direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D69, C69⋊1C6, C69⋊2S3, C32⋊1D23, C23⋊(C3×S3), C3⋊(C3×D23), (C3×C69)⋊2C2, SmallGroup(414,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C69 — C3×D69 |
Generators and relations for C3×D69
G = < a,b,c | a3=b69=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D69
►On 138 points
Generators in S138
(1 24 47)(2 25 48)(3 26 49)(4 27 50)(5 28 51)(6 29 52)(7 30 53)(8 31 54)(9 32 55)(10 33 56)(11 34 57)(12 35 58)(13 36 59)(14 37 60)(15 38 61)(16 39 62)(17 40 63)(18 41 64)(19 42 65)(20 43 66)(21 44 67)(22 45 68)(23 46 69)(70 116 93)(71 117 94)(72 118 95)(73 119 96)(74 120 97)(75 121 98)(76 122 99)(77 123 100)(78 124 101)(79 125 102)(80 126 103)(81 127 104)(82 128 105)(83 129 106)(84 130 107)(85 131 108)(86 132 109)(87 133 110)(88 134 111)(89 135 112)(90 136 113)(91 137 114)(92 138 115)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138)
(1 130)(2 129)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 122)(10 121)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 102)(30 101)(31 100)(32 99)(33 98)(34 97)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 70)(62 138)(63 137)(64 136)(65 135)(66 134)(67 133)(68 132)(69 131)
G:=sub<Sym(138)| (1,24,47)(2,25,48)(3,26,49)(4,27,50)(5,28,51)(6,29,52)(7,30,53)(8,31,54)(9,32,55)(10,33,56)(11,34,57)(12,35,58)(13,36,59)(14,37,60)(15,38,61)(16,39,62)(17,40,63)(18,41,64)(19,42,65)(20,43,66)(21,44,67)(22,45,68)(23,46,69)(70,116,93)(71,117,94)(72,118,95)(73,119,96)(74,120,97)(75,121,98)(76,122,99)(77,123,100)(78,124,101)(79,125,102)(80,126,103)(81,127,104)(82,128,105)(83,129,106)(84,130,107)(85,131,108)(86,132,109)(87,133,110)(88,134,111)(89,135,112)(90,136,113)(91,137,114)(92,138,115), (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)(70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,132)(69,131)>;
G:=Group( (1,24,47)(2,25,48)(3,26,49)(4,27,50)(5,28,51)(6,29,52)(7,30,53)(8,31,54)(9,32,55)(10,33,56)(11,34,57)(12,35,58)(13,36,59)(14,37,60)(15,38,61)(16,39,62)(17,40,63)(18,41,64)(19,42,65)(20,43,66)(21,44,67)(22,45,68)(23,46,69)(70,116,93)(71,117,94)(72,118,95)(73,119,96)(74,120,97)(75,121,98)(76,122,99)(77,123,100)(78,124,101)(79,125,102)(80,126,103)(81,127,104)(82,128,105)(83,129,106)(84,130,107)(85,131,108)(86,132,109)(87,133,110)(88,134,111)(89,135,112)(90,136,113)(91,137,114)(92,138,115), (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)(70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,132)(69,131) );
G=PermutationGroup([[(1,24,47),(2,25,48),(3,26,49),(4,27,50),(5,28,51),(6,29,52),(7,30,53),(8,31,54),(9,32,55),(10,33,56),(11,34,57),(12,35,58),(13,36,59),(14,37,60),(15,38,61),(16,39,62),(17,40,63),(18,41,64),(19,42,65),(20,43,66),(21,44,67),(22,45,68),(23,46,69),(70,116,93),(71,117,94),(72,118,95),(73,119,96),(74,120,97),(75,121,98),(76,122,99),(77,123,100),(78,124,101),(79,125,102),(80,126,103),(81,127,104),(82,128,105),(83,129,106),(84,130,107),(85,131,108),(86,132,109),(87,133,110),(88,134,111),(89,135,112),(90,136,113),(91,137,114),(92,138,115)], [(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),(70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138)], [(1,130),(2,129),(3,128),(4,127),(5,126),(6,125),(7,124),(8,123),(9,122),(10,121),(11,120),(12,119),(13,118),(14,117),(15,116),(16,115),(17,114),(18,113),(19,112),(20,111),(21,110),(22,109),(23,108),(24,107),(25,106),(26,105),(27,104),(28,103),(29,102),(30,101),(31,100),(32,99),(33,98),(34,97),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,70),(62,138),(63,137),(64,136),(65,135),(66,134),(67,133),(68,132),(69,131)]])
108 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 23A | ··· | 23K | 69A | ··· | 69CJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 23 | ··· | 23 | 69 | ··· | 69 |
| size | 1 | 69 | 1 | 1 | 2 | 2 | 2 | 69 | 69 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | D23 | C3×D23 | D69 | C3×D69 |
| kernel | C3×D69 | C3×C69 | D69 | C69 | C69 | C23 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 11 | 22 | 22 | 44 |
Matrix representation of C3×D69 ►in GL2(𝔽139) generated by
| 42 | 0 |
| 0 | 42 |
| 35 | 0 |
| 0 | 4 |
| 0 | 4 |
| 35 | 0 |
G:=sub<GL(2,GF(139))| [42,0,0,42],[35,0,0,4],[0,35,4,0] >;
C3×D69 in GAP, Magma, Sage, TeX
C_3\times D_{69} % in TeX
G:=Group("C3xD69"); // GroupNames label
G:=SmallGroup(414,7);
// by ID
G=gap.SmallGroup(414,7);
# by ID
G:=PCGroup([4,-2,-3,-3,-23,146,6339]);
// Polycyclic
G:=Group<a,b,c|a^3=b^69=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export