direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D51, C51⋊1C6, C51⋊2S3, C32⋊1D17, C17⋊(C3×S3), C3⋊(C3×D17), (C3×C51)⋊2C2, SmallGroup(306,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C3×D51 |
Generators and relations for C3×D51
G = < a,b,c | a3=b51=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D51
►On 102 points
Generators in S102
(1 18 35)(2 19 36)(3 20 37)(4 21 38)(5 22 39)(6 23 40)(7 24 41)(8 25 42)(9 26 43)(10 27 44)(11 28 45)(12 29 46)(13 30 47)(14 31 48)(15 32 49)(16 33 50)(17 34 51)(52 86 69)(53 87 70)(54 88 71)(55 89 72)(56 90 73)(57 91 74)(58 92 75)(59 93 76)(60 94 77)(61 95 78)(62 96 79)(63 97 80)(64 98 81)(65 99 82)(66 100 83)(67 101 84)(68 102 85)
(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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102)
(1 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 84)(17 83)(18 82)(19 81)(20 80)(21 79)(22 78)(23 77)(24 76)(25 75)(26 74)(27 73)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 102)(50 101)(51 100)
G:=sub<Sym(102)| (1,18,35)(2,19,36)(3,20,37)(4,21,38)(5,22,39)(6,23,40)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,28,45)(12,29,46)(13,30,47)(14,31,48)(15,32,49)(16,33,50)(17,34,51)(52,86,69)(53,87,70)(54,88,71)(55,89,72)(56,90,73)(57,91,74)(58,92,75)(59,93,76)(60,94,77)(61,95,78)(62,96,79)(63,97,80)(64,98,81)(65,99,82)(66,100,83)(67,101,84)(68,102,85), (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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,84)(17,83)(18,82)(19,81)(20,80)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,102)(50,101)(51,100)>;
G:=Group( (1,18,35)(2,19,36)(3,20,37)(4,21,38)(5,22,39)(6,23,40)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,28,45)(12,29,46)(13,30,47)(14,31,48)(15,32,49)(16,33,50)(17,34,51)(52,86,69)(53,87,70)(54,88,71)(55,89,72)(56,90,73)(57,91,74)(58,92,75)(59,93,76)(60,94,77)(61,95,78)(62,96,79)(63,97,80)(64,98,81)(65,99,82)(66,100,83)(67,101,84)(68,102,85), (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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,84)(17,83)(18,82)(19,81)(20,80)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,102)(50,101)(51,100) );
G=PermutationGroup([[(1,18,35),(2,19,36),(3,20,37),(4,21,38),(5,22,39),(6,23,40),(7,24,41),(8,25,42),(9,26,43),(10,27,44),(11,28,45),(12,29,46),(13,30,47),(14,31,48),(15,32,49),(16,33,50),(17,34,51),(52,86,69),(53,87,70),(54,88,71),(55,89,72),(56,90,73),(57,91,74),(58,92,75),(59,93,76),(60,94,77),(61,95,78),(62,96,79),(63,97,80),(64,98,81),(65,99,82),(66,100,83),(67,101,84),(68,102,85)], [(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),(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102)], [(1,99),(2,98),(3,97),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,90),(11,89),(12,88),(13,87),(14,86),(15,85),(16,84),(17,83),(18,82),(19,81),(20,80),(21,79),(22,78),(23,77),(24,76),(25,75),(26,74),(27,73),(28,72),(29,71),(30,70),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,102),(50,101),(51,100)]])
81 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 17A | ··· | 17H | 51A | ··· | 51BL |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 17 | ··· | 17 | 51 | ··· | 51 |
| size | 1 | 51 | 1 | 1 | 2 | 2 | 2 | 51 | 51 | 2 | ··· | 2 | 2 | ··· | 2 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | D17 | C3×D17 | D51 | C3×D51 |
| kernel | C3×D51 | C3×C51 | D51 | C51 | C51 | C17 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 8 | 16 | 16 | 32 |
Matrix representation of C3×D51 ►in GL4(𝔽103) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 56 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 29 | 25 |
| 0 | 0 | 27 | 2 |
| 0 | 46 | 0 | 0 |
| 56 | 0 | 0 | 0 |
| 0 | 0 | 56 | 81 |
| 0 | 0 | 91 | 47 |
G:=sub<GL(4,GF(103))| [46,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[56,0,0,0,0,46,0,0,0,0,29,27,0,0,25,2],[0,56,0,0,46,0,0,0,0,0,56,91,0,0,81,47] >;
C3×D51 in GAP, Magma, Sage, TeX
C_3\times D_{51} % in TeX
G:=Group("C3xD51"); // GroupNames label
G:=SmallGroup(306,7);
// by ID
G=gap.SmallGroup(306,7);
# by ID
G:=PCGroup([4,-2,-3,-3,-17,146,4611]);
// Polycyclic
G:=Group<a,b,c|a^3=b^51=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export