metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C21⋊1D8, D28⋊2S3, D12⋊2D7, C42.1D4, C28.1D6, C12.1D14, C84.22C22, C7⋊2(D4⋊S3), C3⋊2(D4⋊D7), C21⋊C8⋊6C2, (C7×D12)⋊4C2, (C3×D28)⋊4C2, C4.15(S3×D7), C6.7(C7⋊D4), C14.7(C3⋊D4), C2.4(C21⋊D4), SmallGroup(336,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C21⋊D8
G = < a,b,c | a21=b8=c2=1, bab-1=a-1, cac=a13, cbc=b-1 >
Smallest permutation representation of C21⋊D8
►On 168 points
Generators in S168
(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)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 101 43 164 24 109 83 127)(2 100 44 163 25 108 84 147)(3 99 45 162 26 107 64 146)(4 98 46 161 27 106 65 145)(5 97 47 160 28 126 66 144)(6 96 48 159 29 125 67 143)(7 95 49 158 30 124 68 142)(8 94 50 157 31 123 69 141)(9 93 51 156 32 122 70 140)(10 92 52 155 33 121 71 139)(11 91 53 154 34 120 72 138)(12 90 54 153 35 119 73 137)(13 89 55 152 36 118 74 136)(14 88 56 151 37 117 75 135)(15 87 57 150 38 116 76 134)(16 86 58 149 39 115 77 133)(17 85 59 148 40 114 78 132)(18 105 60 168 41 113 79 131)(19 104 61 167 42 112 80 130)(20 103 62 166 22 111 81 129)(21 102 63 165 23 110 82 128)
(2 14)(3 6)(4 19)(5 11)(7 16)(9 21)(10 13)(12 18)(17 20)(22 40)(23 32)(25 37)(26 29)(27 42)(28 34)(30 39)(33 36)(35 41)(43 83)(44 75)(45 67)(46 80)(47 72)(48 64)(49 77)(50 69)(51 82)(52 74)(53 66)(54 79)(55 71)(56 84)(57 76)(58 68)(59 81)(60 73)(61 65)(62 78)(63 70)(85 129)(86 142)(87 134)(88 147)(89 139)(90 131)(91 144)(92 136)(93 128)(94 141)(95 133)(96 146)(97 138)(98 130)(99 143)(100 135)(101 127)(102 140)(103 132)(104 145)(105 137)(106 167)(107 159)(108 151)(109 164)(110 156)(111 148)(112 161)(113 153)(114 166)(115 158)(116 150)(117 163)(118 155)(119 168)(120 160)(121 152)(122 165)(123 157)(124 149)(125 162)(126 154)
G:=sub<Sym(168)| (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,101,43,164,24,109,83,127)(2,100,44,163,25,108,84,147)(3,99,45,162,26,107,64,146)(4,98,46,161,27,106,65,145)(5,97,47,160,28,126,66,144)(6,96,48,159,29,125,67,143)(7,95,49,158,30,124,68,142)(8,94,50,157,31,123,69,141)(9,93,51,156,32,122,70,140)(10,92,52,155,33,121,71,139)(11,91,53,154,34,120,72,138)(12,90,54,153,35,119,73,137)(13,89,55,152,36,118,74,136)(14,88,56,151,37,117,75,135)(15,87,57,150,38,116,76,134)(16,86,58,149,39,115,77,133)(17,85,59,148,40,114,78,132)(18,105,60,168,41,113,79,131)(19,104,61,167,42,112,80,130)(20,103,62,166,22,111,81,129)(21,102,63,165,23,110,82,128), (2,14)(3,6)(4,19)(5,11)(7,16)(9,21)(10,13)(12,18)(17,20)(22,40)(23,32)(25,37)(26,29)(27,42)(28,34)(30,39)(33,36)(35,41)(43,83)(44,75)(45,67)(46,80)(47,72)(48,64)(49,77)(50,69)(51,82)(52,74)(53,66)(54,79)(55,71)(56,84)(57,76)(58,68)(59,81)(60,73)(61,65)(62,78)(63,70)(85,129)(86,142)(87,134)(88,147)(89,139)(90,131)(91,144)(92,136)(93,128)(94,141)(95,133)(96,146)(97,138)(98,130)(99,143)(100,135)(101,127)(102,140)(103,132)(104,145)(105,137)(106,167)(107,159)(108,151)(109,164)(110,156)(111,148)(112,161)(113,153)(114,166)(115,158)(116,150)(117,163)(118,155)(119,168)(120,160)(121,152)(122,165)(123,157)(124,149)(125,162)(126,154)>;
G:=Group( (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,101,43,164,24,109,83,127)(2,100,44,163,25,108,84,147)(3,99,45,162,26,107,64,146)(4,98,46,161,27,106,65,145)(5,97,47,160,28,126,66,144)(6,96,48,159,29,125,67,143)(7,95,49,158,30,124,68,142)(8,94,50,157,31,123,69,141)(9,93,51,156,32,122,70,140)(10,92,52,155,33,121,71,139)(11,91,53,154,34,120,72,138)(12,90,54,153,35,119,73,137)(13,89,55,152,36,118,74,136)(14,88,56,151,37,117,75,135)(15,87,57,150,38,116,76,134)(16,86,58,149,39,115,77,133)(17,85,59,148,40,114,78,132)(18,105,60,168,41,113,79,131)(19,104,61,167,42,112,80,130)(20,103,62,166,22,111,81,129)(21,102,63,165,23,110,82,128), (2,14)(3,6)(4,19)(5,11)(7,16)(9,21)(10,13)(12,18)(17,20)(22,40)(23,32)(25,37)(26,29)(27,42)(28,34)(30,39)(33,36)(35,41)(43,83)(44,75)(45,67)(46,80)(47,72)(48,64)(49,77)(50,69)(51,82)(52,74)(53,66)(54,79)(55,71)(56,84)(57,76)(58,68)(59,81)(60,73)(61,65)(62,78)(63,70)(85,129)(86,142)(87,134)(88,147)(89,139)(90,131)(91,144)(92,136)(93,128)(94,141)(95,133)(96,146)(97,138)(98,130)(99,143)(100,135)(101,127)(102,140)(103,132)(104,145)(105,137)(106,167)(107,159)(108,151)(109,164)(110,156)(111,148)(112,161)(113,153)(114,166)(115,158)(116,150)(117,163)(118,155)(119,168)(120,160)(121,152)(122,165)(123,157)(124,149)(125,162)(126,154) );
G=PermutationGroup([[(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),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,101,43,164,24,109,83,127),(2,100,44,163,25,108,84,147),(3,99,45,162,26,107,64,146),(4,98,46,161,27,106,65,145),(5,97,47,160,28,126,66,144),(6,96,48,159,29,125,67,143),(7,95,49,158,30,124,68,142),(8,94,50,157,31,123,69,141),(9,93,51,156,32,122,70,140),(10,92,52,155,33,121,71,139),(11,91,53,154,34,120,72,138),(12,90,54,153,35,119,73,137),(13,89,55,152,36,118,74,136),(14,88,56,151,37,117,75,135),(15,87,57,150,38,116,76,134),(16,86,58,149,39,115,77,133),(17,85,59,148,40,114,78,132),(18,105,60,168,41,113,79,131),(19,104,61,167,42,112,80,130),(20,103,62,166,22,111,81,129),(21,102,63,165,23,110,82,128)], [(2,14),(3,6),(4,19),(5,11),(7,16),(9,21),(10,13),(12,18),(17,20),(22,40),(23,32),(25,37),(26,29),(27,42),(28,34),(30,39),(33,36),(35,41),(43,83),(44,75),(45,67),(46,80),(47,72),(48,64),(49,77),(50,69),(51,82),(52,74),(53,66),(54,79),(55,71),(56,84),(57,76),(58,68),(59,81),(60,73),(61,65),(62,78),(63,70),(85,129),(86,142),(87,134),(88,147),(89,139),(90,131),(91,144),(92,136),(93,128),(94,141),(95,133),(96,146),(97,138),(98,130),(99,143),(100,135),(101,127),(102,140),(103,132),(104,145),(105,137),(106,167),(107,159),(108,151),(109,164),(110,156),(111,148),(112,161),(113,153),(114,166),(115,158),(116,150),(117,163),(118,155),(119,168),(120,160),(121,152),(122,165),(123,157),(124,149),(125,162),(126,154)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 12 | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 28A | 28B | 28C | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 12 | 28 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 42 | 42 | 4 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | D8 | C3⋊D4 | D14 | C7⋊D4 | D4⋊S3 | S3×D7 | D4⋊D7 | C21⋊D4 | C21⋊D8 |
| kernel | C21⋊D8 | C21⋊C8 | C3×D28 | C7×D12 | D28 | C42 | C28 | D12 | C21 | C14 | C12 | C6 | C7 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 1 | 3 | 3 | 3 | 6 |
Matrix representation of C21⋊D8 ►in GL6(𝔽337)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 144 | 34 | 0 | 0 |
| 0 | 0 | 303 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 208 | 0 |
| 0 | 0 | 0 | 0 | 336 | 128 |
| 232 | 206 | 0 | 0 | 0 | 0 |
| 87 | 131 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 303 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 150 | 205 |
| 0 | 0 | 0 | 0 | 76 | 187 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 204 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 303 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 278 | 336 |
G:=sub<GL(6,GF(337))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,144,303,0,0,0,0,34,336,0,0,0,0,0,0,208,336,0,0,0,0,0,128],[232,87,0,0,0,0,206,131,0,0,0,0,0,0,1,303,0,0,0,0,0,336,0,0,0,0,0,0,150,76,0,0,0,0,205,187],[1,204,0,0,0,0,0,336,0,0,0,0,0,0,1,303,0,0,0,0,0,336,0,0,0,0,0,0,1,278,0,0,0,0,0,336] >;
C21⋊D8 in GAP, Magma, Sage, TeX
C_{21}\rtimes D_8 % in TeX
G:=Group("C21:D8"); // GroupNames label
G:=SmallGroup(336,29);
// by ID
G=gap.SmallGroup(336,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,218,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^21=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations
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