direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C51⋊C4, D34.S3, C34⋊Dic3, C102⋊1C4, D17⋊Dic3, D17.2D6, C6⋊(C17⋊C4), C51⋊2(C2×C4), C17⋊(C2×Dic3), (C3×D17)⋊2C4, (C6×D17).2C2, (C3×D17).2C22, C3⋊2(C2×C17⋊C4), SmallGroup(408,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C2×C51⋊C4 |
Generators and relations for C2×C51⋊C4
G = < a,b,c | a2=b51=c4=1, ab=ba, ac=ca, cbc-1=b47 >
Smallest permutation representation of C2×C51⋊C4
►On 102 points
Generators in S102
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)
(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 79)(2 66 17 75)(3 53 33 71)(4 91 49 67)(5 78 14 63)(6 65 30 59)(7 52 46 55)(8 90 11 102)(9 77 27 98)(10 64 43 94)(12 89 24 86)(13 76 40 82)(15 101 21 74)(16 88 37 70)(18 62)(19 100 34 58)(20 87 50 54)(22 61 31 97)(23 99 47 93)(25 73 28 85)(26 60 44 81)(29 72 41 69)(32 84 38 57)(35 96)(36 83 51 92)(39 95 48 80)(42 56 45 68)
G:=sub<Sym(102)| (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78), (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), (1,79)(2,66,17,75)(3,53,33,71)(4,91,49,67)(5,78,14,63)(6,65,30,59)(7,52,46,55)(8,90,11,102)(9,77,27,98)(10,64,43,94)(12,89,24,86)(13,76,40,82)(15,101,21,74)(16,88,37,70)(18,62)(19,100,34,58)(20,87,50,54)(22,61,31,97)(23,99,47,93)(25,73,28,85)(26,60,44,81)(29,72,41,69)(32,84,38,57)(35,96)(36,83,51,92)(39,95,48,80)(42,56,45,68)>;
G:=Group( (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78), (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), (1,79)(2,66,17,75)(3,53,33,71)(4,91,49,67)(5,78,14,63)(6,65,30,59)(7,52,46,55)(8,90,11,102)(9,77,27,98)(10,64,43,94)(12,89,24,86)(13,76,40,82)(15,101,21,74)(16,88,37,70)(18,62)(19,100,34,58)(20,87,50,54)(22,61,31,97)(23,99,47,93)(25,73,28,85)(26,60,44,81)(29,72,41,69)(32,84,38,57)(35,96)(36,83,51,92)(39,95,48,80)(42,56,45,68) );
G=PermutationGroup([[(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78)], [(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)], [(1,79),(2,66,17,75),(3,53,33,71),(4,91,49,67),(5,78,14,63),(6,65,30,59),(7,52,46,55),(8,90,11,102),(9,77,27,98),(10,64,43,94),(12,89,24,86),(13,76,40,82),(15,101,21,74),(16,88,37,70),(18,62),(19,100,34,58),(20,87,50,54),(22,61,31,97),(23,99,47,93),(25,73,28,85),(26,60,44,81),(29,72,41,69),(32,84,38,57),(35,96),(36,83,51,92),(39,95,48,80),(42,56,45,68)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 51A | ··· | 51H | 102A | ··· | 102H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 17 | 17 | 17 | 17 | 34 | 34 | 34 | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 17 | 17 | 2 | 51 | 51 | 51 | 51 | 2 | 34 | 34 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | C17⋊C4 | C2×C17⋊C4 | C51⋊C4 | C2×C51⋊C4 |
| kernel | C2×C51⋊C4 | C51⋊C4 | C6×D17 | C3×D17 | C102 | D34 | D17 | D17 | C34 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 |
Matrix representation of C2×C51⋊C4 ►in GL6(𝔽409)
| 408 | 0 | 0 | 0 | 0 | 0 |
| 0 | 408 | 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 |
| 408 | 408 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 294 | 218 | 269 | 318 |
| 0 | 0 | 91 | 24 | 1 | 408 |
| 0 | 0 | 1 | 115 | 116 | 25 |
| 0 | 0 | 384 | 219 | 269 | 334 |
| 266 | 0 | 0 | 0 | 0 | 0 |
| 143 | 143 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 115 | 116 | 25 |
| 0 | 0 | 318 | 269 | 218 | 294 |
| 0 | 0 | 24 | 166 | 139 | 75 |
G:=sub<GL(6,GF(409))| [408,0,0,0,0,0,0,408,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[408,1,0,0,0,0,408,0,0,0,0,0,0,0,294,91,1,384,0,0,218,24,115,219,0,0,269,1,116,269,0,0,318,408,25,334],[266,143,0,0,0,0,0,143,0,0,0,0,0,0,1,1,318,24,0,0,0,115,269,166,0,0,0,116,218,139,0,0,0,25,294,75] >;
C2×C51⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{51}\rtimes C_4 % in TeX
G:=Group("C2xC51:C4"); // GroupNames label
G:=SmallGroup(408,40);
// by ID
G=gap.SmallGroup(408,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,323,2404,2414]);
// Polycyclic
G:=Group<a,b,c|a^2=b^51=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^47>;
// generators/relations
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