metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊2Dic28, C23.41D28, Dic14.32D4, (C2×C14)⋊1Q16, (C2×C8).5D14, (C2×C4).36D28, C4.124(D4×D7), (C2×C28).47D4, C22⋊C8.4D7, C14.6(C2×Q16), (C2×Dic28)⋊3C2, C28.336(C2×D4), (C2×C56).5C22, C7⋊1(C22⋊Q16), C2.8(C2×Dic28), C14.12C22≀C2, C28.44D4⋊7C2, (C22×C4).89D14, (C22×C14).59D4, (C2×C28).749C23, C28.48D4.4C2, C22.112(C2×D28), C4⋊Dic7.16C22, C2.15(C22⋊D28), C2.15(C8.D14), C14.12(C8.C22), (C22×C28).55C22, (C22×Dic14).3C2, (C2×Dic14).13C22, (C7×C22⋊C8).6C2, (C2×C14).132(C2×D4), (C2×C4).694(C22×D7), SmallGroup(448,273)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊Dic28
G = < a,b,c,d | a2=b2=c56=1, d2=c28, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 764 in 148 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C56, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C22⋊Q16, Dic28, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×C56, C2×Dic14, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C28.44D4, C7×C22⋊C8, C2×Dic28, C28.48D4, C22×Dic14, C22⋊Dic28
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C22≀C2, C2×Q16, C8.C22, D28, C22×D7, C22⋊Q16, Dic28, C2×D28, D4×D7, C22⋊D28, C2×Dic28, C8.D14, C22⋊Dic28
►On 224 points
Generators in S224
(2 200)(4 202)(6 204)(8 206)(10 208)(12 210)(14 212)(16 214)(18 216)(20 218)(22 220)(24 222)(26 224)(28 170)(30 172)(32 174)(34 176)(36 178)(38 180)(40 182)(42 184)(44 186)(46 188)(48 190)(50 192)(52 194)(54 196)(56 198)(58 134)(60 136)(62 138)(64 140)(66 142)(68 144)(70 146)(72 148)(74 150)(76 152)(78 154)(80 156)(82 158)(84 160)(86 162)(88 164)(90 166)(92 168)(94 114)(96 116)(98 118)(100 120)(102 122)(104 124)(106 126)(108 128)(110 130)(112 132)
(1 199)(2 200)(3 201)(4 202)(5 203)(6 204)(7 205)(8 206)(9 207)(10 208)(11 209)(12 210)(13 211)(14 212)(15 213)(16 214)(17 215)(18 216)(19 217)(20 218)(21 219)(22 220)(23 221)(24 222)(25 223)(26 224)(27 169)(28 170)(29 171)(30 172)(31 173)(32 174)(33 175)(34 176)(35 177)(36 178)(37 179)(38 180)(39 181)(40 182)(41 183)(42 184)(43 185)(44 186)(45 187)(46 188)(47 189)(48 190)(49 191)(50 192)(51 193)(52 194)(53 195)(54 196)(55 197)(56 198)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 147)(72 148)(73 149)(74 150)(75 151)(76 152)(77 153)(78 154)(79 155)(80 156)(81 157)(82 158)(83 159)(84 160)(85 161)(86 162)(87 163)(88 164)(89 165)(90 166)(91 167)(92 168)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 105 29 77)(2 104 30 76)(3 103 31 75)(4 102 32 74)(5 101 33 73)(6 100 34 72)(7 99 35 71)(8 98 36 70)(9 97 37 69)(10 96 38 68)(11 95 39 67)(12 94 40 66)(13 93 41 65)(14 92 42 64)(15 91 43 63)(16 90 44 62)(17 89 45 61)(18 88 46 60)(19 87 47 59)(20 86 48 58)(21 85 49 57)(22 84 50 112)(23 83 51 111)(24 82 52 110)(25 81 53 109)(26 80 54 108)(27 79 55 107)(28 78 56 106)(113 183 141 211)(114 182 142 210)(115 181 143 209)(116 180 144 208)(117 179 145 207)(118 178 146 206)(119 177 147 205)(120 176 148 204)(121 175 149 203)(122 174 150 202)(123 173 151 201)(124 172 152 200)(125 171 153 199)(126 170 154 198)(127 169 155 197)(128 224 156 196)(129 223 157 195)(130 222 158 194)(131 221 159 193)(132 220 160 192)(133 219 161 191)(134 218 162 190)(135 217 163 189)(136 216 164 188)(137 215 165 187)(138 214 166 186)(139 213 167 185)(140 212 168 184)
G:=sub<Sym(224)| (2,200)(4,202)(6,204)(8,206)(10,208)(12,210)(14,212)(16,214)(18,216)(20,218)(22,220)(24,222)(26,224)(28,170)(30,172)(32,174)(34,176)(36,178)(38,180)(40,182)(42,184)(44,186)(46,188)(48,190)(50,192)(52,194)(54,196)(56,198)(58,134)(60,136)(62,138)(64,140)(66,142)(68,144)(70,146)(72,148)(74,150)(76,152)(78,154)(80,156)(82,158)(84,160)(86,162)(88,164)(90,166)(92,168)(94,114)(96,116)(98,118)(100,120)(102,122)(104,124)(106,126)(108,128)(110,130)(112,132), (1,199)(2,200)(3,201)(4,202)(5,203)(6,204)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,211)(14,212)(15,213)(16,214)(17,215)(18,216)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,169)(28,170)(29,171)(30,172)(31,173)(32,174)(33,175)(34,176)(35,177)(36,178)(37,179)(38,180)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,188)(47,189)(48,190)(49,191)(50,192)(51,193)(52,194)(53,195)(54,196)(55,197)(56,198)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156)(81,157)(82,158)(83,159)(84,160)(85,161)(86,162)(87,163)(88,164)(89,165)(90,166)(91,167)(92,168)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,105,29,77)(2,104,30,76)(3,103,31,75)(4,102,32,74)(5,101,33,73)(6,100,34,72)(7,99,35,71)(8,98,36,70)(9,97,37,69)(10,96,38,68)(11,95,39,67)(12,94,40,66)(13,93,41,65)(14,92,42,64)(15,91,43,63)(16,90,44,62)(17,89,45,61)(18,88,46,60)(19,87,47,59)(20,86,48,58)(21,85,49,57)(22,84,50,112)(23,83,51,111)(24,82,52,110)(25,81,53,109)(26,80,54,108)(27,79,55,107)(28,78,56,106)(113,183,141,211)(114,182,142,210)(115,181,143,209)(116,180,144,208)(117,179,145,207)(118,178,146,206)(119,177,147,205)(120,176,148,204)(121,175,149,203)(122,174,150,202)(123,173,151,201)(124,172,152,200)(125,171,153,199)(126,170,154,198)(127,169,155,197)(128,224,156,196)(129,223,157,195)(130,222,158,194)(131,221,159,193)(132,220,160,192)(133,219,161,191)(134,218,162,190)(135,217,163,189)(136,216,164,188)(137,215,165,187)(138,214,166,186)(139,213,167,185)(140,212,168,184)>;
G:=Group( (2,200)(4,202)(6,204)(8,206)(10,208)(12,210)(14,212)(16,214)(18,216)(20,218)(22,220)(24,222)(26,224)(28,170)(30,172)(32,174)(34,176)(36,178)(38,180)(40,182)(42,184)(44,186)(46,188)(48,190)(50,192)(52,194)(54,196)(56,198)(58,134)(60,136)(62,138)(64,140)(66,142)(68,144)(70,146)(72,148)(74,150)(76,152)(78,154)(80,156)(82,158)(84,160)(86,162)(88,164)(90,166)(92,168)(94,114)(96,116)(98,118)(100,120)(102,122)(104,124)(106,126)(108,128)(110,130)(112,132), (1,199)(2,200)(3,201)(4,202)(5,203)(6,204)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,211)(14,212)(15,213)(16,214)(17,215)(18,216)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,169)(28,170)(29,171)(30,172)(31,173)(32,174)(33,175)(34,176)(35,177)(36,178)(37,179)(38,180)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,188)(47,189)(48,190)(49,191)(50,192)(51,193)(52,194)(53,195)(54,196)(55,197)(56,198)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156)(81,157)(82,158)(83,159)(84,160)(85,161)(86,162)(87,163)(88,164)(89,165)(90,166)(91,167)(92,168)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,105,29,77)(2,104,30,76)(3,103,31,75)(4,102,32,74)(5,101,33,73)(6,100,34,72)(7,99,35,71)(8,98,36,70)(9,97,37,69)(10,96,38,68)(11,95,39,67)(12,94,40,66)(13,93,41,65)(14,92,42,64)(15,91,43,63)(16,90,44,62)(17,89,45,61)(18,88,46,60)(19,87,47,59)(20,86,48,58)(21,85,49,57)(22,84,50,112)(23,83,51,111)(24,82,52,110)(25,81,53,109)(26,80,54,108)(27,79,55,107)(28,78,56,106)(113,183,141,211)(114,182,142,210)(115,181,143,209)(116,180,144,208)(117,179,145,207)(118,178,146,206)(119,177,147,205)(120,176,148,204)(121,175,149,203)(122,174,150,202)(123,173,151,201)(124,172,152,200)(125,171,153,199)(126,170,154,198)(127,169,155,197)(128,224,156,196)(129,223,157,195)(130,222,158,194)(131,221,159,193)(132,220,160,192)(133,219,161,191)(134,218,162,190)(135,217,163,189)(136,216,164,188)(137,215,165,187)(138,214,166,186)(139,213,167,185)(140,212,168,184) );
G=PermutationGroup([[(2,200),(4,202),(6,204),(8,206),(10,208),(12,210),(14,212),(16,214),(18,216),(20,218),(22,220),(24,222),(26,224),(28,170),(30,172),(32,174),(34,176),(36,178),(38,180),(40,182),(42,184),(44,186),(46,188),(48,190),(50,192),(52,194),(54,196),(56,198),(58,134),(60,136),(62,138),(64,140),(66,142),(68,144),(70,146),(72,148),(74,150),(76,152),(78,154),(80,156),(82,158),(84,160),(86,162),(88,164),(90,166),(92,168),(94,114),(96,116),(98,118),(100,120),(102,122),(104,124),(106,126),(108,128),(110,130),(112,132)], [(1,199),(2,200),(3,201),(4,202),(5,203),(6,204),(7,205),(8,206),(9,207),(10,208),(11,209),(12,210),(13,211),(14,212),(15,213),(16,214),(17,215),(18,216),(19,217),(20,218),(21,219),(22,220),(23,221),(24,222),(25,223),(26,224),(27,169),(28,170),(29,171),(30,172),(31,173),(32,174),(33,175),(34,176),(35,177),(36,178),(37,179),(38,180),(39,181),(40,182),(41,183),(42,184),(43,185),(44,186),(45,187),(46,188),(47,189),(48,190),(49,191),(50,192),(51,193),(52,194),(53,195),(54,196),(55,197),(56,198),(57,133),(58,134),(59,135),(60,136),(61,137),(62,138),(63,139),(64,140),(65,141),(66,142),(67,143),(68,144),(69,145),(70,146),(71,147),(72,148),(73,149),(74,150),(75,151),(76,152),(77,153),(78,154),(79,155),(80,156),(81,157),(82,158),(83,159),(84,160),(85,161),(86,162),(87,163),(88,164),(89,165),(90,166),(91,167),(92,168),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132)], [(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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,105,29,77),(2,104,30,76),(3,103,31,75),(4,102,32,74),(5,101,33,73),(6,100,34,72),(7,99,35,71),(8,98,36,70),(9,97,37,69),(10,96,38,68),(11,95,39,67),(12,94,40,66),(13,93,41,65),(14,92,42,64),(15,91,43,63),(16,90,44,62),(17,89,45,61),(18,88,46,60),(19,87,47,59),(20,86,48,58),(21,85,49,57),(22,84,50,112),(23,83,51,111),(24,82,52,110),(25,81,53,109),(26,80,54,108),(27,79,55,107),(28,78,56,106),(113,183,141,211),(114,182,142,210),(115,181,143,209),(116,180,144,208),(117,179,145,207),(118,178,146,206),(119,177,147,205),(120,176,148,204),(121,175,149,203),(122,174,150,202),(123,173,151,201),(124,172,152,200),(125,171,153,199),(126,170,154,198),(127,169,155,197),(128,224,156,196),(129,223,157,195),(130,222,158,194),(131,221,159,193),(132,220,160,192),(133,219,161,191),(134,218,162,190),(135,217,163,189),(136,216,164,188),(137,215,165,187),(138,214,166,186),(139,213,167,185),(140,212,168,184)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | Q16 | D14 | D14 | D28 | D28 | Dic28 | C8.C22 | D4×D7 | C8.D14 |
| kernel | C22⋊Dic28 | C28.44D4 | C7×C22⋊C8 | C2×Dic28 | C28.48D4 | C22×Dic14 | Dic14 | C2×C28 | C22×C14 | C22⋊C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C14 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 3 | 4 | 6 | 3 | 6 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of C22⋊Dic28 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 32 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 106 | 0 | 0 |
| 81 | 1 | 0 | 0 |
| 0 | 0 | 70 | 29 |
| 0 | 0 | 84 | 101 |
| 1 | 0 | 0 | 0 |
| 32 | 112 | 0 | 0 |
| 0 | 0 | 17 | 80 |
| 0 | 0 | 67 | 96 |
G:=sub<GL(4,GF(113))| [1,32,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,81,0,0,106,1,0,0,0,0,70,84,0,0,29,101],[1,32,0,0,0,112,0,0,0,0,17,67,0,0,80,96] >;
C22⋊Dic28 in GAP, Magma, Sage, TeX
C_2^2\rtimes {\rm Dic}_{28} % in TeX
G:=Group("C2^2:Dic28"); // GroupNames label
G:=SmallGroup(448,273);
// by ID
G=gap.SmallGroup(448,273);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,254,219,226,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^56=1,d^2=c^28,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations