direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D17, C34⋊C6, C102⋊2C2, C51⋊3C22, C17⋊(C2×C6), SmallGroup(204,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C6×D17 |
Generators and relations for C6×D17
G = < a,b,c | a6=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D17
►On 102 points
Generators in S102
(1 84 47 54 33 92)(2 85 48 55 34 93)(3 69 49 56 18 94)(4 70 50 57 19 95)(5 71 51 58 20 96)(6 72 35 59 21 97)(7 73 36 60 22 98)(8 74 37 61 23 99)(9 75 38 62 24 100)(10 76 39 63 25 101)(11 77 40 64 26 102)(12 78 41 65 27 86)(13 79 42 66 28 87)(14 80 43 67 29 88)(15 81 44 68 30 89)(16 82 45 52 31 90)(17 83 46 53 32 91)
(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 53)(2 52)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 85)(32 84)(33 83)(34 82)(35 86)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)
G:=sub<Sym(102)| (1,84,47,54,33,92)(2,85,48,55,34,93)(3,69,49,56,18,94)(4,70,50,57,19,95)(5,71,51,58,20,96)(6,72,35,59,21,97)(7,73,36,60,22,98)(8,74,37,61,23,99)(9,75,38,62,24,100)(10,76,39,63,25,101)(11,77,40,64,26,102)(12,78,41,65,27,86)(13,79,42,66,28,87)(14,80,43,67,29,88)(15,81,44,68,30,89)(16,82,45,52,31,90)(17,83,46,53,32,91), (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), (1,53)(2,52)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,85)(32,84)(33,83)(34,82)(35,86)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)>;
G:=Group( (1,84,47,54,33,92)(2,85,48,55,34,93)(3,69,49,56,18,94)(4,70,50,57,19,95)(5,71,51,58,20,96)(6,72,35,59,21,97)(7,73,36,60,22,98)(8,74,37,61,23,99)(9,75,38,62,24,100)(10,76,39,63,25,101)(11,77,40,64,26,102)(12,78,41,65,27,86)(13,79,42,66,28,87)(14,80,43,67,29,88)(15,81,44,68,30,89)(16,82,45,52,31,90)(17,83,46,53,32,91), (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), (1,53)(2,52)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,85)(32,84)(33,83)(34,82)(35,86)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87) );
G=PermutationGroup([[(1,84,47,54,33,92),(2,85,48,55,34,93),(3,69,49,56,18,94),(4,70,50,57,19,95),(5,71,51,58,20,96),(6,72,35,59,21,97),(7,73,36,60,22,98),(8,74,37,61,23,99),(9,75,38,62,24,100),(10,76,39,63,25,101),(11,77,40,64,26,102),(12,78,41,65,27,86),(13,79,42,66,28,87),(14,80,43,67,29,88),(15,81,44,68,30,89),(16,82,45,52,31,90),(17,83,46,53,32,91)], [(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)], [(1,53),(2,52),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,81),(19,80),(20,79),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,70),(30,69),(31,85),(32,84),(33,83),(34,82),(35,86),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87)]])
C6×D17 is a maximal subgroup of
C51⋊D4 C3⋊D68
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 102A | ··· | 102P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D17 | D34 | C3×D17 | C6×D17 |
| kernel | C6×D17 | C3×D17 | C102 | D34 | D17 | C34 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 8 | 8 | 16 | 16 |
Matrix representation of C6×D17 ►in GL3(𝔽103) generated by
| 102 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 0 | 97 | 1 |
| 0 | 68 | 40 |
| 1 | 0 | 0 |
| 0 | 40 | 102 |
| 0 | 54 | 63 |
G:=sub<GL(3,GF(103))| [102,0,0,0,46,0,0,0,46],[1,0,0,0,97,68,0,1,40],[1,0,0,0,40,54,0,102,63] >;
C6×D17 in GAP, Magma, Sage, TeX
C_6\times D_{17} % in TeX
G:=Group("C6xD17"); // GroupNames label
G:=SmallGroup(204,9);
// by ID
G=gap.SmallGroup(204,9);
# by ID
G:=PCGroup([4,-2,-2,-3,-17,3075]);
// Polycyclic
G:=Group<a,b,c|a^6=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export