metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊2D4, C23.17D28, C8⋊1(C7⋊D4), C7⋊6(C8⋊D4), C56⋊1C4⋊18C2, (C2×C8).77D14, (C2×C4).51D28, C28.420(C2×D4), (C2×C28).297D4, (C2×M4(2))⋊1D7, C2.D56⋊41C2, C28⋊7D4.17C2, (C14×M4(2))⋊1C2, (C2×C56).63C22, C28.230(C4○D4), C4.114(C4○D28), C28.44D4⋊41C2, C28.48D4⋊41C2, C2.22(C8⋊D14), C14.73(C4⋊D4), C2.21(C28⋊7D4), C14.22(C8⋊C22), (C2×C28).775C23, (C2×D28).20C22, C22.134(C2×D28), (C22×C4).142D14, (C22×C14).103D4, C4⋊Dic7.26C22, C2.22(C8.D14), C14.22(C8.C22), (C22×C28).304C22, (C2×Dic14).19C22, (C2×C56⋊C2)⋊2C2, C4.113(C2×C7⋊D4), (C2×C14).165(C2×D4), (C2×C4).724(C22×D7), SmallGroup(448,668)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊2D4
G = < a,b,c | a56=b4=c2=1, bab-1=a-1, cac=a27, cbc=b-1 >
Subgroups: 708 in 120 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C56, C56, Dic14, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊D4, C56⋊C2, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×C56, C7×M4(2), C2×Dic14, C2×D28, C2×C7⋊D4, C22×C28, C28.44D4, C56⋊1C4, C2.D56, C2×C56⋊C2, C28.48D4, C28⋊7D4, C14×M4(2), C56⋊2D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C8.C22, D28, C7⋊D4, C22×D7, C8⋊D4, C2×D28, C4○D28, C2×C7⋊D4, C8⋊D14, C8.D14, C28⋊7D4, C56⋊2D4
►On 224 points
Generators in S224
(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 68 149 179)(2 67 150 178)(3 66 151 177)(4 65 152 176)(5 64 153 175)(6 63 154 174)(7 62 155 173)(8 61 156 172)(9 60 157 171)(10 59 158 170)(11 58 159 169)(12 57 160 224)(13 112 161 223)(14 111 162 222)(15 110 163 221)(16 109 164 220)(17 108 165 219)(18 107 166 218)(19 106 167 217)(20 105 168 216)(21 104 113 215)(22 103 114 214)(23 102 115 213)(24 101 116 212)(25 100 117 211)(26 99 118 210)(27 98 119 209)(28 97 120 208)(29 96 121 207)(30 95 122 206)(31 94 123 205)(32 93 124 204)(33 92 125 203)(34 91 126 202)(35 90 127 201)(36 89 128 200)(37 88 129 199)(38 87 130 198)(39 86 131 197)(40 85 132 196)(41 84 133 195)(42 83 134 194)(43 82 135 193)(44 81 136 192)(45 80 137 191)(46 79 138 190)(47 78 139 189)(48 77 140 188)(49 76 141 187)(50 75 142 186)(51 74 143 185)(52 73 144 184)(53 72 145 183)(54 71 146 182)(55 70 147 181)(56 69 148 180)
(1 179)(2 206)(3 177)(4 204)(5 175)(6 202)(7 173)(8 200)(9 171)(10 198)(11 169)(12 196)(13 223)(14 194)(15 221)(16 192)(17 219)(18 190)(19 217)(20 188)(21 215)(22 186)(23 213)(24 184)(25 211)(26 182)(27 209)(28 180)(29 207)(30 178)(31 205)(32 176)(33 203)(34 174)(35 201)(36 172)(37 199)(38 170)(39 197)(40 224)(41 195)(42 222)(43 193)(44 220)(45 191)(46 218)(47 189)(48 216)(49 187)(50 214)(51 185)(52 212)(53 183)(54 210)(55 181)(56 208)(57 132)(58 159)(59 130)(60 157)(61 128)(62 155)(63 126)(64 153)(65 124)(66 151)(67 122)(68 149)(69 120)(70 147)(71 118)(72 145)(73 116)(74 143)(75 114)(76 141)(77 168)(78 139)(79 166)(80 137)(81 164)(82 135)(83 162)(84 133)(85 160)(86 131)(87 158)(88 129)(89 156)(90 127)(91 154)(92 125)(93 152)(94 123)(95 150)(96 121)(97 148)(98 119)(99 146)(100 117)(101 144)(102 115)(103 142)(104 113)(105 140)(106 167)(107 138)(108 165)(109 136)(110 163)(111 134)(112 161)
G:=sub<Sym(224)| (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), (1,68,149,179)(2,67,150,178)(3,66,151,177)(4,65,152,176)(5,64,153,175)(6,63,154,174)(7,62,155,173)(8,61,156,172)(9,60,157,171)(10,59,158,170)(11,58,159,169)(12,57,160,224)(13,112,161,223)(14,111,162,222)(15,110,163,221)(16,109,164,220)(17,108,165,219)(18,107,166,218)(19,106,167,217)(20,105,168,216)(21,104,113,215)(22,103,114,214)(23,102,115,213)(24,101,116,212)(25,100,117,211)(26,99,118,210)(27,98,119,209)(28,97,120,208)(29,96,121,207)(30,95,122,206)(31,94,123,205)(32,93,124,204)(33,92,125,203)(34,91,126,202)(35,90,127,201)(36,89,128,200)(37,88,129,199)(38,87,130,198)(39,86,131,197)(40,85,132,196)(41,84,133,195)(42,83,134,194)(43,82,135,193)(44,81,136,192)(45,80,137,191)(46,79,138,190)(47,78,139,189)(48,77,140,188)(49,76,141,187)(50,75,142,186)(51,74,143,185)(52,73,144,184)(53,72,145,183)(54,71,146,182)(55,70,147,181)(56,69,148,180), (1,179)(2,206)(3,177)(4,204)(5,175)(6,202)(7,173)(8,200)(9,171)(10,198)(11,169)(12,196)(13,223)(14,194)(15,221)(16,192)(17,219)(18,190)(19,217)(20,188)(21,215)(22,186)(23,213)(24,184)(25,211)(26,182)(27,209)(28,180)(29,207)(30,178)(31,205)(32,176)(33,203)(34,174)(35,201)(36,172)(37,199)(38,170)(39,197)(40,224)(41,195)(42,222)(43,193)(44,220)(45,191)(46,218)(47,189)(48,216)(49,187)(50,214)(51,185)(52,212)(53,183)(54,210)(55,181)(56,208)(57,132)(58,159)(59,130)(60,157)(61,128)(62,155)(63,126)(64,153)(65,124)(66,151)(67,122)(68,149)(69,120)(70,147)(71,118)(72,145)(73,116)(74,143)(75,114)(76,141)(77,168)(78,139)(79,166)(80,137)(81,164)(82,135)(83,162)(84,133)(85,160)(86,131)(87,158)(88,129)(89,156)(90,127)(91,154)(92,125)(93,152)(94,123)(95,150)(96,121)(97,148)(98,119)(99,146)(100,117)(101,144)(102,115)(103,142)(104,113)(105,140)(106,167)(107,138)(108,165)(109,136)(110,163)(111,134)(112,161)>;
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)(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), (1,68,149,179)(2,67,150,178)(3,66,151,177)(4,65,152,176)(5,64,153,175)(6,63,154,174)(7,62,155,173)(8,61,156,172)(9,60,157,171)(10,59,158,170)(11,58,159,169)(12,57,160,224)(13,112,161,223)(14,111,162,222)(15,110,163,221)(16,109,164,220)(17,108,165,219)(18,107,166,218)(19,106,167,217)(20,105,168,216)(21,104,113,215)(22,103,114,214)(23,102,115,213)(24,101,116,212)(25,100,117,211)(26,99,118,210)(27,98,119,209)(28,97,120,208)(29,96,121,207)(30,95,122,206)(31,94,123,205)(32,93,124,204)(33,92,125,203)(34,91,126,202)(35,90,127,201)(36,89,128,200)(37,88,129,199)(38,87,130,198)(39,86,131,197)(40,85,132,196)(41,84,133,195)(42,83,134,194)(43,82,135,193)(44,81,136,192)(45,80,137,191)(46,79,138,190)(47,78,139,189)(48,77,140,188)(49,76,141,187)(50,75,142,186)(51,74,143,185)(52,73,144,184)(53,72,145,183)(54,71,146,182)(55,70,147,181)(56,69,148,180), (1,179)(2,206)(3,177)(4,204)(5,175)(6,202)(7,173)(8,200)(9,171)(10,198)(11,169)(12,196)(13,223)(14,194)(15,221)(16,192)(17,219)(18,190)(19,217)(20,188)(21,215)(22,186)(23,213)(24,184)(25,211)(26,182)(27,209)(28,180)(29,207)(30,178)(31,205)(32,176)(33,203)(34,174)(35,201)(36,172)(37,199)(38,170)(39,197)(40,224)(41,195)(42,222)(43,193)(44,220)(45,191)(46,218)(47,189)(48,216)(49,187)(50,214)(51,185)(52,212)(53,183)(54,210)(55,181)(56,208)(57,132)(58,159)(59,130)(60,157)(61,128)(62,155)(63,126)(64,153)(65,124)(66,151)(67,122)(68,149)(69,120)(70,147)(71,118)(72,145)(73,116)(74,143)(75,114)(76,141)(77,168)(78,139)(79,166)(80,137)(81,164)(82,135)(83,162)(84,133)(85,160)(86,131)(87,158)(88,129)(89,156)(90,127)(91,154)(92,125)(93,152)(94,123)(95,150)(96,121)(97,148)(98,119)(99,146)(100,117)(101,144)(102,115)(103,142)(104,113)(105,140)(106,167)(107,138)(108,165)(109,136)(110,163)(111,134)(112,161) );
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),(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)], [(1,68,149,179),(2,67,150,178),(3,66,151,177),(4,65,152,176),(5,64,153,175),(6,63,154,174),(7,62,155,173),(8,61,156,172),(9,60,157,171),(10,59,158,170),(11,58,159,169),(12,57,160,224),(13,112,161,223),(14,111,162,222),(15,110,163,221),(16,109,164,220),(17,108,165,219),(18,107,166,218),(19,106,167,217),(20,105,168,216),(21,104,113,215),(22,103,114,214),(23,102,115,213),(24,101,116,212),(25,100,117,211),(26,99,118,210),(27,98,119,209),(28,97,120,208),(29,96,121,207),(30,95,122,206),(31,94,123,205),(32,93,124,204),(33,92,125,203),(34,91,126,202),(35,90,127,201),(36,89,128,200),(37,88,129,199),(38,87,130,198),(39,86,131,197),(40,85,132,196),(41,84,133,195),(42,83,134,194),(43,82,135,193),(44,81,136,192),(45,80,137,191),(46,79,138,190),(47,78,139,189),(48,77,140,188),(49,76,141,187),(50,75,142,186),(51,74,143,185),(52,73,144,184),(53,72,145,183),(54,71,146,182),(55,70,147,181),(56,69,148,180)], [(1,179),(2,206),(3,177),(4,204),(5,175),(6,202),(7,173),(8,200),(9,171),(10,198),(11,169),(12,196),(13,223),(14,194),(15,221),(16,192),(17,219),(18,190),(19,217),(20,188),(21,215),(22,186),(23,213),(24,184),(25,211),(26,182),(27,209),(28,180),(29,207),(30,178),(31,205),(32,176),(33,203),(34,174),(35,201),(36,172),(37,199),(38,170),(39,197),(40,224),(41,195),(42,222),(43,193),(44,220),(45,191),(46,218),(47,189),(48,216),(49,187),(50,214),(51,185),(52,212),(53,183),(54,210),(55,181),(56,208),(57,132),(58,159),(59,130),(60,157),(61,128),(62,155),(63,126),(64,153),(65,124),(66,151),(67,122),(68,149),(69,120),(70,147),(71,118),(72,145),(73,116),(74,143),(75,114),(76,141),(77,168),(78,139),(79,166),(80,137),(81,164),(82,135),(83,162),(84,133),(85,160),(86,131),(87,158),(88,129),(89,156),(90,127),(91,154),(92,125),(93,152),(94,123),(95,150),(96,121),(97,148),(98,119),(99,146),(100,117),(101,144),(102,115),(103,142),(104,113),(105,140),(106,167),(107,138),(108,165),(109,136),(110,163),(111,134),(112,161)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 56 | 2 | 2 | 4 | 56 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D28 | D28 | C4○D28 | C8⋊C22 | C8.C22 | C8⋊D14 | C8.D14 |
| kernel | C56⋊2D4 | C28.44D4 | C56⋊1C4 | C2.D56 | C2×C56⋊C2 | C28.48D4 | C28⋊7D4 | C14×M4(2) | C56 | C2×C28 | C22×C14 | C2×M4(2) | C28 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 6 | 6 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C56⋊2D4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 49 | 0 | 80 | 35 |
| 0 | 0 | 65 | 30 | 0 | 111 |
| 0 | 0 | 99 | 14 | 0 | 112 |
| 0 | 0 | 111 | 81 | 1 | 34 |
| 39 | 87 | 0 | 0 | 0 | 0 |
| 102 | 74 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 52 | 0 | 0 |
| 0 | 0 | 59 | 53 | 0 | 0 |
| 0 | 0 | 111 | 81 | 1 | 34 |
| 0 | 0 | 99 | 14 | 0 | 112 |
| 39 | 87 | 0 | 0 | 0 | 0 |
| 15 | 74 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 52 | 0 | 0 |
| 0 | 0 | 59 | 53 | 0 | 0 |
| 0 | 0 | 61 | 85 | 112 | 79 |
| 0 | 0 | 34 | 97 | 0 | 1 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,49,65,99,111,0,0,0,30,14,81,0,0,80,0,0,1,0,0,35,111,112,34],[39,102,0,0,0,0,87,74,0,0,0,0,0,0,60,59,111,99,0,0,52,53,81,14,0,0,0,0,1,0,0,0,0,0,34,112],[39,15,0,0,0,0,87,74,0,0,0,0,0,0,60,59,61,34,0,0,52,53,85,97,0,0,0,0,112,0,0,0,0,0,79,1] >;
C56⋊2D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_2D_4 % in TeX
G:=Group("C56:2D4"); // GroupNames label
G:=SmallGroup(448,668);
// by ID
G=gap.SmallGroup(448,668);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,387,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^27,c*b*c=b^-1>;
// generators/relations