direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D19, C38⋊3C6, C114⋊2C2, C57⋊3C22, C19⋊3(C2×C6), SmallGroup(228,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C6×D19 |
Generators and relations for C6×D19
G = < a,b,c | a6=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D19
►On 114 points
Generators in S114
(1 95 48 60 21 103)(2 77 49 61 22 104)(3 78 50 62 23 105)(4 79 51 63 24 106)(5 80 52 64 25 107)(6 81 53 65 26 108)(7 82 54 66 27 109)(8 83 55 67 28 110)(9 84 56 68 29 111)(10 85 57 69 30 112)(11 86 39 70 31 113)(12 87 40 71 32 114)(13 88 41 72 33 96)(14 89 42 73 34 97)(15 90 43 74 35 98)(16 91 44 75 36 99)(17 92 45 76 37 100)(18 93 46 58 38 101)(19 94 47 59 20 102)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)
(1 59)(2 58)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 95)(21 94)(22 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 84)(32 83)(33 82)(34 81)(35 80)(36 79)(37 78)(38 77)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 114)(56 113)(57 112)
G:=sub<Sym(114)| (1,95,48,60,21,103)(2,77,49,61,22,104)(3,78,50,62,23,105)(4,79,51,63,24,106)(5,80,52,64,25,107)(6,81,53,65,26,108)(7,82,54,66,27,109)(8,83,55,67,28,110)(9,84,56,68,29,111)(10,85,57,69,30,112)(11,86,39,70,31,113)(12,87,40,71,32,114)(13,88,41,72,33,96)(14,89,42,73,34,97)(15,90,43,74,35,98)(16,91,44,75,36,99)(17,92,45,76,37,100)(18,93,46,58,38,101)(19,94,47,59,20,102), (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)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,59)(2,58)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,114)(56,113)(57,112)>;
G:=Group( (1,95,48,60,21,103)(2,77,49,61,22,104)(3,78,50,62,23,105)(4,79,51,63,24,106)(5,80,52,64,25,107)(6,81,53,65,26,108)(7,82,54,66,27,109)(8,83,55,67,28,110)(9,84,56,68,29,111)(10,85,57,69,30,112)(11,86,39,70,31,113)(12,87,40,71,32,114)(13,88,41,72,33,96)(14,89,42,73,34,97)(15,90,43,74,35,98)(16,91,44,75,36,99)(17,92,45,76,37,100)(18,93,46,58,38,101)(19,94,47,59,20,102), (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)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,59)(2,58)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,114)(56,113)(57,112) );
G=PermutationGroup([[(1,95,48,60,21,103),(2,77,49,61,22,104),(3,78,50,62,23,105),(4,79,51,63,24,106),(5,80,52,64,25,107),(6,81,53,65,26,108),(7,82,54,66,27,109),(8,83,55,67,28,110),(9,84,56,68,29,111),(10,85,57,69,30,112),(11,86,39,70,31,113),(12,87,40,71,32,114),(13,88,41,72,33,96),(14,89,42,73,34,97),(15,90,43,74,35,98),(16,91,44,75,36,99),(17,92,45,76,37,100),(18,93,46,58,38,101),(19,94,47,59,20,102)], [(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),(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)], [(1,59),(2,58),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,60),(20,95),(21,94),(22,93),(23,92),(24,91),(25,90),(26,89),(27,88),(28,87),(29,86),(30,85),(31,84),(32,83),(33,82),(34,81),(35,80),(36,79),(37,78),(38,77),(39,111),(40,110),(41,109),(42,108),(43,107),(44,106),(45,105),(46,104),(47,103),(48,102),(49,101),(50,100),(51,99),(52,98),(53,97),(54,96),(55,114),(56,113),(57,112)]])
C6×D19 is a maximal subgroup of
C57⋊D4 C3⋊D76
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 114A | ··· | 114R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D19 | D38 | C3×D19 | C6×D19 |
| kernel | C6×D19 | C3×D19 | C114 | D38 | D19 | C38 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 9 | 9 | 18 | 18 |
Matrix representation of C6×D19 ►in GL2(𝔽37) generated by
| 11 | 0 |
| 0 | 11 |
| 11 | 4 |
| 34 | 36 |
| 1 | 0 |
| 34 | 36 |
G:=sub<GL(2,GF(37))| [11,0,0,11],[11,34,4,36],[1,34,0,36] >;
C6×D19 in GAP, Magma, Sage, TeX
C_6\times D_{19} % in TeX
G:=Group("C6xD19"); // GroupNames label
G:=SmallGroup(228,12);
// by ID
G=gap.SmallGroup(228,12);
# by ID
G:=PCGroup([4,-2,-2,-3,-19,3459]);
// Polycyclic
G:=Group<a,b,c|a^6=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export