direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C3⋊D28, C6⋊2D28, C42⋊2D4, D14⋊6D6, Dic3⋊4D14, D42⋊8C22, C42.22C23, C21⋊5(C2×D4), C3⋊3(C2×D28), C14⋊1(C3⋊D4), (C2×Dic3)⋊4D7, (C2×C14).17D6, (C2×C6).17D14, (C6×D7)⋊6C22, (C22×D7)⋊3S3, (Dic3×C14)⋊6C2, (C22×D21)⋊4C2, C22.15(S3×D7), C6.22(C22×D7), C14.22(C22×S3), (C2×C42).16C22, (C7×Dic3)⋊7C22, (C2×C6×D7)⋊2C2, C7⋊1(C2×C3⋊D4), C2.22(C2×S3×D7), SmallGroup(336,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C3⋊D28
G = < a,b,c,d | a2=b3=c28=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 732 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C7, C2×C4, D4, C23, Dic3, D6, C2×C6, C2×C6, D7, C14, C14, C2×D4, C21, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C28, D14, D14, C2×C14, C3×D7, D21, C42, C42, C2×C3⋊D4, D28, C2×C28, C22×D7, C22×D7, C7×Dic3, C6×D7, C6×D7, D42, D42, C2×C42, C2×D28, C3⋊D28, Dic3×C14, C2×C6×D7, C22×D21, C2×C3⋊D28
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C3⋊D4, C22×S3, D14, C2×C3⋊D4, D28, C22×D7, S3×D7, C2×D28, C3⋊D28, C2×S3×D7, C2×C3⋊D28
►On 168 points
Generators in S168
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 168)(26 141)(27 142)(28 143)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)(101 136)(102 137)(103 138)(104 139)(105 140)(106 113)(107 114)(108 115)(109 116)(110 117)(111 118)(112 119)
(1 111 81)(2 82 112)(3 85 83)(4 84 86)(5 87 57)(6 58 88)(7 89 59)(8 60 90)(9 91 61)(10 62 92)(11 93 63)(12 64 94)(13 95 65)(14 66 96)(15 97 67)(16 68 98)(17 99 69)(18 70 100)(19 101 71)(20 72 102)(21 103 73)(22 74 104)(23 105 75)(24 76 106)(25 107 77)(26 78 108)(27 109 79)(28 80 110)(29 168 114)(30 115 141)(31 142 116)(32 117 143)(33 144 118)(34 119 145)(35 146 120)(36 121 147)(37 148 122)(38 123 149)(39 150 124)(40 125 151)(41 152 126)(42 127 153)(43 154 128)(44 129 155)(45 156 130)(46 131 157)(47 158 132)(48 133 159)(49 160 134)(50 135 161)(51 162 136)(52 137 163)(53 164 138)(54 139 165)(55 166 140)(56 113 167)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 128)(30 127)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(37 120)(38 119)(39 118)(40 117)(41 116)(42 115)(43 114)(44 113)(45 140)(46 139)(47 138)(48 137)(49 136)(50 135)(51 134)(52 133)(53 132)(54 131)(55 130)(56 129)(57 85)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)
G:=sub<Sym(168)| (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,161)(19,162)(20,163)(21,164)(22,165)(23,166)(24,167)(25,168)(26,141)(27,142)(28,143)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,137)(103,138)(104,139)(105,140)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)(112,119), (1,111,81)(2,82,112)(3,85,83)(4,84,86)(5,87,57)(6,58,88)(7,89,59)(8,60,90)(9,91,61)(10,62,92)(11,93,63)(12,64,94)(13,95,65)(14,66,96)(15,97,67)(16,68,98)(17,99,69)(18,70,100)(19,101,71)(20,72,102)(21,103,73)(22,74,104)(23,105,75)(24,76,106)(25,107,77)(26,78,108)(27,109,79)(28,80,110)(29,168,114)(30,115,141)(31,142,116)(32,117,143)(33,144,118)(34,119,145)(35,146,120)(36,121,147)(37,148,122)(38,123,149)(39,150,124)(40,125,151)(41,152,126)(42,127,153)(43,154,128)(44,129,155)(45,156,130)(46,131,157)(47,158,132)(48,133,159)(49,160,134)(50,135,161)(51,162,136)(52,137,163)(53,164,138)(54,139,165)(55,166,140)(56,113,167), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,128)(30,127)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,120)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,85)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)>;
G:=Group( (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,161)(19,162)(20,163)(21,164)(22,165)(23,166)(24,167)(25,168)(26,141)(27,142)(28,143)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,137)(103,138)(104,139)(105,140)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)(112,119), (1,111,81)(2,82,112)(3,85,83)(4,84,86)(5,87,57)(6,58,88)(7,89,59)(8,60,90)(9,91,61)(10,62,92)(11,93,63)(12,64,94)(13,95,65)(14,66,96)(15,97,67)(16,68,98)(17,99,69)(18,70,100)(19,101,71)(20,72,102)(21,103,73)(22,74,104)(23,105,75)(24,76,106)(25,107,77)(26,78,108)(27,109,79)(28,80,110)(29,168,114)(30,115,141)(31,142,116)(32,117,143)(33,144,118)(34,119,145)(35,146,120)(36,121,147)(37,148,122)(38,123,149)(39,150,124)(40,125,151)(41,152,126)(42,127,153)(43,154,128)(44,129,155)(45,156,130)(46,131,157)(47,158,132)(48,133,159)(49,160,134)(50,135,161)(51,162,136)(52,137,163)(53,164,138)(54,139,165)(55,166,140)(56,113,167), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,128)(30,127)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,120)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,85)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162) );
G=PermutationGroup([[(1,144),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,151),(9,152),(10,153),(11,154),(12,155),(13,156),(14,157),(15,158),(16,159),(17,160),(18,161),(19,162),(20,163),(21,164),(22,165),(23,166),(24,167),(25,168),(26,141),(27,142),(28,143),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(85,120),(86,121),(87,122),(88,123),(89,124),(90,125),(91,126),(92,127),(93,128),(94,129),(95,130),(96,131),(97,132),(98,133),(99,134),(100,135),(101,136),(102,137),(103,138),(104,139),(105,140),(106,113),(107,114),(108,115),(109,116),(110,117),(111,118),(112,119)], [(1,111,81),(2,82,112),(3,85,83),(4,84,86),(5,87,57),(6,58,88),(7,89,59),(8,60,90),(9,91,61),(10,62,92),(11,93,63),(12,64,94),(13,95,65),(14,66,96),(15,97,67),(16,68,98),(17,99,69),(18,70,100),(19,101,71),(20,72,102),(21,103,73),(22,74,104),(23,105,75),(24,76,106),(25,107,77),(26,78,108),(27,109,79),(28,80,110),(29,168,114),(30,115,141),(31,142,116),(32,117,143),(33,144,118),(34,119,145),(35,146,120),(36,121,147),(37,148,122),(38,123,149),(39,150,124),(40,125,151),(41,152,126),(42,127,153),(43,154,128),(44,129,155),(45,156,130),(46,131,157),(47,158,132),(48,133,159),(49,160,134),(50,135,161),(51,162,136),(52,137,163),(53,164,138),(54,139,165),(55,166,140),(56,113,167)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,128),(30,127),(31,126),(32,125),(33,124),(34,123),(35,122),(36,121),(37,120),(38,119),(39,118),(40,117),(41,116),(42,115),(43,114),(44,113),(45,140),(46,139),(47,138),(48,137),(49,136),(50,135),(51,134),(52,133),(53,132),(54,131),(55,130),(56,129),(57,85),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86),(141,153),(142,152),(143,151),(144,150),(145,149),(146,148),(154,168),(155,167),(156,166),(157,165),(158,164),(159,163),(160,162)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 7A | 7B | 7C | 14A | ··· | 14I | 21A | 21B | 21C | 28A | ··· | 28L | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 42 | 42 | 2 | 6 | 6 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D7 | C3⋊D4 | D14 | D14 | D28 | S3×D7 | C3⋊D28 | C2×S3×D7 |
| kernel | C2×C3⋊D28 | C3⋊D28 | Dic3×C14 | C2×C6×D7 | C22×D21 | C22×D7 | C42 | D14 | C2×C14 | C2×Dic3 | C14 | Dic3 | C2×C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 12 | 3 | 6 | 3 |
Matrix representation of C2×C3⋊D28 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 1 |
| 0 | 0 | 336 | 0 |
| 275 | 219 | 0 | 0 |
| 228 | 48 | 0 | 0 |
| 0 | 0 | 0 | 336 |
| 0 | 0 | 336 | 0 |
| 336 | 0 | 0 | 0 |
| 178 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,1,0,0,0,0,336,336,0,0,1,0],[275,228,0,0,219,48,0,0,0,0,0,336,0,0,336,0],[336,178,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C2×C3⋊D28 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes D_{28} % in TeX
G:=Group("C2xC3:D28"); // GroupNames label
G:=SmallGroup(336,158);
// by ID
G=gap.SmallGroup(336,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,55,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^28=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations