metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14.4D6, D6.3D14, Dic7.6D6, C42.17C23, Dic3.5D14, Dic21.14C22, C21⋊Q8⋊5C2, C7⋊D4⋊1S3, C3⋊D4⋊1D7, C21⋊9(C4○D4), C21⋊D4⋊4C2, (C2×C14).1D6, (C2×C6).2D14, C3⋊4(D4⋊2D7), C7⋊4(D4⋊2S3), (S3×Dic7)⋊3C2, (Dic3×D7)⋊4C2, C22.3(S3×D7), (C2×Dic21)⋊8C2, (C6×D7).4C22, C6.17(C22×D7), (S3×C14).3C22, C14.17(C22×S3), (C2×C42).11C22, (C3×Dic7).6C22, (C7×Dic3).5C22, C2.19(C2×S3×D7), (C7×C3⋊D4)⋊1C2, (C3×C7⋊D4)⋊2C2, SmallGroup(336,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.C23
G = < a,b,c,d | a6=b2=1, c14=d2=a3, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c13 >
Subgroups: 396 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, D6, C2×C6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, S3×C7, C3×D7, C42, C42, D4⋊2S3, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7⋊D4, C7×D4, C7×Dic3, C3×Dic7, Dic21, C6×D7, S3×C14, C2×C42, D4⋊2D7, Dic3×D7, S3×Dic7, C21⋊D4, C21⋊Q8, C3×C7⋊D4, C7×C3⋊D4, C2×Dic21, C42.C23
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, D4⋊2S3, C22×D7, S3×D7, D4⋊2D7, C2×S3×D7, C42.C23
►On 168 points
Generators in S168
(1 72 37 15 58 51)(2 52 59 16 38 73)(3 74 39 17 60 53)(4 54 61 18 40 75)(5 76 41 19 62 55)(6 56 63 20 42 77)(7 78 43 21 64 29)(8 30 65 22 44 79)(9 80 45 23 66 31)(10 32 67 24 46 81)(11 82 47 25 68 33)(12 34 69 26 48 83)(13 84 49 27 70 35)(14 36 71 28 50 57)(85 115 143 99 129 157)(86 158 130 100 144 116)(87 117 145 101 131 159)(88 160 132 102 146 118)(89 119 147 103 133 161)(90 162 134 104 148 120)(91 121 149 105 135 163)(92 164 136 106 150 122)(93 123 151 107 137 165)(94 166 138 108 152 124)(95 125 153 109 139 167)(96 168 140 110 154 126)(97 127 155 111 113 141)(98 142 114 112 156 128)
(1 163)(2 136)(3 165)(4 138)(5 167)(6 140)(7 141)(8 114)(9 143)(10 116)(11 145)(12 118)(13 147)(14 120)(15 149)(16 122)(17 151)(18 124)(19 153)(20 126)(21 155)(22 128)(23 157)(24 130)(25 159)(26 132)(27 161)(28 134)(29 97)(30 142)(31 99)(32 144)(33 101)(34 146)(35 103)(36 148)(37 105)(38 150)(39 107)(40 152)(41 109)(42 154)(43 111)(44 156)(45 85)(46 158)(47 87)(48 160)(49 89)(50 162)(51 91)(52 164)(53 93)(54 166)(55 95)(56 168)(57 90)(58 121)(59 92)(60 123)(61 94)(62 125)(63 96)(64 127)(65 98)(66 129)(67 100)(68 131)(69 102)(70 133)(71 104)(72 135)(73 106)(74 137)(75 108)(76 139)(77 110)(78 113)(79 112)(80 115)(81 86)(82 117)(83 88)(84 119)
(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 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 30 43 44)(31 56 45 42)(32 41 46 55)(33 54 47 40)(34 39 48 53)(35 52 49 38)(36 37 50 51)(57 58 71 72)(59 84 73 70)(60 69 74 83)(61 82 75 68)(62 67 76 81)(63 80 77 66)(64 65 78 79)(85 96 99 110)(86 109 100 95)(87 94 101 108)(88 107 102 93)(89 92 103 106)(90 105 104 91)(97 112 111 98)(113 128 127 114)(115 126 129 140)(116 139 130 125)(117 124 131 138)(118 137 132 123)(119 122 133 136)(120 135 134 121)(141 156 155 142)(143 154 157 168)(144 167 158 153)(145 152 159 166)(146 165 160 151)(147 150 161 164)(148 163 162 149)
G:=sub<Sym(168)| (1,72,37,15,58,51)(2,52,59,16,38,73)(3,74,39,17,60,53)(4,54,61,18,40,75)(5,76,41,19,62,55)(6,56,63,20,42,77)(7,78,43,21,64,29)(8,30,65,22,44,79)(9,80,45,23,66,31)(10,32,67,24,46,81)(11,82,47,25,68,33)(12,34,69,26,48,83)(13,84,49,27,70,35)(14,36,71,28,50,57)(85,115,143,99,129,157)(86,158,130,100,144,116)(87,117,145,101,131,159)(88,160,132,102,146,118)(89,119,147,103,133,161)(90,162,134,104,148,120)(91,121,149,105,135,163)(92,164,136,106,150,122)(93,123,151,107,137,165)(94,166,138,108,152,124)(95,125,153,109,139,167)(96,168,140,110,154,126)(97,127,155,111,113,141)(98,142,114,112,156,128), (1,163)(2,136)(3,165)(4,138)(5,167)(6,140)(7,141)(8,114)(9,143)(10,116)(11,145)(12,118)(13,147)(14,120)(15,149)(16,122)(17,151)(18,124)(19,153)(20,126)(21,155)(22,128)(23,157)(24,130)(25,159)(26,132)(27,161)(28,134)(29,97)(30,142)(31,99)(32,144)(33,101)(34,146)(35,103)(36,148)(37,105)(38,150)(39,107)(40,152)(41,109)(42,154)(43,111)(44,156)(45,85)(46,158)(47,87)(48,160)(49,89)(50,162)(51,91)(52,164)(53,93)(54,166)(55,95)(56,168)(57,90)(58,121)(59,92)(60,123)(61,94)(62,125)(63,96)(64,127)(65,98)(66,129)(67,100)(68,131)(69,102)(70,133)(71,104)(72,135)(73,106)(74,137)(75,108)(76,139)(77,110)(78,113)(79,112)(80,115)(81,86)(82,117)(83,88)(84,119), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,30,43,44)(31,56,45,42)(32,41,46,55)(33,54,47,40)(34,39,48,53)(35,52,49,38)(36,37,50,51)(57,58,71,72)(59,84,73,70)(60,69,74,83)(61,82,75,68)(62,67,76,81)(63,80,77,66)(64,65,78,79)(85,96,99,110)(86,109,100,95)(87,94,101,108)(88,107,102,93)(89,92,103,106)(90,105,104,91)(97,112,111,98)(113,128,127,114)(115,126,129,140)(116,139,130,125)(117,124,131,138)(118,137,132,123)(119,122,133,136)(120,135,134,121)(141,156,155,142)(143,154,157,168)(144,167,158,153)(145,152,159,166)(146,165,160,151)(147,150,161,164)(148,163,162,149)>;
G:=Group( (1,72,37,15,58,51)(2,52,59,16,38,73)(3,74,39,17,60,53)(4,54,61,18,40,75)(5,76,41,19,62,55)(6,56,63,20,42,77)(7,78,43,21,64,29)(8,30,65,22,44,79)(9,80,45,23,66,31)(10,32,67,24,46,81)(11,82,47,25,68,33)(12,34,69,26,48,83)(13,84,49,27,70,35)(14,36,71,28,50,57)(85,115,143,99,129,157)(86,158,130,100,144,116)(87,117,145,101,131,159)(88,160,132,102,146,118)(89,119,147,103,133,161)(90,162,134,104,148,120)(91,121,149,105,135,163)(92,164,136,106,150,122)(93,123,151,107,137,165)(94,166,138,108,152,124)(95,125,153,109,139,167)(96,168,140,110,154,126)(97,127,155,111,113,141)(98,142,114,112,156,128), (1,163)(2,136)(3,165)(4,138)(5,167)(6,140)(7,141)(8,114)(9,143)(10,116)(11,145)(12,118)(13,147)(14,120)(15,149)(16,122)(17,151)(18,124)(19,153)(20,126)(21,155)(22,128)(23,157)(24,130)(25,159)(26,132)(27,161)(28,134)(29,97)(30,142)(31,99)(32,144)(33,101)(34,146)(35,103)(36,148)(37,105)(38,150)(39,107)(40,152)(41,109)(42,154)(43,111)(44,156)(45,85)(46,158)(47,87)(48,160)(49,89)(50,162)(51,91)(52,164)(53,93)(54,166)(55,95)(56,168)(57,90)(58,121)(59,92)(60,123)(61,94)(62,125)(63,96)(64,127)(65,98)(66,129)(67,100)(68,131)(69,102)(70,133)(71,104)(72,135)(73,106)(74,137)(75,108)(76,139)(77,110)(78,113)(79,112)(80,115)(81,86)(82,117)(83,88)(84,119), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,30,43,44)(31,56,45,42)(32,41,46,55)(33,54,47,40)(34,39,48,53)(35,52,49,38)(36,37,50,51)(57,58,71,72)(59,84,73,70)(60,69,74,83)(61,82,75,68)(62,67,76,81)(63,80,77,66)(64,65,78,79)(85,96,99,110)(86,109,100,95)(87,94,101,108)(88,107,102,93)(89,92,103,106)(90,105,104,91)(97,112,111,98)(113,128,127,114)(115,126,129,140)(116,139,130,125)(117,124,131,138)(118,137,132,123)(119,122,133,136)(120,135,134,121)(141,156,155,142)(143,154,157,168)(144,167,158,153)(145,152,159,166)(146,165,160,151)(147,150,161,164)(148,163,162,149) );
G=PermutationGroup([[(1,72,37,15,58,51),(2,52,59,16,38,73),(3,74,39,17,60,53),(4,54,61,18,40,75),(5,76,41,19,62,55),(6,56,63,20,42,77),(7,78,43,21,64,29),(8,30,65,22,44,79),(9,80,45,23,66,31),(10,32,67,24,46,81),(11,82,47,25,68,33),(12,34,69,26,48,83),(13,84,49,27,70,35),(14,36,71,28,50,57),(85,115,143,99,129,157),(86,158,130,100,144,116),(87,117,145,101,131,159),(88,160,132,102,146,118),(89,119,147,103,133,161),(90,162,134,104,148,120),(91,121,149,105,135,163),(92,164,136,106,150,122),(93,123,151,107,137,165),(94,166,138,108,152,124),(95,125,153,109,139,167),(96,168,140,110,154,126),(97,127,155,111,113,141),(98,142,114,112,156,128)], [(1,163),(2,136),(3,165),(4,138),(5,167),(6,140),(7,141),(8,114),(9,143),(10,116),(11,145),(12,118),(13,147),(14,120),(15,149),(16,122),(17,151),(18,124),(19,153),(20,126),(21,155),(22,128),(23,157),(24,130),(25,159),(26,132),(27,161),(28,134),(29,97),(30,142),(31,99),(32,144),(33,101),(34,146),(35,103),(36,148),(37,105),(38,150),(39,107),(40,152),(41,109),(42,154),(43,111),(44,156),(45,85),(46,158),(47,87),(48,160),(49,89),(50,162),(51,91),(52,164),(53,93),(54,166),(55,95),(56,168),(57,90),(58,121),(59,92),(60,123),(61,94),(62,125),(63,96),(64,127),(65,98),(66,129),(67,100),(68,131),(69,102),(70,133),(71,104),(72,135),(73,106),(74,137),(75,108),(76,139),(77,110),(78,113),(79,112),(80,115),(81,86),(82,117),(83,88),(84,119)], [(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)], [(1,14,15,28),(2,27,16,13),(3,12,17,26),(4,25,18,11),(5,10,19,24),(6,23,20,9),(7,8,21,22),(29,30,43,44),(31,56,45,42),(32,41,46,55),(33,54,47,40),(34,39,48,53),(35,52,49,38),(36,37,50,51),(57,58,71,72),(59,84,73,70),(60,69,74,83),(61,82,75,68),(62,67,76,81),(63,80,77,66),(64,65,78,79),(85,96,99,110),(86,109,100,95),(87,94,101,108),(88,107,102,93),(89,92,103,106),(90,105,104,91),(97,112,111,98),(113,128,127,114),(115,126,129,140),(116,139,130,125),(117,124,131,138),(118,137,132,123),(119,122,133,136),(120,135,134,121),(141,156,155,142),(143,154,157,168),(144,167,158,153),(145,152,159,166),(146,165,160,151),(147,150,161,164),(148,163,162,149)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 7A | 7B | 7C | 12 | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 21A | 21B | 21C | 28A | 28B | 28C | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 6 | 14 | 2 | 6 | 14 | 21 | 21 | 42 | 2 | 4 | 28 | 2 | 2 | 2 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 4 | 4 | 4 | 12 | 12 | 12 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D7 | C4○D4 | D14 | D14 | D14 | D4⋊2S3 | S3×D7 | D4⋊2D7 | C2×S3×D7 | C42.C23 |
| kernel | C42.C23 | Dic3×D7 | S3×Dic7 | C21⋊D4 | C21⋊Q8 | C3×C7⋊D4 | C7×C3⋊D4 | C2×Dic21 | C7⋊D4 | Dic7 | D14 | C2×C14 | C3⋊D4 | C21 | Dic3 | D6 | C2×C6 | C7 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 1 | 3 | 3 | 3 | 6 |
Matrix representation of C42.C23 ►in GL6(𝔽337)
| 1 | 290 | 0 | 0 | 0 | 0 |
| 208 | 335 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 0 | 336 |
| 198 | 39 | 0 | 0 | 0 | 0 |
| 49 | 139 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 75 |
| 0 | 0 | 0 | 0 | 18 | 148 |
| 254 | 246 | 0 | 0 | 0 | 0 |
| 309 | 83 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 193 | 0 | 0 |
| 0 | 0 | 288 | 192 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 0 |
| 0 | 0 | 0 | 0 | 18 | 148 |
| 254 | 246 | 0 | 0 | 0 | 0 |
| 309 | 83 | 0 | 0 | 0 | 0 |
| 0 | 0 | 144 | 336 | 0 | 0 |
| 0 | 0 | 178 | 193 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 0 |
| 0 | 0 | 0 | 0 | 0 | 189 |
G:=sub<GL(6,GF(337))| [1,208,0,0,0,0,290,335,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336],[198,49,0,0,0,0,39,139,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,189,18,0,0,0,0,75,148],[254,309,0,0,0,0,246,83,0,0,0,0,0,0,35,288,0,0,0,0,193,192,0,0,0,0,0,0,189,18,0,0,0,0,0,148],[254,309,0,0,0,0,246,83,0,0,0,0,0,0,144,178,0,0,0,0,336,193,0,0,0,0,0,0,189,0,0,0,0,0,0,189] >;
C42.C23 in GAP, Magma, Sage, TeX
C_{42}.C_2^3 % in TeX
G:=Group("C42.C2^3"); // GroupNames label
G:=SmallGroup(336,153);
// by ID
G=gap.SmallGroup(336,153);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,116,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=1,c^14=d^2=a^3,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^13>;
// generators/relations