direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C7⋊D7, C21⋊2D7, C72⋊12C6, (C7×C21)⋊4C2, C7⋊3(C3×D7), SmallGroup(294,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C3×C7⋊D7 |
Generators and relations for C3×C7⋊D7
G = < a,b,c,d | a3=b7=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C3×C7⋊D7
►On 147 points
Generators in S147
(1 16 106)(2 17 107)(3 18 108)(4 19 109)(5 20 110)(6 21 111)(7 15 112)(8 105 56)(9 99 50)(10 100 51)(11 101 52)(12 102 53)(13 103 54)(14 104 55)(22 119 70)(23 113 64)(24 114 65)(25 115 66)(26 116 67)(27 117 68)(28 118 69)(29 126 77)(30 120 71)(31 121 72)(32 122 73)(33 123 74)(34 124 75)(35 125 76)(36 133 84)(37 127 78)(38 128 79)(39 129 80)(40 130 81)(41 131 82)(42 132 83)(43 140 91)(44 134 85)(45 135 86)(46 136 87)(47 137 88)(48 138 89)(49 139 90)(57 146 97)(58 147 98)(59 141 92)(60 142 93)(61 143 94)(62 144 95)(63 145 96)
(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)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)
(1 58 83 68 53 87 72)(2 59 84 69 54 88 73)(3 60 78 70 55 89 74)(4 61 79 64 56 90 75)(5 62 80 65 50 91 76)(6 63 81 66 51 85 77)(7 57 82 67 52 86 71)(8 49 34 19 143 38 23)(9 43 35 20 144 39 24)(10 44 29 21 145 40 25)(11 45 30 15 146 41 26)(12 46 31 16 147 42 27)(13 47 32 17 141 36 28)(14 48 33 18 142 37 22)(92 133 118 103 137 122 107)(93 127 119 104 138 123 108)(94 128 113 105 139 124 109)(95 129 114 99 140 125 110)(96 130 115 100 134 126 111)(97 131 116 101 135 120 112)(98 132 117 102 136 121 106)
(1 72)(2 71)(3 77)(4 76)(5 75)(6 74)(7 73)(8 39)(9 38)(10 37)(11 36)(12 42)(13 41)(14 40)(15 32)(16 31)(17 30)(18 29)(19 35)(20 34)(21 33)(22 25)(23 24)(26 28)(43 143)(44 142)(45 141)(46 147)(47 146)(48 145)(49 144)(50 79)(51 78)(52 84)(53 83)(54 82)(55 81)(56 80)(57 88)(58 87)(59 86)(60 85)(61 91)(62 90)(63 89)(64 65)(66 70)(67 69)(92 135)(93 134)(94 140)(95 139)(96 138)(97 137)(98 136)(99 128)(100 127)(101 133)(102 132)(103 131)(104 130)(105 129)(106 121)(107 120)(108 126)(109 125)(110 124)(111 123)(112 122)(113 114)(115 119)(116 118)
G:=sub<Sym(147)| (1,16,106)(2,17,107)(3,18,108)(4,19,109)(5,20,110)(6,21,111)(7,15,112)(8,105,56)(9,99,50)(10,100,51)(11,101,52)(12,102,53)(13,103,54)(14,104,55)(22,119,70)(23,113,64)(24,114,65)(25,115,66)(26,116,67)(27,117,68)(28,118,69)(29,126,77)(30,120,71)(31,121,72)(32,122,73)(33,123,74)(34,124,75)(35,125,76)(36,133,84)(37,127,78)(38,128,79)(39,129,80)(40,130,81)(41,131,82)(42,132,83)(43,140,91)(44,134,85)(45,135,86)(46,136,87)(47,137,88)(48,138,89)(49,139,90)(57,146,97)(58,147,98)(59,141,92)(60,142,93)(61,143,94)(62,144,95)(63,145,96), (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)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147), (1,58,83,68,53,87,72)(2,59,84,69,54,88,73)(3,60,78,70,55,89,74)(4,61,79,64,56,90,75)(5,62,80,65,50,91,76)(6,63,81,66,51,85,77)(7,57,82,67,52,86,71)(8,49,34,19,143,38,23)(9,43,35,20,144,39,24)(10,44,29,21,145,40,25)(11,45,30,15,146,41,26)(12,46,31,16,147,42,27)(13,47,32,17,141,36,28)(14,48,33,18,142,37,22)(92,133,118,103,137,122,107)(93,127,119,104,138,123,108)(94,128,113,105,139,124,109)(95,129,114,99,140,125,110)(96,130,115,100,134,126,111)(97,131,116,101,135,120,112)(98,132,117,102,136,121,106), (1,72)(2,71)(3,77)(4,76)(5,75)(6,74)(7,73)(8,39)(9,38)(10,37)(11,36)(12,42)(13,41)(14,40)(15,32)(16,31)(17,30)(18,29)(19,35)(20,34)(21,33)(22,25)(23,24)(26,28)(43,143)(44,142)(45,141)(46,147)(47,146)(48,145)(49,144)(50,79)(51,78)(52,84)(53,83)(54,82)(55,81)(56,80)(57,88)(58,87)(59,86)(60,85)(61,91)(62,90)(63,89)(64,65)(66,70)(67,69)(92,135)(93,134)(94,140)(95,139)(96,138)(97,137)(98,136)(99,128)(100,127)(101,133)(102,132)(103,131)(104,130)(105,129)(106,121)(107,120)(108,126)(109,125)(110,124)(111,123)(112,122)(113,114)(115,119)(116,118)>;
G:=Group( (1,16,106)(2,17,107)(3,18,108)(4,19,109)(5,20,110)(6,21,111)(7,15,112)(8,105,56)(9,99,50)(10,100,51)(11,101,52)(12,102,53)(13,103,54)(14,104,55)(22,119,70)(23,113,64)(24,114,65)(25,115,66)(26,116,67)(27,117,68)(28,118,69)(29,126,77)(30,120,71)(31,121,72)(32,122,73)(33,123,74)(34,124,75)(35,125,76)(36,133,84)(37,127,78)(38,128,79)(39,129,80)(40,130,81)(41,131,82)(42,132,83)(43,140,91)(44,134,85)(45,135,86)(46,136,87)(47,137,88)(48,138,89)(49,139,90)(57,146,97)(58,147,98)(59,141,92)(60,142,93)(61,143,94)(62,144,95)(63,145,96), (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)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147), (1,58,83,68,53,87,72)(2,59,84,69,54,88,73)(3,60,78,70,55,89,74)(4,61,79,64,56,90,75)(5,62,80,65,50,91,76)(6,63,81,66,51,85,77)(7,57,82,67,52,86,71)(8,49,34,19,143,38,23)(9,43,35,20,144,39,24)(10,44,29,21,145,40,25)(11,45,30,15,146,41,26)(12,46,31,16,147,42,27)(13,47,32,17,141,36,28)(14,48,33,18,142,37,22)(92,133,118,103,137,122,107)(93,127,119,104,138,123,108)(94,128,113,105,139,124,109)(95,129,114,99,140,125,110)(96,130,115,100,134,126,111)(97,131,116,101,135,120,112)(98,132,117,102,136,121,106), (1,72)(2,71)(3,77)(4,76)(5,75)(6,74)(7,73)(8,39)(9,38)(10,37)(11,36)(12,42)(13,41)(14,40)(15,32)(16,31)(17,30)(18,29)(19,35)(20,34)(21,33)(22,25)(23,24)(26,28)(43,143)(44,142)(45,141)(46,147)(47,146)(48,145)(49,144)(50,79)(51,78)(52,84)(53,83)(54,82)(55,81)(56,80)(57,88)(58,87)(59,86)(60,85)(61,91)(62,90)(63,89)(64,65)(66,70)(67,69)(92,135)(93,134)(94,140)(95,139)(96,138)(97,137)(98,136)(99,128)(100,127)(101,133)(102,132)(103,131)(104,130)(105,129)(106,121)(107,120)(108,126)(109,125)(110,124)(111,123)(112,122)(113,114)(115,119)(116,118) );
G=PermutationGroup([[(1,16,106),(2,17,107),(3,18,108),(4,19,109),(5,20,110),(6,21,111),(7,15,112),(8,105,56),(9,99,50),(10,100,51),(11,101,52),(12,102,53),(13,103,54),(14,104,55),(22,119,70),(23,113,64),(24,114,65),(25,115,66),(26,116,67),(27,117,68),(28,118,69),(29,126,77),(30,120,71),(31,121,72),(32,122,73),(33,123,74),(34,124,75),(35,125,76),(36,133,84),(37,127,78),(38,128,79),(39,129,80),(40,130,81),(41,131,82),(42,132,83),(43,140,91),(44,134,85),(45,135,86),(46,136,87),(47,137,88),(48,138,89),(49,139,90),(57,146,97),(58,147,98),(59,141,92),(60,142,93),(61,143,94),(62,144,95),(63,145,96)], [(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),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147)], [(1,58,83,68,53,87,72),(2,59,84,69,54,88,73),(3,60,78,70,55,89,74),(4,61,79,64,56,90,75),(5,62,80,65,50,91,76),(6,63,81,66,51,85,77),(7,57,82,67,52,86,71),(8,49,34,19,143,38,23),(9,43,35,20,144,39,24),(10,44,29,21,145,40,25),(11,45,30,15,146,41,26),(12,46,31,16,147,42,27),(13,47,32,17,141,36,28),(14,48,33,18,142,37,22),(92,133,118,103,137,122,107),(93,127,119,104,138,123,108),(94,128,113,105,139,124,109),(95,129,114,99,140,125,110),(96,130,115,100,134,126,111),(97,131,116,101,135,120,112),(98,132,117,102,136,121,106)], [(1,72),(2,71),(3,77),(4,76),(5,75),(6,74),(7,73),(8,39),(9,38),(10,37),(11,36),(12,42),(13,41),(14,40),(15,32),(16,31),(17,30),(18,29),(19,35),(20,34),(21,33),(22,25),(23,24),(26,28),(43,143),(44,142),(45,141),(46,147),(47,146),(48,145),(49,144),(50,79),(51,78),(52,84),(53,83),(54,82),(55,81),(56,80),(57,88),(58,87),(59,86),(60,85),(61,91),(62,90),(63,89),(64,65),(66,70),(67,69),(92,135),(93,134),(94,140),(95,139),(96,138),(97,137),(98,136),(99,128),(100,127),(101,133),(102,132),(103,131),(104,130),(105,129),(106,121),(107,120),(108,126),(109,125),(110,124),(111,123),(112,122),(113,114),(115,119),(116,118)]])
78 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7X | 21A | ··· | 21AV |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 49 | 1 | 1 | 49 | 49 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D7 | C3×D7 |
| kernel | C3×C7⋊D7 | C7×C21 | C7⋊D7 | C72 | C21 | C7 |
| # reps | 1 | 1 | 2 | 2 | 24 | 48 |
Matrix representation of C3×C7⋊D7 ►in GL4(𝔽43) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 42 | 16 | 0 | 0 |
| 37 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 1 | 0 | 0 |
| 5 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 42 | 15 |
| 34 | 42 | 0 | 0 |
| 37 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(43))| [1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[42,37,0,0,16,9,0,0,0,0,1,0,0,0,0,1],[28,5,0,0,1,34,0,0,0,0,0,42,0,0,1,15],[34,37,0,0,42,9,0,0,0,0,0,1,0,0,1,0] >;
C3×C7⋊D7 in GAP, Magma, Sage, TeX
C_3\times C_7\rtimes D_7
% in TeX
G:=Group("C3xC7:D7"); // GroupNames label
G:=SmallGroup(294,19);
// by ID
G=gap.SmallGroup(294,19);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,434,4035]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^7=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export