metabelian, supersoluble, monomial, A-group
Aliases: C7⋊D21, C21⋊1D7, C72⋊3S3, C3⋊(C7⋊D7), (C7×C21)⋊1C2, SmallGroup(294,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C7×C21 — C7⋊D21 |
Generators and relations for C7⋊D21
G = < a,b,c | a7=b21=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C7⋊D21
►On 147 points
Generators in S147
(1 76 63 110 87 140 27)(2 77 43 111 88 141 28)(3 78 44 112 89 142 29)(4 79 45 113 90 143 30)(5 80 46 114 91 144 31)(6 81 47 115 92 145 32)(7 82 48 116 93 146 33)(8 83 49 117 94 147 34)(9 84 50 118 95 127 35)(10 64 51 119 96 128 36)(11 65 52 120 97 129 37)(12 66 53 121 98 130 38)(13 67 54 122 99 131 39)(14 68 55 123 100 132 40)(15 69 56 124 101 133 41)(16 70 57 125 102 134 42)(17 71 58 126 103 135 22)(18 72 59 106 104 136 23)(19 73 60 107 105 137 24)(20 74 61 108 85 138 25)(21 75 62 109 86 139 26)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 84)(40 83)(41 82)(42 81)(43 138)(44 137)(45 136)(46 135)(47 134)(48 133)(49 132)(50 131)(51 130)(52 129)(53 128)(54 127)(55 147)(56 146)(57 145)(58 144)(59 143)(60 142)(61 141)(62 140)(63 139)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)
G:=sub<Sym(147)| (1,76,63,110,87,140,27)(2,77,43,111,88,141,28)(3,78,44,112,89,142,29)(4,79,45,113,90,143,30)(5,80,46,114,91,144,31)(6,81,47,115,92,145,32)(7,82,48,116,93,146,33)(8,83,49,117,94,147,34)(9,84,50,118,95,127,35)(10,64,51,119,96,128,36)(11,65,52,120,97,129,37)(12,66,53,121,98,130,38)(13,67,54,122,99,131,39)(14,68,55,123,100,132,40)(15,69,56,124,101,133,41)(16,70,57,125,102,134,42)(17,71,58,126,103,135,22)(18,72,59,106,104,136,23)(19,73,60,107,105,137,24)(20,74,61,108,85,138,25)(21,75,62,109,86,139,26), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,84)(40,83)(41,82)(42,81)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,147)(56,146)(57,145)(58,144)(59,143)(60,142)(61,141)(62,140)(63,139)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)>;
G:=Group( (1,76,63,110,87,140,27)(2,77,43,111,88,141,28)(3,78,44,112,89,142,29)(4,79,45,113,90,143,30)(5,80,46,114,91,144,31)(6,81,47,115,92,145,32)(7,82,48,116,93,146,33)(8,83,49,117,94,147,34)(9,84,50,118,95,127,35)(10,64,51,119,96,128,36)(11,65,52,120,97,129,37)(12,66,53,121,98,130,38)(13,67,54,122,99,131,39)(14,68,55,123,100,132,40)(15,69,56,124,101,133,41)(16,70,57,125,102,134,42)(17,71,58,126,103,135,22)(18,72,59,106,104,136,23)(19,73,60,107,105,137,24)(20,74,61,108,85,138,25)(21,75,62,109,86,139,26), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,84)(40,83)(41,82)(42,81)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,147)(56,146)(57,145)(58,144)(59,143)(60,142)(61,141)(62,140)(63,139)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112) );
G=PermutationGroup([[(1,76,63,110,87,140,27),(2,77,43,111,88,141,28),(3,78,44,112,89,142,29),(4,79,45,113,90,143,30),(5,80,46,114,91,144,31),(6,81,47,115,92,145,32),(7,82,48,116,93,146,33),(8,83,49,117,94,147,34),(9,84,50,118,95,127,35),(10,64,51,119,96,128,36),(11,65,52,120,97,129,37),(12,66,53,121,98,130,38),(13,67,54,122,99,131,39),(14,68,55,123,100,132,40),(15,69,56,124,101,133,41),(16,70,57,125,102,134,42),(17,71,58,126,103,135,22),(18,72,59,106,104,136,23),(19,73,60,107,105,137,24),(20,74,61,108,85,138,25),(21,75,62,109,86,139,26)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,84),(40,83),(41,82),(42,81),(43,138),(44,137),(45,136),(46,135),(47,134),(48,133),(49,132),(50,131),(51,130),(52,129),(53,128),(54,127),(55,147),(56,146),(57,145),(58,144),(59,143),(60,142),(61,141),(62,140),(63,139),(85,111),(86,110),(87,109),(88,108),(89,107),(90,106),(91,126),(92,125),(93,124),(94,123),(95,122),(96,121),(97,120),(98,119),(99,118),(100,117),(101,116),(102,115),(103,114),(104,113),(105,112)]])
75 conjugacy classes
| class | 1 | 2 | 3 | 7A | ··· | 7X | 21A | ··· | 21AV |
| order | 1 | 2 | 3 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 147 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D7 | D21 |
| kernel | C7⋊D21 | C7×C21 | C72 | C21 | C7 |
| # reps | 1 | 1 | 1 | 24 | 48 |
Matrix representation of C7⋊D21 ►in GL4(𝔽43) generated by
| 0 | 1 | 0 | 0 |
| 42 | 19 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 3 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 37 | 19 |
| 0 | 0 | 24 | 17 |
| 34 | 2 | 0 | 0 |
| 3 | 9 | 0 | 0 |
| 0 | 0 | 37 | 19 |
| 0 | 0 | 14 | 6 |
G:=sub<GL(4,GF(43))| [0,42,0,0,1,19,0,0,0,0,1,0,0,0,0,1],[20,40,0,0,3,34,0,0,0,0,37,24,0,0,19,17],[34,3,0,0,2,9,0,0,0,0,37,14,0,0,19,6] >;
C7⋊D21 in GAP, Magma, Sage, TeX
C_7\rtimes D_{21} % in TeX
G:=Group("C7:D21"); // GroupNames label
G:=SmallGroup(294,22);
// by ID
G=gap.SmallGroup(294,22);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,33,434,4035]);
// Polycyclic
G:=Group<a,b,c|a^7=b^21=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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