metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊18D4, Dic7⋊2M4(2), C8⋊9(C7⋊D4), C7⋊5(C8⋊6D4), D14⋊C8⋊39C2, C14.80(C4×D4), Dic7⋊C8⋊41C2, D14⋊C4.17C4, (C8×Dic7)⋊31C2, C28.443(C2×D4), (C2×C8).276D14, (C2×M4(2))⋊8D7, C23.20(C4×D7), C14.33(C8○D4), (C14×M4(2))⋊7C2, Dic7⋊C4.17C4, C2.22(D7×M4(2)), C23.D7.13C4, C28.254(C4○D4), C4.138(C4○D28), C28.55D4⋊30C2, C2.18(D28.C4), (C2×C56).318C22, (C2×C28).868C23, (C22×C4).137D14, C14.33(C2×M4(2)), (C22×C28).375C22, (C4×Dic7).286C22, (C2×C4).51(C4×D7), C2.25(C4×C7⋊D4), (C2×C8⋊D7)⋊25C2, (C2×C7⋊D4).11C4, (C4×C7⋊D4).17C2, C4.134(C2×C7⋊D4), C22.147(C2×C4×D7), (C2×C28).188(C2×C4), (C2×C7⋊C8).324C22, (C2×C4×D7).185C22, (C22×C14).68(C2×C4), (C2×Dic7).68(C2×C4), (C22×D7).25(C2×C4), (C2×C4).810(C22×D7), (C2×C14).138(C22×C4), SmallGroup(448,662)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊18D4
G = < a,b,c | a56=b4=c2=1, bab-1=a41, cac=a13, cbc=b-1 >
Subgroups: 484 in 122 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C2×M4(2), C7⋊C8, C56, C56, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊6D4, C8⋊D7, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C56, C7×M4(2), C2×C4×D7, C2×C7⋊D4, C22×C28, C8×Dic7, Dic7⋊C8, D14⋊C8, C28.55D4, C2×C8⋊D7, C4×C7⋊D4, C14×M4(2), C56⋊18D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, M4(2), C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×M4(2), C8○D4, C4×D7, C7⋊D4, C22×D7, C8⋊6D4, C2×C4×D7, C4○D28, C2×C7⋊D4, D7×M4(2), D28.C4, C4×C7⋊D4, C56⋊18D4
►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 86 155 176)(2 71 156 217)(3 112 157 202)(4 97 158 187)(5 82 159 172)(6 67 160 213)(7 108 161 198)(8 93 162 183)(9 78 163 224)(10 63 164 209)(11 104 165 194)(12 89 166 179)(13 74 167 220)(14 59 168 205)(15 100 113 190)(16 85 114 175)(17 70 115 216)(18 111 116 201)(19 96 117 186)(20 81 118 171)(21 66 119 212)(22 107 120 197)(23 92 121 182)(24 77 122 223)(25 62 123 208)(26 103 124 193)(27 88 125 178)(28 73 126 219)(29 58 127 204)(30 99 128 189)(31 84 129 174)(32 69 130 215)(33 110 131 200)(34 95 132 185)(35 80 133 170)(36 65 134 211)(37 106 135 196)(38 91 136 181)(39 76 137 222)(40 61 138 207)(41 102 139 192)(42 87 140 177)(43 72 141 218)(44 57 142 203)(45 98 143 188)(46 83 144 173)(47 68 145 214)(48 109 146 199)(49 94 147 184)(50 79 148 169)(51 64 149 210)(52 105 150 195)(53 90 151 180)(54 75 152 221)(55 60 153 206)(56 101 154 191)
(2 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 191)(58 204)(59 217)(60 174)(61 187)(62 200)(63 213)(64 170)(65 183)(66 196)(67 209)(68 222)(69 179)(70 192)(71 205)(72 218)(73 175)(74 188)(75 201)(76 214)(77 171)(78 184)(79 197)(80 210)(81 223)(82 180)(83 193)(84 206)(85 219)(86 176)(87 189)(88 202)(89 215)(90 172)(91 185)(92 198)(93 211)(94 224)(95 181)(96 194)(97 207)(98 220)(99 177)(100 190)(101 203)(102 216)(103 173)(104 186)(105 199)(106 212)(107 169)(108 182)(109 195)(110 208)(111 221)(112 178)(114 126)(115 139)(116 152)(117 165)(118 122)(119 135)(120 148)(121 161)(123 131)(124 144)(125 157)(128 140)(129 153)(130 166)(132 136)(133 149)(134 162)(137 145)(138 158)(142 154)(143 167)(146 150)(147 163)(151 159)(156 168)(160 164)
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), (1,86,155,176)(2,71,156,217)(3,112,157,202)(4,97,158,187)(5,82,159,172)(6,67,160,213)(7,108,161,198)(8,93,162,183)(9,78,163,224)(10,63,164,209)(11,104,165,194)(12,89,166,179)(13,74,167,220)(14,59,168,205)(15,100,113,190)(16,85,114,175)(17,70,115,216)(18,111,116,201)(19,96,117,186)(20,81,118,171)(21,66,119,212)(22,107,120,197)(23,92,121,182)(24,77,122,223)(25,62,123,208)(26,103,124,193)(27,88,125,178)(28,73,126,219)(29,58,127,204)(30,99,128,189)(31,84,129,174)(32,69,130,215)(33,110,131,200)(34,95,132,185)(35,80,133,170)(36,65,134,211)(37,106,135,196)(38,91,136,181)(39,76,137,222)(40,61,138,207)(41,102,139,192)(42,87,140,177)(43,72,141,218)(44,57,142,203)(45,98,143,188)(46,83,144,173)(47,68,145,214)(48,109,146,199)(49,94,147,184)(50,79,148,169)(51,64,149,210)(52,105,150,195)(53,90,151,180)(54,75,152,221)(55,60,153,206)(56,101,154,191), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,191)(58,204)(59,217)(60,174)(61,187)(62,200)(63,213)(64,170)(65,183)(66,196)(67,209)(68,222)(69,179)(70,192)(71,205)(72,218)(73,175)(74,188)(75,201)(76,214)(77,171)(78,184)(79,197)(80,210)(81,223)(82,180)(83,193)(84,206)(85,219)(86,176)(87,189)(88,202)(89,215)(90,172)(91,185)(92,198)(93,211)(94,224)(95,181)(96,194)(97,207)(98,220)(99,177)(100,190)(101,203)(102,216)(103,173)(104,186)(105,199)(106,212)(107,169)(108,182)(109,195)(110,208)(111,221)(112,178)(114,126)(115,139)(116,152)(117,165)(118,122)(119,135)(120,148)(121,161)(123,131)(124,144)(125,157)(128,140)(129,153)(130,166)(132,136)(133,149)(134,162)(137,145)(138,158)(142,154)(143,167)(146,150)(147,163)(151,159)(156,168)(160,164)>;
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), (1,86,155,176)(2,71,156,217)(3,112,157,202)(4,97,158,187)(5,82,159,172)(6,67,160,213)(7,108,161,198)(8,93,162,183)(9,78,163,224)(10,63,164,209)(11,104,165,194)(12,89,166,179)(13,74,167,220)(14,59,168,205)(15,100,113,190)(16,85,114,175)(17,70,115,216)(18,111,116,201)(19,96,117,186)(20,81,118,171)(21,66,119,212)(22,107,120,197)(23,92,121,182)(24,77,122,223)(25,62,123,208)(26,103,124,193)(27,88,125,178)(28,73,126,219)(29,58,127,204)(30,99,128,189)(31,84,129,174)(32,69,130,215)(33,110,131,200)(34,95,132,185)(35,80,133,170)(36,65,134,211)(37,106,135,196)(38,91,136,181)(39,76,137,222)(40,61,138,207)(41,102,139,192)(42,87,140,177)(43,72,141,218)(44,57,142,203)(45,98,143,188)(46,83,144,173)(47,68,145,214)(48,109,146,199)(49,94,147,184)(50,79,148,169)(51,64,149,210)(52,105,150,195)(53,90,151,180)(54,75,152,221)(55,60,153,206)(56,101,154,191), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,191)(58,204)(59,217)(60,174)(61,187)(62,200)(63,213)(64,170)(65,183)(66,196)(67,209)(68,222)(69,179)(70,192)(71,205)(72,218)(73,175)(74,188)(75,201)(76,214)(77,171)(78,184)(79,197)(80,210)(81,223)(82,180)(83,193)(84,206)(85,219)(86,176)(87,189)(88,202)(89,215)(90,172)(91,185)(92,198)(93,211)(94,224)(95,181)(96,194)(97,207)(98,220)(99,177)(100,190)(101,203)(102,216)(103,173)(104,186)(105,199)(106,212)(107,169)(108,182)(109,195)(110,208)(111,221)(112,178)(114,126)(115,139)(116,152)(117,165)(118,122)(119,135)(120,148)(121,161)(123,131)(124,144)(125,157)(128,140)(129,153)(130,166)(132,136)(133,149)(134,162)(137,145)(138,158)(142,154)(143,167)(146,150)(147,163)(151,159)(156,168)(160,164) );
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)], [(1,86,155,176),(2,71,156,217),(3,112,157,202),(4,97,158,187),(5,82,159,172),(6,67,160,213),(7,108,161,198),(8,93,162,183),(9,78,163,224),(10,63,164,209),(11,104,165,194),(12,89,166,179),(13,74,167,220),(14,59,168,205),(15,100,113,190),(16,85,114,175),(17,70,115,216),(18,111,116,201),(19,96,117,186),(20,81,118,171),(21,66,119,212),(22,107,120,197),(23,92,121,182),(24,77,122,223),(25,62,123,208),(26,103,124,193),(27,88,125,178),(28,73,126,219),(29,58,127,204),(30,99,128,189),(31,84,129,174),(32,69,130,215),(33,110,131,200),(34,95,132,185),(35,80,133,170),(36,65,134,211),(37,106,135,196),(38,91,136,181),(39,76,137,222),(40,61,138,207),(41,102,139,192),(42,87,140,177),(43,72,141,218),(44,57,142,203),(45,98,143,188),(46,83,144,173),(47,68,145,214),(48,109,146,199),(49,94,147,184),(50,79,148,169),(51,64,149,210),(52,105,150,195),(53,90,151,180),(54,75,152,221),(55,60,153,206),(56,101,154,191)], [(2,14),(3,27),(4,40),(5,53),(6,10),(7,23),(8,36),(9,49),(11,19),(12,32),(13,45),(16,28),(17,41),(18,54),(20,24),(21,37),(22,50),(25,33),(26,46),(30,42),(31,55),(34,38),(35,51),(39,47),(44,56),(48,52),(57,191),(58,204),(59,217),(60,174),(61,187),(62,200),(63,213),(64,170),(65,183),(66,196),(67,209),(68,222),(69,179),(70,192),(71,205),(72,218),(73,175),(74,188),(75,201),(76,214),(77,171),(78,184),(79,197),(80,210),(81,223),(82,180),(83,193),(84,206),(85,219),(86,176),(87,189),(88,202),(89,215),(90,172),(91,185),(92,198),(93,211),(94,224),(95,181),(96,194),(97,207),(98,220),(99,177),(100,190),(101,203),(102,216),(103,173),(104,186),(105,199),(106,212),(107,169),(108,182),(109,195),(110,208),(111,221),(112,178),(114,126),(115,139),(116,152),(117,165),(118,122),(119,135),(120,148),(121,161),(123,131),(124,144),(125,157),(128,140),(129,153),(130,166),(132,136),(133,149),(134,162),(137,145),(138,158),(142,154),(143,167),(146,150),(147,163),(151,159),(156,168),(160,164)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 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 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | M4(2) | C4○D4 | D14 | D14 | C8○D4 | C7⋊D4 | C4×D7 | C4×D7 | C4○D28 | D7×M4(2) | D28.C4 |
| kernel | C56⋊18D4 | C8×Dic7 | Dic7⋊C8 | D14⋊C8 | C28.55D4 | C2×C8⋊D7 | C4×C7⋊D4 | C14×M4(2) | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | C56 | C2×M4(2) | Dic7 | C28 | C2×C8 | C22×C4 | C14 | C8 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 2 | 6 | 3 | 4 | 12 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of C56⋊18D4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 1 | 0 | 0 |
| 0 | 0 | 87 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 1 | 0 | 0 |
| 0 | 0 | 54 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 1 | 0 | 0 |
| 0 | 0 | 54 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,87,0,0,0,0,1,25,0,0,0,0,0,0,0,98,0,0,0,0,1,0],[0,1,0,0,0,0,112,0,0,0,0,0,0,0,88,54,0,0,0,0,1,25,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,88,54,0,0,0,0,1,25,0,0,0,0,0,0,1,0,0,0,0,0,0,112] >;
C56⋊18D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{18}D_4 % in TeX
G:=Group("C56:18D4"); // GroupNames label
G:=SmallGroup(448,662);
// by ID
G=gap.SmallGroup(448,662);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,758,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^41,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations