direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C28, C84⋊13C22, C42.49C23, C42⋊6(C2×C4), C6⋊1(C2×C28), C12⋊3(C2×C14), (C2×C12)⋊5C14, (C2×C84)⋊13C2, C3⋊1(C22×C28), C21⋊7(C22×C4), D6.4(C2×C14), (C2×C14).37D6, Dic3⋊3(C2×C14), (C2×Dic3)⋊5C14, C22.9(S3×C14), C6.2(C22×C14), (Dic3×C14)⋊11C2, (C22×S3).2C14, C14.39(C22×S3), (C2×C42).48C22, (S3×C14).15C22, (C7×Dic3)⋊10C22, C2.1(S3×C2×C14), (S3×C2×C14).4C2, (C2×C6).9(C2×C14), SmallGroup(336,185)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C28 |
Generators and relations for S3×C2×C28
G = < a,b,c,d | a2=b28=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 184 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C7, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C14, C14, C14, C22×C4, C21, C4×S3, C2×Dic3, C2×C12, C22×S3, C28, C28, C2×C14, C2×C14, S3×C7, C42, C42, S3×C2×C4, C2×C28, C2×C28, C22×C14, C7×Dic3, C84, S3×C14, C2×C42, C22×C28, S3×C28, Dic3×C14, C2×C84, S3×C2×C14, S3×C2×C28
Quotients: C1, C2, C4, C22, S3, C7, C2×C4, C23, D6, C14, C22×C4, C4×S3, C22×S3, C28, C2×C14, S3×C7, S3×C2×C4, C2×C28, C22×C14, S3×C14, C22×C28, S3×C28, S3×C2×C14, S3×C2×C28
►On 168 points
Generators in S168
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 85)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 166)(122 167)(123 168)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(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 93 154)(2 94 155)(3 95 156)(4 96 157)(5 97 158)(6 98 159)(7 99 160)(8 100 161)(9 101 162)(10 102 163)(11 103 164)(12 104 165)(13 105 166)(14 106 167)(15 107 168)(16 108 141)(17 109 142)(18 110 143)(19 111 144)(20 112 145)(21 85 146)(22 86 147)(23 87 148)(24 88 149)(25 89 150)(26 90 151)(27 91 152)(28 92 153)(29 130 70)(30 131 71)(31 132 72)(32 133 73)(33 134 74)(34 135 75)(35 136 76)(36 137 77)(37 138 78)(38 139 79)(39 140 80)(40 113 81)(41 114 82)(42 115 83)(43 116 84)(44 117 57)(45 118 58)(46 119 59)(47 120 60)(48 121 61)(49 122 62)(50 123 63)(51 124 64)(52 125 65)(53 126 66)(54 127 67)(55 128 68)(56 129 69)
(29 130)(30 131)(31 132)(32 133)(33 134)(34 135)(35 136)(36 137)(37 138)(38 139)(39 140)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)(97 158)(98 159)(99 160)(100 161)(101 162)(102 163)(103 164)(104 165)(105 166)(106 167)(107 168)(108 141)(109 142)(110 143)(111 144)(112 145)
G:=sub<Sym(168)| (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,85)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (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), (1,93,154)(2,94,155)(3,95,156)(4,96,157)(5,97,158)(6,98,159)(7,99,160)(8,100,161)(9,101,162)(10,102,163)(11,103,164)(12,104,165)(13,105,166)(14,106,167)(15,107,168)(16,108,141)(17,109,142)(18,110,143)(19,111,144)(20,112,145)(21,85,146)(22,86,147)(23,87,148)(24,88,149)(25,89,150)(26,90,151)(27,91,152)(28,92,153)(29,130,70)(30,131,71)(31,132,72)(32,133,73)(33,134,74)(34,135,75)(35,136,76)(36,137,77)(37,138,78)(38,139,79)(39,140,80)(40,113,81)(41,114,82)(42,115,83)(43,116,84)(44,117,57)(45,118,58)(46,119,59)(47,120,60)(48,121,61)(49,122,62)(50,123,63)(51,124,64)(52,125,65)(53,126,66)(54,127,67)(55,128,68)(56,129,69), (29,130)(30,131)(31,132)(32,133)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157)(97,158)(98,159)(99,160)(100,161)(101,162)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,141)(109,142)(110,143)(111,144)(112,145)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,85)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (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), (1,93,154)(2,94,155)(3,95,156)(4,96,157)(5,97,158)(6,98,159)(7,99,160)(8,100,161)(9,101,162)(10,102,163)(11,103,164)(12,104,165)(13,105,166)(14,106,167)(15,107,168)(16,108,141)(17,109,142)(18,110,143)(19,111,144)(20,112,145)(21,85,146)(22,86,147)(23,87,148)(24,88,149)(25,89,150)(26,90,151)(27,91,152)(28,92,153)(29,130,70)(30,131,71)(31,132,72)(32,133,73)(33,134,74)(34,135,75)(35,136,76)(36,137,77)(37,138,78)(38,139,79)(39,140,80)(40,113,81)(41,114,82)(42,115,83)(43,116,84)(44,117,57)(45,118,58)(46,119,59)(47,120,60)(48,121,61)(49,122,62)(50,123,63)(51,124,64)(52,125,65)(53,126,66)(54,127,67)(55,128,68)(56,129,69), (29,130)(30,131)(31,132)(32,133)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157)(97,158)(98,159)(99,160)(100,161)(101,162)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,141)(109,142)(110,143)(111,144)(112,145) );
G=PermutationGroup([[(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,98),(42,99),(43,100),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,85),(113,158),(114,159),(115,160),(116,161),(117,162),(118,163),(119,164),(120,165),(121,166),(122,167),(123,168),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,157)], [(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)], [(1,93,154),(2,94,155),(3,95,156),(4,96,157),(5,97,158),(6,98,159),(7,99,160),(8,100,161),(9,101,162),(10,102,163),(11,103,164),(12,104,165),(13,105,166),(14,106,167),(15,107,168),(16,108,141),(17,109,142),(18,110,143),(19,111,144),(20,112,145),(21,85,146),(22,86,147),(23,87,148),(24,88,149),(25,89,150),(26,90,151),(27,91,152),(28,92,153),(29,130,70),(30,131,71),(31,132,72),(32,133,73),(33,134,74),(34,135,75),(35,136,76),(36,137,77),(37,138,78),(38,139,79),(39,140,80),(40,113,81),(41,114,82),(42,115,83),(43,116,84),(44,117,57),(45,118,58),(46,119,59),(47,120,60),(48,121,61),(49,122,62),(50,123,63),(51,124,64),(52,125,65),(53,126,66),(54,127,67),(55,128,68),(56,129,69)], [(29,130),(30,131),(31,132),(32,133),(33,134),(34,135),(35,136),(36,137),(37,138),(38,139),(39,140),(40,113),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(85,146),(86,147),(87,148),(88,149),(89,150),(90,151),(91,152),(92,153),(93,154),(94,155),(95,156),(96,157),(97,158),(98,159),(99,160),(100,161),(101,162),(102,163),(103,164),(104,165),(105,166),(106,167),(107,168),(108,141),(109,142),(110,143),(111,144),(112,145)]])
168 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 14A | ··· | 14R | 14S | ··· | 14AP | 21A | ··· | 21F | 28A | ··· | 28X | 28Y | ··· | 28AV | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
168 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | S3 | D6 | D6 | C4×S3 | S3×C7 | S3×C14 | S3×C14 | S3×C28 |
| kernel | S3×C2×C28 | S3×C28 | Dic3×C14 | C2×C84 | S3×C2×C14 | S3×C14 | S3×C2×C4 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C2×C28 | C28 | C2×C14 | C14 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 6 | 24 | 6 | 6 | 6 | 48 | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 |
Matrix representation of S3×C2×C28 ►in GL3(𝔽337) generated by
| 336 | 0 | 0 |
| 0 | 336 | 0 |
| 0 | 0 | 336 |
| 36 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 1 | 0 | 0 |
| 0 | 336 | 336 |
| 0 | 1 | 0 |
| 336 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 336 | 336 |
G:=sub<GL(3,GF(337))| [336,0,0,0,336,0,0,0,336],[36,0,0,0,42,0,0,0,42],[1,0,0,0,336,1,0,336,0],[336,0,0,0,1,336,0,0,336] >;
S3×C2×C28 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{28} % in TeX
G:=Group("S3xC2xC28"); // GroupNames label
G:=SmallGroup(336,185);
// by ID
G=gap.SmallGroup(336,185);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-3,266,8069]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations