metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊7D4, C8⋊2D28, C4.Q8⋊3D7, C7⋊2(C8⋊2D4), (C2×D56)⋊24C2, C4⋊D28⋊6C2, C4⋊C4.39D14, C4.51(C2×D28), (C2×C8).61D14, C28.131(C2×D4), C14.D8⋊16C2, C28.30(C4○D4), C4.4(Q8⋊2D7), (C2×Dic7).42D4, (C22×D7).24D4, C22.217(D4×D7), C2.17(C4⋊D28), C14.44(C4⋊D4), C2.22(D56⋊C2), C14.70(C8⋊C22), (C2×C28).281C23, (C2×C56).110C22, (C2×D28).75C22, (C7×C4.Q8)⋊3C2, (C2×C8⋊D7)⋊2C2, (C2×C7⋊C8).58C22, (C2×C4×D7).33C22, (C2×C14).286(C2×D4), (C7×C4⋊C4).74C22, (C2×C4).384(C22×D7), SmallGroup(448,399)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊7D4
G = < a,b,c | a56=b4=c2=1, bab-1=a43, cac=a-1, cbc=b-1 >
Subgroups: 972 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C8⋊2D4, C8⋊D7, D56, C2×C7⋊C8, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C2×D28, C14.D8, C7×C4.Q8, C4⋊D28, C2×C8⋊D7, C2×D56, C56⋊7D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, D28, C22×D7, C8⋊2D4, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, D56⋊C2, C56⋊7D4
►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 71 192 141)(2 58 193 128)(3 101 194 115)(4 88 195 158)(5 75 196 145)(6 62 197 132)(7 105 198 119)(8 92 199 162)(9 79 200 149)(10 66 201 136)(11 109 202 123)(12 96 203 166)(13 83 204 153)(14 70 205 140)(15 57 206 127)(16 100 207 114)(17 87 208 157)(18 74 209 144)(19 61 210 131)(20 104 211 118)(21 91 212 161)(22 78 213 148)(23 65 214 135)(24 108 215 122)(25 95 216 165)(26 82 217 152)(27 69 218 139)(28 112 219 126)(29 99 220 113)(30 86 221 156)(31 73 222 143)(32 60 223 130)(33 103 224 117)(34 90 169 160)(35 77 170 147)(36 64 171 134)(37 107 172 121)(38 94 173 164)(39 81 174 151)(40 68 175 138)(41 111 176 125)(42 98 177 168)(43 85 178 155)(44 72 179 142)(45 59 180 129)(46 102 181 116)(47 89 182 159)(48 76 183 146)(49 63 184 133)(50 106 185 120)(51 93 186 163)(52 80 187 150)(53 67 188 137)(54 110 189 124)(55 97 190 167)(56 84 191 154)
(1 141)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 124)(19 123)(20 122)(21 121)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 168)(31 167)(32 166)(33 165)(34 164)(35 163)(36 162)(37 161)(38 160)(39 159)(40 158)(41 157)(42 156)(43 155)(44 154)(45 153)(46 152)(47 151)(48 150)(49 149)(50 148)(51 147)(52 146)(53 145)(54 144)(55 143)(56 142)(57 206)(58 205)(59 204)(60 203)(61 202)(62 201)(63 200)(64 199)(65 198)(66 197)(67 196)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 187)(77 186)(78 185)(79 184)(80 183)(81 182)(82 181)(83 180)(84 179)(85 178)(86 177)(87 176)(88 175)(89 174)(90 173)(91 172)(92 171)(93 170)(94 169)(95 224)(96 223)(97 222)(98 221)(99 220)(100 219)(101 218)(102 217)(103 216)(104 215)(105 214)(106 213)(107 212)(108 211)(109 210)(110 209)(111 208)(112 207)
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), (1,71,192,141)(2,58,193,128)(3,101,194,115)(4,88,195,158)(5,75,196,145)(6,62,197,132)(7,105,198,119)(8,92,199,162)(9,79,200,149)(10,66,201,136)(11,109,202,123)(12,96,203,166)(13,83,204,153)(14,70,205,140)(15,57,206,127)(16,100,207,114)(17,87,208,157)(18,74,209,144)(19,61,210,131)(20,104,211,118)(21,91,212,161)(22,78,213,148)(23,65,214,135)(24,108,215,122)(25,95,216,165)(26,82,217,152)(27,69,218,139)(28,112,219,126)(29,99,220,113)(30,86,221,156)(31,73,222,143)(32,60,223,130)(33,103,224,117)(34,90,169,160)(35,77,170,147)(36,64,171,134)(37,107,172,121)(38,94,173,164)(39,81,174,151)(40,68,175,138)(41,111,176,125)(42,98,177,168)(43,85,178,155)(44,72,179,142)(45,59,180,129)(46,102,181,116)(47,89,182,159)(48,76,183,146)(49,63,184,133)(50,106,185,120)(51,93,186,163)(52,80,187,150)(53,67,188,137)(54,110,189,124)(55,97,190,167)(56,84,191,154), (1,141)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,121)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,168)(31,167)(32,166)(33,165)(34,164)(35,163)(36,162)(37,161)(38,160)(39,159)(40,158)(41,157)(42,156)(43,155)(44,154)(45,153)(46,152)(47,151)(48,150)(49,149)(50,148)(51,147)(52,146)(53,145)(54,144)(55,143)(56,142)(57,206)(58,205)(59,204)(60,203)(61,202)(62,201)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,224)(96,223)(97,222)(98,221)(99,220)(100,219)(101,218)(102,217)(103,216)(104,215)(105,214)(106,213)(107,212)(108,211)(109,210)(110,209)(111,208)(112,207)>;
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), (1,71,192,141)(2,58,193,128)(3,101,194,115)(4,88,195,158)(5,75,196,145)(6,62,197,132)(7,105,198,119)(8,92,199,162)(9,79,200,149)(10,66,201,136)(11,109,202,123)(12,96,203,166)(13,83,204,153)(14,70,205,140)(15,57,206,127)(16,100,207,114)(17,87,208,157)(18,74,209,144)(19,61,210,131)(20,104,211,118)(21,91,212,161)(22,78,213,148)(23,65,214,135)(24,108,215,122)(25,95,216,165)(26,82,217,152)(27,69,218,139)(28,112,219,126)(29,99,220,113)(30,86,221,156)(31,73,222,143)(32,60,223,130)(33,103,224,117)(34,90,169,160)(35,77,170,147)(36,64,171,134)(37,107,172,121)(38,94,173,164)(39,81,174,151)(40,68,175,138)(41,111,176,125)(42,98,177,168)(43,85,178,155)(44,72,179,142)(45,59,180,129)(46,102,181,116)(47,89,182,159)(48,76,183,146)(49,63,184,133)(50,106,185,120)(51,93,186,163)(52,80,187,150)(53,67,188,137)(54,110,189,124)(55,97,190,167)(56,84,191,154), (1,141)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,121)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,168)(31,167)(32,166)(33,165)(34,164)(35,163)(36,162)(37,161)(38,160)(39,159)(40,158)(41,157)(42,156)(43,155)(44,154)(45,153)(46,152)(47,151)(48,150)(49,149)(50,148)(51,147)(52,146)(53,145)(54,144)(55,143)(56,142)(57,206)(58,205)(59,204)(60,203)(61,202)(62,201)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,224)(96,223)(97,222)(98,221)(99,220)(100,219)(101,218)(102,217)(103,216)(104,215)(105,214)(106,213)(107,212)(108,211)(109,210)(110,209)(111,208)(112,207) );
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)], [(1,71,192,141),(2,58,193,128),(3,101,194,115),(4,88,195,158),(5,75,196,145),(6,62,197,132),(7,105,198,119),(8,92,199,162),(9,79,200,149),(10,66,201,136),(11,109,202,123),(12,96,203,166),(13,83,204,153),(14,70,205,140),(15,57,206,127),(16,100,207,114),(17,87,208,157),(18,74,209,144),(19,61,210,131),(20,104,211,118),(21,91,212,161),(22,78,213,148),(23,65,214,135),(24,108,215,122),(25,95,216,165),(26,82,217,152),(27,69,218,139),(28,112,219,126),(29,99,220,113),(30,86,221,156),(31,73,222,143),(32,60,223,130),(33,103,224,117),(34,90,169,160),(35,77,170,147),(36,64,171,134),(37,107,172,121),(38,94,173,164),(39,81,174,151),(40,68,175,138),(41,111,176,125),(42,98,177,168),(43,85,178,155),(44,72,179,142),(45,59,180,129),(46,102,181,116),(47,89,182,159),(48,76,183,146),(49,63,184,133),(50,106,185,120),(51,93,186,163),(52,80,187,150),(53,67,188,137),(54,110,189,124),(55,97,190,167),(56,84,191,154)], [(1,141),(2,140),(3,139),(4,138),(5,137),(6,136),(7,135),(8,134),(9,133),(10,132),(11,131),(12,130),(13,129),(14,128),(15,127),(16,126),(17,125),(18,124),(19,123),(20,122),(21,121),(22,120),(23,119),(24,118),(25,117),(26,116),(27,115),(28,114),(29,113),(30,168),(31,167),(32,166),(33,165),(34,164),(35,163),(36,162),(37,161),(38,160),(39,159),(40,158),(41,157),(42,156),(43,155),(44,154),(45,153),(46,152),(47,151),(48,150),(49,149),(50,148),(51,147),(52,146),(53,145),(54,144),(55,143),(56,142),(57,206),(58,205),(59,204),(60,203),(61,202),(62,201),(63,200),(64,199),(65,198),(66,197),(67,196),(68,195),(69,194),(70,193),(71,192),(72,191),(73,190),(74,189),(75,188),(76,187),(77,186),(78,185),(79,184),(80,183),(81,182),(82,181),(83,180),(84,179),(85,178),(86,177),(87,176),(88,175),(89,174),(90,173),(91,172),(92,171),(93,170),(94,169),(95,224),(96,223),(97,222),(98,221),(99,220),(100,219),(101,218),(102,217),(103,216),(104,215),(105,214),(106,213),(107,212),(108,211),(109,210),(110,209),(111,208),(112,207)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 56 | 56 | 2 | 2 | 8 | 8 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D28 | C8⋊C22 | Q8⋊2D7 | D4×D7 | D56⋊C2 |
| kernel | C56⋊7D4 | C14.D8 | C7×C4.Q8 | C4⋊D28 | C2×C8⋊D7 | C2×D56 | C56 | C2×Dic7 | C22×D7 | C4.Q8 | C28 | C4⋊C4 | C2×C8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56⋊7D4 ►in GL8(𝔽113)
| 0 | 16 | 96 | 91 | 0 | 0 | 0 | 0 |
| 97 | 45 | 22 | 20 | 0 | 0 | 0 | 0 |
| 96 | 91 | 0 | 97 | 0 | 0 | 0 | 0 |
| 22 | 20 | 16 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 86 | 106 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 98 |
| 0 | 0 | 0 | 0 | 70 | 60 | 86 | 106 |
| 0 | 0 | 0 | 0 | 53 | 64 | 7 | 98 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 22 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 0 | 22 |
| 0 | 0 | 0 | 0 | 28 | 0 | 96 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 96 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 22 | 0 |
| 0 | 0 | 0 | 0 | 100 | 96 | 43 | 91 |
| 0 | 0 | 0 | 0 | 28 | 0 | 96 | 0 |
| 0 | 0 | 0 | 0 | 65 | 85 | 13 | 17 |
G:=sub<GL(8,GF(113))| [0,97,96,22,0,0,0,0,16,45,91,20,0,0,0,0,96,22,0,16,0,0,0,0,91,20,97,68,0,0,0,0,0,0,0,0,0,0,70,53,0,0,0,0,0,0,60,64,0,0,0,0,86,7,86,7,0,0,0,0,106,98,106,98],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17,0,28,0,0,0,0,0,0,17,0,28,0,0,0,0,22,0,96,0,0,0,0,0,0,22,0,96],[0,0,1,24,0,0,0,0,0,0,0,112,0,0,0,0,1,24,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,17,100,28,65,0,0,0,0,0,96,0,85,0,0,0,0,22,43,96,13,0,0,0,0,0,91,0,17] >;
C56⋊7D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_7D_4 % in TeX
G:=Group("C56:7D4"); // GroupNames label
G:=SmallGroup(448,399);
// by ID
G=gap.SmallGroup(448,399);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,254,555,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^43,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations