direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D56, C21⋊5D8, C56⋊9C6, C24⋊3D7, C168⋊3C2, D28⋊7C6, C6.14D28, C42.24D4, C12.53D14, C84.60C22, C7⋊4(C3×D8), C8⋊1(C3×D7), C4.9(C6×D7), (C3×D28)⋊7C2, C2.4(C3×D28), C28.32(C2×C6), C14.18(C3×D4), SmallGroup(336,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D56
G = < a,b,c | a3=b56=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D56
►On 168 points
Generators in S168
(1 116 110)(2 117 111)(3 118 112)(4 119 57)(5 120 58)(6 121 59)(7 122 60)(8 123 61)(9 124 62)(10 125 63)(11 126 64)(12 127 65)(13 128 66)(14 129 67)(15 130 68)(16 131 69)(17 132 70)(18 133 71)(19 134 72)(20 135 73)(21 136 74)(22 137 75)(23 138 76)(24 139 77)(25 140 78)(26 141 79)(27 142 80)(28 143 81)(29 144 82)(30 145 83)(31 146 84)(32 147 85)(33 148 86)(34 149 87)(35 150 88)(36 151 89)(37 152 90)(38 153 91)(39 154 92)(40 155 93)(41 156 94)(42 157 95)(43 158 96)(44 159 97)(45 160 98)(46 161 99)(47 162 100)(48 163 101)(49 164 102)(50 165 103)(51 166 104)(52 167 105)(53 168 106)(54 113 107)(55 114 108)(56 115 109)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(57 106)(58 105)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)(107 112)(108 111)(109 110)(113 118)(114 117)(115 116)(119 168)(120 167)(121 166)(122 165)(123 164)(124 163)(125 162)(126 161)(127 160)(128 159)(129 158)(130 157)(131 156)(132 155)(133 154)(134 153)(135 152)(136 151)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)
G:=sub<Sym(168)| (1,116,110)(2,117,111)(3,118,112)(4,119,57)(5,120,58)(6,121,59)(7,122,60)(8,123,61)(9,124,62)(10,125,63)(11,126,64)(12,127,65)(13,128,66)(14,129,67)(15,130,68)(16,131,69)(17,132,70)(18,133,71)(19,134,72)(20,135,73)(21,136,74)(22,137,75)(23,138,76)(24,139,77)(25,140,78)(26,141,79)(27,142,80)(28,143,81)(29,144,82)(30,145,83)(31,146,84)(32,147,85)(33,148,86)(34,149,87)(35,150,88)(36,151,89)(37,152,90)(38,153,91)(39,154,92)(40,155,93)(41,156,94)(42,157,95)(43,158,96)(44,159,97)(45,160,98)(46,161,99)(47,162,100)(48,163,101)(49,164,102)(50,165,103)(51,166,104)(52,167,105)(53,168,106)(54,113,107)(55,114,108)(56,115,109), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(107,112)(108,111)(109,110)(113,118)(114,117)(115,116)(119,168)(120,167)(121,166)(122,165)(123,164)(124,163)(125,162)(126,161)(127,160)(128,159)(129,158)(130,157)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)>;
G:=Group( (1,116,110)(2,117,111)(3,118,112)(4,119,57)(5,120,58)(6,121,59)(7,122,60)(8,123,61)(9,124,62)(10,125,63)(11,126,64)(12,127,65)(13,128,66)(14,129,67)(15,130,68)(16,131,69)(17,132,70)(18,133,71)(19,134,72)(20,135,73)(21,136,74)(22,137,75)(23,138,76)(24,139,77)(25,140,78)(26,141,79)(27,142,80)(28,143,81)(29,144,82)(30,145,83)(31,146,84)(32,147,85)(33,148,86)(34,149,87)(35,150,88)(36,151,89)(37,152,90)(38,153,91)(39,154,92)(40,155,93)(41,156,94)(42,157,95)(43,158,96)(44,159,97)(45,160,98)(46,161,99)(47,162,100)(48,163,101)(49,164,102)(50,165,103)(51,166,104)(52,167,105)(53,168,106)(54,113,107)(55,114,108)(56,115,109), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(107,112)(108,111)(109,110)(113,118)(114,117)(115,116)(119,168)(120,167)(121,166)(122,165)(123,164)(124,163)(125,162)(126,161)(127,160)(128,159)(129,158)(130,157)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144) );
G=PermutationGroup([[(1,116,110),(2,117,111),(3,118,112),(4,119,57),(5,120,58),(6,121,59),(7,122,60),(8,123,61),(9,124,62),(10,125,63),(11,126,64),(12,127,65),(13,128,66),(14,129,67),(15,130,68),(16,131,69),(17,132,70),(18,133,71),(19,134,72),(20,135,73),(21,136,74),(22,137,75),(23,138,76),(24,139,77),(25,140,78),(26,141,79),(27,142,80),(28,143,81),(29,144,82),(30,145,83),(31,146,84),(32,147,85),(33,148,86),(34,149,87),(35,150,88),(36,151,89),(37,152,90),(38,153,91),(39,154,92),(40,155,93),(41,156,94),(42,157,95),(43,158,96),(44,159,97),(45,160,98),(46,161,99),(47,162,100),(48,163,101),(49,164,102),(50,165,103),(51,166,104),(52,167,105),(53,168,106),(54,113,107),(55,114,108),(56,115,109)], [(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(57,106),(58,105),(59,104),(60,103),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82),(107,112),(108,111),(109,110),(113,118),(114,117),(115,116),(119,168),(120,167),(121,166),(122,165),(123,164),(124,163),(125,162),(126,161),(127,160),(128,159),(129,158),(130,157),(131,156),(132,155),(133,154),(134,153),(135,152),(136,151),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 28 | 28 | 1 | 1 | 2 | 1 | 1 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D7 | D8 | C3×D4 | D14 | C3×D7 | C3×D8 | D28 | C6×D7 | D56 | C3×D28 | C3×D56 |
| kernel | C3×D56 | C168 | C3×D28 | D56 | C56 | D28 | C42 | C24 | C21 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 3 | 2 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C3×D56 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 208 | 0 |
| 0 | 0 | 0 | 208 |
| 299 | 18 | 0 | 0 |
| 319 | 177 | 0 | 0 |
| 0 | 0 | 324 | 13 |
| 0 | 0 | 324 | 324 |
| 299 | 18 | 0 | 0 |
| 313 | 38 | 0 | 0 |
| 0 | 0 | 324 | 324 |
| 0 | 0 | 324 | 13 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,208,0,0,0,0,208],[299,319,0,0,18,177,0,0,0,0,324,324,0,0,13,324],[299,313,0,0,18,38,0,0,0,0,324,324,0,0,324,13] >;
C3×D56 in GAP, Magma, Sage, TeX
C_3\times D_{56} % in TeX
G:=Group("C3xD56"); // GroupNames label
G:=SmallGroup(336,61);
// by ID
G=gap.SmallGroup(336,61);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,223,867,69,10373]);
// Polycyclic
G:=Group<a,b,c|a^3=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export