metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.3D6, C42.4D4, C21⋊1SD16, Dic6⋊2D7, D28.2S3, C12.3D14, C84.23C22, C3⋊2(Q8⋊D7), C21⋊C8⋊7C2, C7⋊2(D4.S3), C4.16(S3×D7), (C7×Dic6)⋊4C2, (C3×D28).2C2, C6.8(C7⋊D4), C14.8(C3⋊D4), C2.5(C21⋊D4), SmallGroup(336,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.D6
G = < a,b,c | a42=c2=1, b4=a21, bab-1=a-1, cac=a13, cbc=b3 >
Smallest permutation representation of C28.D6
►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)
(1 106 78 148 22 85 57 127)(2 105 79 147 23 126 58 168)(3 104 80 146 24 125 59 167)(4 103 81 145 25 124 60 166)(5 102 82 144 26 123 61 165)(6 101 83 143 27 122 62 164)(7 100 84 142 28 121 63 163)(8 99 43 141 29 120 64 162)(9 98 44 140 30 119 65 161)(10 97 45 139 31 118 66 160)(11 96 46 138 32 117 67 159)(12 95 47 137 33 116 68 158)(13 94 48 136 34 115 69 157)(14 93 49 135 35 114 70 156)(15 92 50 134 36 113 71 155)(16 91 51 133 37 112 72 154)(17 90 52 132 38 111 73 153)(18 89 53 131 39 110 74 152)(19 88 54 130 40 109 75 151)(20 87 55 129 41 108 76 150)(21 86 56 128 42 107 77 149)
(2 14)(3 27)(4 40)(5 11)(6 24)(7 37)(9 21)(10 34)(12 18)(13 31)(16 28)(17 41)(19 25)(20 38)(23 35)(26 32)(30 42)(33 39)(43 64)(44 77)(45 48)(46 61)(47 74)(49 58)(50 71)(51 84)(52 55)(53 68)(54 81)(56 65)(57 78)(59 62)(60 75)(63 72)(66 69)(67 82)(70 79)(73 76)(80 83)(85 127)(86 140)(87 153)(88 166)(89 137)(90 150)(91 163)(92 134)(93 147)(94 160)(95 131)(96 144)(97 157)(98 128)(99 141)(100 154)(101 167)(102 138)(103 151)(104 164)(105 135)(106 148)(107 161)(108 132)(109 145)(110 158)(111 129)(112 142)(113 155)(114 168)(115 139)(116 152)(117 165)(118 136)(119 149)(120 162)(121 133)(122 146)(123 159)(124 130)(125 143)(126 156)
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), (1,106,78,148,22,85,57,127)(2,105,79,147,23,126,58,168)(3,104,80,146,24,125,59,167)(4,103,81,145,25,124,60,166)(5,102,82,144,26,123,61,165)(6,101,83,143,27,122,62,164)(7,100,84,142,28,121,63,163)(8,99,43,141,29,120,64,162)(9,98,44,140,30,119,65,161)(10,97,45,139,31,118,66,160)(11,96,46,138,32,117,67,159)(12,95,47,137,33,116,68,158)(13,94,48,136,34,115,69,157)(14,93,49,135,35,114,70,156)(15,92,50,134,36,113,71,155)(16,91,51,133,37,112,72,154)(17,90,52,132,38,111,73,153)(18,89,53,131,39,110,74,152)(19,88,54,130,40,109,75,151)(20,87,55,129,41,108,76,150)(21,86,56,128,42,107,77,149), (2,14)(3,27)(4,40)(5,11)(6,24)(7,37)(9,21)(10,34)(12,18)(13,31)(16,28)(17,41)(19,25)(20,38)(23,35)(26,32)(30,42)(33,39)(43,64)(44,77)(45,48)(46,61)(47,74)(49,58)(50,71)(51,84)(52,55)(53,68)(54,81)(56,65)(57,78)(59,62)(60,75)(63,72)(66,69)(67,82)(70,79)(73,76)(80,83)(85,127)(86,140)(87,153)(88,166)(89,137)(90,150)(91,163)(92,134)(93,147)(94,160)(95,131)(96,144)(97,157)(98,128)(99,141)(100,154)(101,167)(102,138)(103,151)(104,164)(105,135)(106,148)(107,161)(108,132)(109,145)(110,158)(111,129)(112,142)(113,155)(114,168)(115,139)(116,152)(117,165)(118,136)(119,149)(120,162)(121,133)(122,146)(123,159)(124,130)(125,143)(126,156)>;
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), (1,106,78,148,22,85,57,127)(2,105,79,147,23,126,58,168)(3,104,80,146,24,125,59,167)(4,103,81,145,25,124,60,166)(5,102,82,144,26,123,61,165)(6,101,83,143,27,122,62,164)(7,100,84,142,28,121,63,163)(8,99,43,141,29,120,64,162)(9,98,44,140,30,119,65,161)(10,97,45,139,31,118,66,160)(11,96,46,138,32,117,67,159)(12,95,47,137,33,116,68,158)(13,94,48,136,34,115,69,157)(14,93,49,135,35,114,70,156)(15,92,50,134,36,113,71,155)(16,91,51,133,37,112,72,154)(17,90,52,132,38,111,73,153)(18,89,53,131,39,110,74,152)(19,88,54,130,40,109,75,151)(20,87,55,129,41,108,76,150)(21,86,56,128,42,107,77,149), (2,14)(3,27)(4,40)(5,11)(6,24)(7,37)(9,21)(10,34)(12,18)(13,31)(16,28)(17,41)(19,25)(20,38)(23,35)(26,32)(30,42)(33,39)(43,64)(44,77)(45,48)(46,61)(47,74)(49,58)(50,71)(51,84)(52,55)(53,68)(54,81)(56,65)(57,78)(59,62)(60,75)(63,72)(66,69)(67,82)(70,79)(73,76)(80,83)(85,127)(86,140)(87,153)(88,166)(89,137)(90,150)(91,163)(92,134)(93,147)(94,160)(95,131)(96,144)(97,157)(98,128)(99,141)(100,154)(101,167)(102,138)(103,151)(104,164)(105,135)(106,148)(107,161)(108,132)(109,145)(110,158)(111,129)(112,142)(113,155)(114,168)(115,139)(116,152)(117,165)(118,136)(119,149)(120,162)(121,133)(122,146)(123,159)(124,130)(125,143)(126,156) );
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)], [(1,106,78,148,22,85,57,127),(2,105,79,147,23,126,58,168),(3,104,80,146,24,125,59,167),(4,103,81,145,25,124,60,166),(5,102,82,144,26,123,61,165),(6,101,83,143,27,122,62,164),(7,100,84,142,28,121,63,163),(8,99,43,141,29,120,64,162),(9,98,44,140,30,119,65,161),(10,97,45,139,31,118,66,160),(11,96,46,138,32,117,67,159),(12,95,47,137,33,116,68,158),(13,94,48,136,34,115,69,157),(14,93,49,135,35,114,70,156),(15,92,50,134,36,113,71,155),(16,91,51,133,37,112,72,154),(17,90,52,132,38,111,73,153),(18,89,53,131,39,110,74,152),(19,88,54,130,40,109,75,151),(20,87,55,129,41,108,76,150),(21,86,56,128,42,107,77,149)], [(2,14),(3,27),(4,40),(5,11),(6,24),(7,37),(9,21),(10,34),(12,18),(13,31),(16,28),(17,41),(19,25),(20,38),(23,35),(26,32),(30,42),(33,39),(43,64),(44,77),(45,48),(46,61),(47,74),(49,58),(50,71),(51,84),(52,55),(53,68),(54,81),(56,65),(57,78),(59,62),(60,75),(63,72),(66,69),(67,82),(70,79),(73,76),(80,83),(85,127),(86,140),(87,153),(88,166),(89,137),(90,150),(91,163),(92,134),(93,147),(94,160),(95,131),(96,144),(97,157),(98,128),(99,141),(100,154),(101,167),(102,138),(103,151),(104,164),(105,135),(106,148),(107,161),(108,132),(109,145),(110,158),(111,129),(112,142),(113,155),(114,168),(115,139),(116,152),(117,165),(118,136),(119,149),(120,162),(121,133),(122,146),(123,159),(124,130),(125,143),(126,156)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 12 | 14A | 14B | 14C | 21A | 21B | 21C | 28A | 28B | 28C | 28D | ··· | 28I | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 28 | ··· | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 28 | 2 | 2 | 12 | 2 | 28 | 28 | 2 | 2 | 2 | 42 | 42 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 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 | SD16 | C3⋊D4 | D14 | C7⋊D4 | D4.S3 | S3×D7 | Q8⋊D7 | C21⋊D4 | C28.D6 |
| kernel | C28.D6 | C21⋊C8 | C3×D28 | C7×Dic6 | D28 | C42 | C28 | Dic6 | 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 C28.D6 ►in GL6(𝔽337)
| 304 | 336 | 0 | 0 | 0 | 0 |
| 305 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 128 | 0 | 0 | 0 |
| 0 | 0 | 273 | 208 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 0 | 336 |
| 33 | 1 | 0 | 0 | 0 | 0 |
| 260 | 304 | 0 | 0 | 0 | 0 |
| 0 | 0 | 209 | 160 | 0 | 0 |
| 0 | 0 | 203 | 128 | 0 | 0 |
| 0 | 0 | 0 | 0 | 179 | 158 |
| 0 | 0 | 0 | 0 | 70 | 299 |
| 33 | 1 | 0 | 0 | 0 | 0 |
| 260 | 304 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 69 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 133 | 336 |
G:=sub<GL(6,GF(337))| [304,305,0,0,0,0,336,336,0,0,0,0,0,0,128,273,0,0,0,0,0,208,0,0,0,0,0,0,336,0,0,0,0,0,0,336],[33,260,0,0,0,0,1,304,0,0,0,0,0,0,209,203,0,0,0,0,160,128,0,0,0,0,0,0,179,70,0,0,0,0,158,299],[33,260,0,0,0,0,1,304,0,0,0,0,0,0,1,69,0,0,0,0,0,336,0,0,0,0,0,0,1,133,0,0,0,0,0,336] >;
C28.D6 in GAP, Magma, Sage, TeX
C_{28}.D_6 % in TeX
G:=Group("C28.D6"); // GroupNames label
G:=SmallGroup(336,32);
// by ID
G=gap.SmallGroup(336,32);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,55,218,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^42=c^2=1,b^4=a^21,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^3>;
// generators/relations
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