direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C51, C3⋊C102, C51⋊3C6, C32⋊1C34, (C3×C51)⋊4C2, SmallGroup(306,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C51 |
Generators and relations for S3×C51
G = < a,b,c | a51=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C51
►On 102 points
Generators in S102
(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 35 18)(2 36 19)(3 37 20)(4 38 21)(5 39 22)(6 40 23)(7 41 24)(8 42 25)(9 43 26)(10 44 27)(11 45 28)(12 46 29)(13 47 30)(14 48 31)(15 49 32)(16 50 33)(17 51 34)(52 69 86)(53 70 87)(54 71 88)(55 72 89)(56 73 90)(57 74 91)(58 75 92)(59 76 93)(60 77 94)(61 78 95)(62 79 96)(63 80 97)(64 81 98)(65 82 99)(66 83 100)(67 84 101)(68 85 102)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)
G:=sub<Sym(102)| (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), (1,35,18)(2,36,19)(3,37,20)(4,38,21)(5,39,22)(6,40,23)(7,41,24)(8,42,25)(9,43,26)(10,44,27)(11,45,28)(12,46,29)(13,47,30)(14,48,31)(15,49,32)(16,50,33)(17,51,34)(52,69,86)(53,70,87)(54,71,88)(55,72,89)(56,73,90)(57,74,91)(58,75,92)(59,76,93)(60,77,94)(61,78,95)(62,79,96)(63,80,97)(64,81,98)(65,82,99)(66,83,100)(67,84,101)(68,85,102), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)>;
G:=Group( (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), (1,35,18)(2,36,19)(3,37,20)(4,38,21)(5,39,22)(6,40,23)(7,41,24)(8,42,25)(9,43,26)(10,44,27)(11,45,28)(12,46,29)(13,47,30)(14,48,31)(15,49,32)(16,50,33)(17,51,34)(52,69,86)(53,70,87)(54,71,88)(55,72,89)(56,73,90)(57,74,91)(58,75,92)(59,76,93)(60,77,94)(61,78,95)(62,79,96)(63,80,97)(64,81,98)(65,82,99)(66,83,100)(67,84,101)(68,85,102), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81) );
G=PermutationGroup([[(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)], [(1,35,18),(2,36,19),(3,37,20),(4,38,21),(5,39,22),(6,40,23),(7,41,24),(8,42,25),(9,43,26),(10,44,27),(11,45,28),(12,46,29),(13,47,30),(14,48,31),(15,49,32),(16,50,33),(17,51,34),(52,69,86),(53,70,87),(54,71,88),(55,72,89),(56,73,90),(57,74,91),(58,75,92),(59,76,93),(60,77,94),(61,78,95),(62,79,96),(63,80,97),(64,81,98),(65,82,99),(66,83,100),(67,84,101),(68,85,102)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81)]])
153 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51AF | 51AG | ··· | 51CB | 102A | ··· | 102AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
153 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C17 | C34 | C51 | C102 | S3 | C3×S3 | S3×C17 | S3×C51 |
| kernel | S3×C51 | C3×C51 | S3×C17 | C51 | C3×S3 | C32 | S3 | C3 | C51 | C17 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 16 | 16 | 32 | 32 | 1 | 2 | 16 | 32 |
Matrix representation of S3×C51 ►in GL2(𝔽103) generated by
| 19 | 0 |
| 0 | 19 |
| 46 | 99 |
| 0 | 56 |
| 85 | 18 |
| 45 | 18 |
G:=sub<GL(2,GF(103))| [19,0,0,19],[46,0,99,56],[85,45,18,18] >;
S3×C51 in GAP, Magma, Sage, TeX
S_3\times C_{51} % in TeX
G:=Group("S3xC51"); // GroupNames label
G:=SmallGroup(306,6);
// by ID
G=gap.SmallGroup(306,6);
# by ID
G:=PCGroup([4,-2,-3,-17,-3,3267]);
// Polycyclic
G:=Group<a,b,c|a^51=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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