direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7×D4⋊S3, C21⋊9D8, D12⋊2C14, C28.36D6, C42.47D4, C84.43C22, D4⋊(S3×C7), C3⋊2(C7×D8), C3⋊C8⋊1C14, (C7×D4)⋊4S3, C6.7(C7×D4), (D4×C21)⋊7C2, (C3×D4)⋊1C14, (C7×D12)⋊8C2, C4.1(S3×C14), C12.1(C2×C14), C14.23(C3⋊D4), (C7×C3⋊C8)⋊8C2, C2.4(C7×C3⋊D4), SmallGroup(336,85)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4⋊S3
G = < a,b,c,d,e | a7=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Smallest permutation representation of C7×D4⋊S3
►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)
(1 135 55 19)(2 136 56 20)(3 137 50 21)(4 138 51 15)(5 139 52 16)(6 140 53 17)(7 134 54 18)(8 106 144 45)(9 107 145 46)(10 108 146 47)(11 109 147 48)(12 110 141 49)(13 111 142 43)(14 112 143 44)(22 60 158 121)(23 61 159 122)(24 62 160 123)(25 63 161 124)(26 57 155 125)(27 58 156 126)(28 59 157 120)(29 68 149 99)(30 69 150 100)(31 70 151 101)(32 64 152 102)(33 65 153 103)(34 66 154 104)(35 67 148 105)(36 127 165 73)(37 128 166 74)(38 129 167 75)(39 130 168 76)(40 131 162 77)(41 132 163 71)(42 133 164 72)(78 86 116 93)(79 87 117 94)(80 88 118 95)(81 89 119 96)(82 90 113 97)(83 91 114 98)(84 85 115 92)
(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 17)(9 18)(10 19)(11 20)(12 21)(13 15)(14 16)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 159)(30 160)(31 161)(32 155)(33 156)(34 157)(35 158)(36 89)(37 90)(38 91)(39 85)(40 86)(41 87)(42 88)(50 110)(51 111)(52 112)(53 106)(54 107)(55 108)(56 109)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 117)(72 118)(73 119)(74 113)(75 114)(76 115)(77 116)(78 131)(79 132)(80 133)(81 127)(82 128)(83 129)(84 130)(92 168)(93 162)(94 163)(95 164)(96 165)(97 166)(98 167)(99 122)(100 123)(101 124)(102 125)(103 126)(104 120)(105 121)(134 145)(135 146)(136 147)(137 141)(138 142)(139 143)(140 144)
(1 118 35)(2 119 29)(3 113 30)(4 114 31)(5 115 32)(6 116 33)(7 117 34)(8 40 126)(9 41 120)(10 42 121)(11 36 122)(12 37 123)(13 38 124)(14 39 125)(15 91 101)(16 85 102)(17 86 103)(18 87 104)(19 88 105)(20 89 99)(21 90 100)(22 108 133)(23 109 127)(24 110 128)(25 111 129)(26 112 130)(27 106 131)(28 107 132)(43 75 161)(44 76 155)(45 77 156)(46 71 157)(47 72 158)(48 73 159)(49 74 160)(50 82 150)(51 83 151)(52 84 152)(53 78 153)(54 79 154)(55 80 148)(56 81 149)(57 143 168)(58 144 162)(59 145 163)(60 146 164)(61 147 165)(62 141 166)(63 142 167)(64 139 92)(65 140 93)(66 134 94)(67 135 95)(68 136 96)(69 137 97)(70 138 98)
(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 138)(16 139)(17 140)(18 134)(19 135)(20 136)(21 137)(22 42)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 119)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 141)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(78 153)(79 154)(80 148)(81 149)(82 150)(83 151)(84 152)(92 102)(93 103)(94 104)(95 105)(96 99)(97 100)(98 101)(120 132)(121 133)(122 127)(123 128)(124 129)(125 130)(126 131)(155 168)(156 162)(157 163)(158 164)(159 165)(160 166)(161 167)
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), (1,135,55,19)(2,136,56,20)(3,137,50,21)(4,138,51,15)(5,139,52,16)(6,140,53,17)(7,134,54,18)(8,106,144,45)(9,107,145,46)(10,108,146,47)(11,109,147,48)(12,110,141,49)(13,111,142,43)(14,112,143,44)(22,60,158,121)(23,61,159,122)(24,62,160,123)(25,63,161,124)(26,57,155,125)(27,58,156,126)(28,59,157,120)(29,68,149,99)(30,69,150,100)(31,70,151,101)(32,64,152,102)(33,65,153,103)(34,66,154,104)(35,67,148,105)(36,127,165,73)(37,128,166,74)(38,129,167,75)(39,130,168,76)(40,131,162,77)(41,132,163,71)(42,133,164,72)(78,86,116,93)(79,87,117,94)(80,88,118,95)(81,89,119,96)(82,90,113,97)(83,91,114,98)(84,85,115,92), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,17)(9,18)(10,19)(11,20)(12,21)(13,15)(14,16)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,159)(30,160)(31,161)(32,155)(33,156)(34,157)(35,158)(36,89)(37,90)(38,91)(39,85)(40,86)(41,87)(42,88)(50,110)(51,111)(52,112)(53,106)(54,107)(55,108)(56,109)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,117)(72,118)(73,119)(74,113)(75,114)(76,115)(77,116)(78,131)(79,132)(80,133)(81,127)(82,128)(83,129)(84,130)(92,168)(93,162)(94,163)(95,164)(96,165)(97,166)(98,167)(99,122)(100,123)(101,124)(102,125)(103,126)(104,120)(105,121)(134,145)(135,146)(136,147)(137,141)(138,142)(139,143)(140,144), (1,118,35)(2,119,29)(3,113,30)(4,114,31)(5,115,32)(6,116,33)(7,117,34)(8,40,126)(9,41,120)(10,42,121)(11,36,122)(12,37,123)(13,38,124)(14,39,125)(15,91,101)(16,85,102)(17,86,103)(18,87,104)(19,88,105)(20,89,99)(21,90,100)(22,108,133)(23,109,127)(24,110,128)(25,111,129)(26,112,130)(27,106,131)(28,107,132)(43,75,161)(44,76,155)(45,77,156)(46,71,157)(47,72,158)(48,73,159)(49,74,160)(50,82,150)(51,83,151)(52,84,152)(53,78,153)(54,79,154)(55,80,148)(56,81,149)(57,143,168)(58,144,162)(59,145,163)(60,146,164)(61,147,165)(62,141,166)(63,142,167)(64,139,92)(65,140,93)(66,134,94)(67,135,95)(68,136,96)(69,137,97)(70,138,98), (8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,138)(16,139)(17,140)(18,134)(19,135)(20,136)(21,137)(22,42)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,119)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,141)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(78,153)(79,154)(80,148)(81,149)(82,150)(83,151)(84,152)(92,102)(93,103)(94,104)(95,105)(96,99)(97,100)(98,101)(120,132)(121,133)(122,127)(123,128)(124,129)(125,130)(126,131)(155,168)(156,162)(157,163)(158,164)(159,165)(160,166)(161,167)>;
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), (1,135,55,19)(2,136,56,20)(3,137,50,21)(4,138,51,15)(5,139,52,16)(6,140,53,17)(7,134,54,18)(8,106,144,45)(9,107,145,46)(10,108,146,47)(11,109,147,48)(12,110,141,49)(13,111,142,43)(14,112,143,44)(22,60,158,121)(23,61,159,122)(24,62,160,123)(25,63,161,124)(26,57,155,125)(27,58,156,126)(28,59,157,120)(29,68,149,99)(30,69,150,100)(31,70,151,101)(32,64,152,102)(33,65,153,103)(34,66,154,104)(35,67,148,105)(36,127,165,73)(37,128,166,74)(38,129,167,75)(39,130,168,76)(40,131,162,77)(41,132,163,71)(42,133,164,72)(78,86,116,93)(79,87,117,94)(80,88,118,95)(81,89,119,96)(82,90,113,97)(83,91,114,98)(84,85,115,92), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,17)(9,18)(10,19)(11,20)(12,21)(13,15)(14,16)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,159)(30,160)(31,161)(32,155)(33,156)(34,157)(35,158)(36,89)(37,90)(38,91)(39,85)(40,86)(41,87)(42,88)(50,110)(51,111)(52,112)(53,106)(54,107)(55,108)(56,109)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,117)(72,118)(73,119)(74,113)(75,114)(76,115)(77,116)(78,131)(79,132)(80,133)(81,127)(82,128)(83,129)(84,130)(92,168)(93,162)(94,163)(95,164)(96,165)(97,166)(98,167)(99,122)(100,123)(101,124)(102,125)(103,126)(104,120)(105,121)(134,145)(135,146)(136,147)(137,141)(138,142)(139,143)(140,144), (1,118,35)(2,119,29)(3,113,30)(4,114,31)(5,115,32)(6,116,33)(7,117,34)(8,40,126)(9,41,120)(10,42,121)(11,36,122)(12,37,123)(13,38,124)(14,39,125)(15,91,101)(16,85,102)(17,86,103)(18,87,104)(19,88,105)(20,89,99)(21,90,100)(22,108,133)(23,109,127)(24,110,128)(25,111,129)(26,112,130)(27,106,131)(28,107,132)(43,75,161)(44,76,155)(45,77,156)(46,71,157)(47,72,158)(48,73,159)(49,74,160)(50,82,150)(51,83,151)(52,84,152)(53,78,153)(54,79,154)(55,80,148)(56,81,149)(57,143,168)(58,144,162)(59,145,163)(60,146,164)(61,147,165)(62,141,166)(63,142,167)(64,139,92)(65,140,93)(66,134,94)(67,135,95)(68,136,96)(69,137,97)(70,138,98), (8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,138)(16,139)(17,140)(18,134)(19,135)(20,136)(21,137)(22,42)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,119)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,141)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(78,153)(79,154)(80,148)(81,149)(82,150)(83,151)(84,152)(92,102)(93,103)(94,104)(95,105)(96,99)(97,100)(98,101)(120,132)(121,133)(122,127)(123,128)(124,129)(125,130)(126,131)(155,168)(156,162)(157,163)(158,164)(159,165)(160,166)(161,167) );
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)], [(1,135,55,19),(2,136,56,20),(3,137,50,21),(4,138,51,15),(5,139,52,16),(6,140,53,17),(7,134,54,18),(8,106,144,45),(9,107,145,46),(10,108,146,47),(11,109,147,48),(12,110,141,49),(13,111,142,43),(14,112,143,44),(22,60,158,121),(23,61,159,122),(24,62,160,123),(25,63,161,124),(26,57,155,125),(27,58,156,126),(28,59,157,120),(29,68,149,99),(30,69,150,100),(31,70,151,101),(32,64,152,102),(33,65,153,103),(34,66,154,104),(35,67,148,105),(36,127,165,73),(37,128,166,74),(38,129,167,75),(39,130,168,76),(40,131,162,77),(41,132,163,71),(42,133,164,72),(78,86,116,93),(79,87,117,94),(80,88,118,95),(81,89,119,96),(82,90,113,97),(83,91,114,98),(84,85,115,92)], [(1,47),(2,48),(3,49),(4,43),(5,44),(6,45),(7,46),(8,17),(9,18),(10,19),(11,20),(12,21),(13,15),(14,16),(22,148),(23,149),(24,150),(25,151),(26,152),(27,153),(28,154),(29,159),(30,160),(31,161),(32,155),(33,156),(34,157),(35,158),(36,89),(37,90),(38,91),(39,85),(40,86),(41,87),(42,88),(50,110),(51,111),(52,112),(53,106),(54,107),(55,108),(56,109),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,117),(72,118),(73,119),(74,113),(75,114),(76,115),(77,116),(78,131),(79,132),(80,133),(81,127),(82,128),(83,129),(84,130),(92,168),(93,162),(94,163),(95,164),(96,165),(97,166),(98,167),(99,122),(100,123),(101,124),(102,125),(103,126),(104,120),(105,121),(134,145),(135,146),(136,147),(137,141),(138,142),(139,143),(140,144)], [(1,118,35),(2,119,29),(3,113,30),(4,114,31),(5,115,32),(6,116,33),(7,117,34),(8,40,126),(9,41,120),(10,42,121),(11,36,122),(12,37,123),(13,38,124),(14,39,125),(15,91,101),(16,85,102),(17,86,103),(18,87,104),(19,88,105),(20,89,99),(21,90,100),(22,108,133),(23,109,127),(24,110,128),(25,111,129),(26,112,130),(27,106,131),(28,107,132),(43,75,161),(44,76,155),(45,77,156),(46,71,157),(47,72,158),(48,73,159),(49,74,160),(50,82,150),(51,83,151),(52,84,152),(53,78,153),(54,79,154),(55,80,148),(56,81,149),(57,143,168),(58,144,162),(59,145,163),(60,146,164),(61,147,165),(62,141,166),(63,142,167),(64,139,92),(65,140,93),(66,134,94),(67,135,95),(68,136,96),(69,137,97),(70,138,98)], [(8,106),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,138),(16,139),(17,140),(18,134),(19,135),(20,136),(21,137),(22,42),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,119),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(43,142),(44,143),(45,144),(46,145),(47,146),(48,147),(49,141),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(78,153),(79,154),(80,148),(81,149),(82,150),(83,151),(84,152),(92,102),(93,103),(94,104),(95,105),(96,99),(97,100),(98,101),(120,132),(121,133),(122,127),(123,128),(124,129),(125,130),(126,131),(155,168),(156,162),(157,163),(158,164),(159,165),(160,166),(161,167)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | ··· | 7F | 8A | 8B | 12 | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42F | 42G | ··· | 42R | 56A | ··· | 56L | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 7 | ··· | 7 | 8 | 8 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 6 | 6 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | S3 | D4 | D6 | D8 | C3⋊D4 | S3×C7 | C7×D4 | S3×C14 | C7×D8 | C7×C3⋊D4 | D4⋊S3 | C7×D4⋊S3 |
| kernel | C7×D4⋊S3 | C7×C3⋊C8 | C7×D12 | D4×C21 | D4⋊S3 | C3⋊C8 | D12 | C3×D4 | C7×D4 | C42 | C28 | C21 | C14 | D4 | C6 | C4 | C3 | C2 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 12 | 12 | 1 | 6 |
Matrix representation of C7×D4⋊S3 ►in GL4(𝔽337) generated by
| 79 | 0 | 0 | 0 |
| 0 | 79 | 0 | 0 |
| 0 | 0 | 79 | 0 |
| 0 | 0 | 0 | 79 |
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 335 |
| 0 | 0 | 1 | 336 |
| 139 | 59 | 0 | 0 |
| 278 | 198 | 0 | 0 |
| 0 | 0 | 0 | 311 |
| 0 | 0 | 324 | 0 |
| 336 | 1 | 0 | 0 |
| 336 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 336 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 336 |
G:=sub<GL(4,GF(337))| [79,0,0,0,0,79,0,0,0,0,79,0,0,0,0,79],[336,0,0,0,0,336,0,0,0,0,1,1,0,0,335,336],[139,278,0,0,59,198,0,0,0,0,0,324,0,0,311,0],[336,336,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,336,336,0,0,0,0,1,1,0,0,0,336] >;
C7×D4⋊S3 in GAP, Magma, Sage, TeX
C_7\times D_4\rtimes S_3
% in TeX
G:=Group("C7xD4:S3"); // GroupNames label
G:=SmallGroup(336,85);
// by ID
G=gap.SmallGroup(336,85);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,361,2019,1017,69,8069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations
Export