζ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ8 | 1 | 1 | ζ73 | ζ74 | ζ7 | ζ75 | ζ72 | ζ76 | ζ73 | ζ74 | ζ7 | ζ75 | ζ72 | ζ76 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ9 | 1 | -1 | ζ76 | ζ7 | ζ72 | ζ73 | ζ74 | ζ75 | -ζ76 | -ζ7 | -ζ72 | -ζ73 | -ζ74 | -ζ75 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 14 |
| ρ10 | 1 | 1 | ζ72 | ζ75 | ζ73 | ζ7 | ζ76 | ζ74 | ζ72 | ζ75 | ζ73 | ζ7 | ζ76 | ζ74 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ11 | 1 | 1 | ζ76 | ζ7 | ζ72 | ζ73 | ζ74 | ζ75 | ζ76 | ζ7 | ζ72 | ζ73 | ζ74 | ζ75 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ12 | 1 | -1 | ζ73 | ζ74 | ζ7 | ζ75 | ζ72 | ζ76 | -ζ73 | -ζ74 | -ζ7 | -ζ75 | -ζ72 | -ζ76 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 14 |
| ρ13 | 1 | 1 | ζ75 | ζ72 | ζ74 | ζ76 | ζ7 | ζ73 | ζ75 | ζ72 | ζ74 | ζ76 | ζ7 | ζ73 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ14 | 1 | 1 | ζ7 | ζ76 | ζ75 | ζ74 | ζ73 | ζ72 | ζ7 | ζ76 | ζ75 | ζ74 | ζ73 | ζ72 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ15 | 7 | -7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | -ζ2925-ζ2924-ζ2923-ζ2920-ζ2916-ζ297-ζ29 | -ζ2921-ζ2919-ζ2917-ζ2914-ζ2911-ζ293-ζ292 | -ζ2928-ζ2922-ζ2913-ζ299-ζ296-ζ295-ζ294 | -ζ2927-ζ2926-ζ2918-ζ2915-ζ2912-ζ2910-ζ298 | complex faithful |
| ρ16 | 7 | -7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | -ζ2921-ζ2919-ζ2917-ζ2914-ζ2911-ζ293-ζ292 | -ζ2928-ζ2922-ζ2913-ζ299-ζ296-ζ295-ζ294 | -ζ2927-ζ2926-ζ2918-ζ2915-ζ2912-ζ2910-ζ298 | -ζ2925-ζ2924-ζ2923-ζ2920-ζ2916-ζ297-ζ29 | complex faithful |
| ρ17 | 7 | -7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | -ζ2928-ζ2922-ζ2913-ζ299-ζ296-ζ295-ζ294 | -ζ2927-ζ2926-ζ2918-ζ2915-ζ2912-ζ2910-ζ298 | -ζ2925-ζ2924-ζ2923-ζ2920-ζ2916-ζ297-ζ29 | -ζ2921-ζ2919-ζ2917-ζ2914-ζ2911-ζ293-ζ292 | complex faithful |
| ρ18 | 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | complex lifted from C29⋊C7 |
| ρ19 | 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | complex lifted from C29⋊C7 |
| ρ20 | 7 | -7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | -ζ2927-ζ2926-ζ2918-ζ2915-ζ2912-ζ2910-ζ298 | -ζ2925-ζ2924-ζ2923-ζ2920-ζ2916-ζ297-ζ29 | -ζ2921-ζ2919-ζ2917-ζ2914-ζ2911-ζ293-ζ292 | -ζ2928-ζ2922-ζ2913-ζ299-ζ296-ζ295-ζ294 | complex faithful |
| ρ21 | 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | complex lifted from C29⋊C7 |
| ρ22 | 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 | ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 | ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 | ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 | complex lifted from C29⋊C7 |
►On 58 points
Generators in S58
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)
(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)
(2 17 25 8 26 24 21)(3 4 20 15 22 18 12)(5 7 10 29 14 6 23)(9 13 19 28 27 11 16)(31 46 54 37 55 53 50)(32 33 49 44 51 47 41)(34 36 39 58 43 35 52)(38 42 48 57 56 40 45)
G:=sub<Sym(58)| (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58), (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), (2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16)(31,46,54,37,55,53,50)(32,33,49,44,51,47,41)(34,36,39,58,43,35,52)(38,42,48,57,56,40,45)>;
G:=Group( (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58), (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), (2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16)(31,46,54,37,55,53,50)(32,33,49,44,51,47,41)(34,36,39,58,43,35,52)(38,42,48,57,56,40,45) );
G=PermutationGroup([[(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58)], [(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)], [(2,17,25,8,26,24,21),(3,4,20,15,22,18,12),(5,7,10,29,14,6,23),(9,13,19,28,27,11,16),(31,46,54,37,55,53,50),(32,33,49,44,51,47,41),(34,36,39,58,43,35,52),(38,42,48,57,56,40,45)]])
Matrix representation of C2×C29⋊C7 ►in GL8(𝔽2437)
| 2436 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 485 | 744 | 1424 | 1281 | 1950 | 2174 | 1 |
| 0 | 486 | 744 | 1424 | 1281 | 1950 | 2174 | 1 |
| 0 | 485 | 745 | 1424 | 1281 | 1950 | 2174 | 1 |
| 0 | 485 | 744 | 1425 | 1281 | 1950 | 2174 | 1 |
| 0 | 485 | 744 | 1424 | 1282 | 1950 | 2174 | 1 |
| 0 | 485 | 744 | 1424 | 1281 | 1951 | 2174 | 1 |
| 0 | 485 | 744 | 1424 | 1281 | 1950 | 2175 | 1 |
| 492 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 231 | 2091 | 870 | 491 | 139 | 1134 | 2210 |
| 0 | 2133 | 2405 | 382 | 74 | 2330 | 1191 | 1909 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1981 | 36 | 949 | 783 | 1279 | 1794 | 411 |
| 0 | 232 | 1126 | 1493 | 2330 | 2392 | 1386 | 2026 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 151 | 414 | 591 | 1996 | 1312 | 1570 | 1500 |
G:=sub<GL(8,GF(2437))| [2436,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,485,486,485,485,485,485,485,0,744,744,745,744,744,744,744,0,1424,1424,1424,1425,1424,1424,1424,0,1281,1281,1281,1281,1282,1281,1281,0,1950,1950,1950,1950,1950,1951,1950,0,2174,2174,2174,2174,2174,2174,2175,0,1,1,1,1,1,1,1],[492,0,0,0,0,0,0,0,0,231,2133,0,1981,232,0,151,0,2091,2405,1,36,1126,0,414,0,870,382,0,949,1493,0,591,0,491,74,0,783,2330,1,1996,0,139,2330,0,1279,2392,0,1312,0,1134,1191,0,1794,1386,0,1570,0,2210,1909,0,411,2026,0,1500] >;
C2×C29⋊C7 in GAP, Magma, Sage, TeX
C_2\times C_{29}\rtimes C_7 % in TeX
G:=Group("C2xC29:C7"); // GroupNames label
G:=SmallGroup(406,2);
// by ID
G=gap.SmallGroup(406,2);
# by ID
G:=PCGroup([3,-2,-7,-29,1013]);
// Polycyclic
G:=Group<a,b,c|a^2=b^29=c^7=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^20>;
// generators/relations
Export
Subgroup lattice of C2×C29⋊C7 in TeX
Character table of C2×C29⋊C7 in TeX