metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊6D4, D14⋊3D8, (C2×D8)⋊5D7, (C14×D8)⋊6C2, C7⋊5(C8⋊7D4), C2.29(D7×D8), C28⋊2D4⋊4C2, C8⋊10(C7⋊D4), C56⋊1C4⋊23C2, C14.46(C2×D8), (C2×D4).64D14, C28.166(C2×D4), (C2×C8).238D14, C14.34(C4○D8), C28.93(C4○D4), D4⋊Dic7⋊29C2, (C2×C56).90C22, (C22×D7).58D4, C22.257(D4×D7), C2.18(D8⋊3D7), C4.28(D4⋊2D7), C2.16(C28⋊2D4), (C2×C28).434C23, (C2×Dic7).112D4, (D4×C14).83C22, C14.109(C4⋊D4), C4⋊Dic7.165C22, (D7×C2×C8)⋊3C2, C4.79(C2×C7⋊D4), (C2×C14).347(C2×D4), (C2×C7⋊C8).271C22, (C2×C4×D7).239C22, (C2×C4).524(C22×D7), SmallGroup(448,691)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊6D4
G = < a,b,c | a56=b4=c2=1, bab-1=a-1, cac=a41, cbc=b-1 >
Subgroups: 708 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, 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, D8, C22×C4, C2×D4, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×C14, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C8⋊7D4, C8×D7, C2×C7⋊C8, C4⋊Dic7, C23.D7, C2×C56, C7×D8, C2×C4×D7, C2×C7⋊D4, D4×C14, C56⋊1C4, D4⋊Dic7, D7×C2×C8, C28⋊2D4, C14×D8, C56⋊6D4
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C4⋊D4, C2×D8, C4○D8, C7⋊D4, C22×D7, C8⋊7D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D7×D8, D8⋊3D7, C28⋊2D4, C56⋊6D4
►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 66 212 119)(2 65 213 118)(3 64 214 117)(4 63 215 116)(5 62 216 115)(6 61 217 114)(7 60 218 113)(8 59 219 168)(9 58 220 167)(10 57 221 166)(11 112 222 165)(12 111 223 164)(13 110 224 163)(14 109 169 162)(15 108 170 161)(16 107 171 160)(17 106 172 159)(18 105 173 158)(19 104 174 157)(20 103 175 156)(21 102 176 155)(22 101 177 154)(23 100 178 153)(24 99 179 152)(25 98 180 151)(26 97 181 150)(27 96 182 149)(28 95 183 148)(29 94 184 147)(30 93 185 146)(31 92 186 145)(32 91 187 144)(33 90 188 143)(34 89 189 142)(35 88 190 141)(36 87 191 140)(37 86 192 139)(38 85 193 138)(39 84 194 137)(40 83 195 136)(41 82 196 135)(42 81 197 134)(43 80 198 133)(44 79 199 132)(45 78 200 131)(46 77 201 130)(47 76 202 129)(48 75 203 128)(49 74 204 127)(50 73 205 126)(51 72 206 125)(52 71 207 124)(53 70 208 123)(54 69 209 122)(55 68 210 121)(56 67 211 120)
(2 42)(3 27)(4 12)(5 53)(6 38)(7 23)(9 49)(10 34)(11 19)(13 45)(14 30)(16 56)(17 41)(18 26)(20 52)(21 37)(24 48)(25 33)(28 44)(31 55)(32 40)(35 51)(39 47)(46 54)(57 142)(58 127)(59 168)(60 153)(61 138)(62 123)(63 164)(64 149)(65 134)(66 119)(67 160)(68 145)(69 130)(70 115)(71 156)(72 141)(73 126)(74 167)(75 152)(76 137)(77 122)(78 163)(79 148)(80 133)(81 118)(82 159)(83 144)(84 129)(85 114)(86 155)(87 140)(88 125)(89 166)(90 151)(91 136)(92 121)(93 162)(94 147)(95 132)(96 117)(97 158)(98 143)(99 128)(100 113)(101 154)(102 139)(103 124)(104 165)(105 150)(106 135)(107 120)(108 161)(109 146)(110 131)(111 116)(112 157)(169 185)(171 211)(172 196)(173 181)(174 222)(175 207)(176 192)(178 218)(179 203)(180 188)(182 214)(183 199)(186 210)(187 195)(189 221)(190 206)(193 217)(194 202)(197 213)(200 224)(201 209)(204 220)(208 216)(215 223)
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), (1,66,212,119)(2,65,213,118)(3,64,214,117)(4,63,215,116)(5,62,216,115)(6,61,217,114)(7,60,218,113)(8,59,219,168)(9,58,220,167)(10,57,221,166)(11,112,222,165)(12,111,223,164)(13,110,224,163)(14,109,169,162)(15,108,170,161)(16,107,171,160)(17,106,172,159)(18,105,173,158)(19,104,174,157)(20,103,175,156)(21,102,176,155)(22,101,177,154)(23,100,178,153)(24,99,179,152)(25,98,180,151)(26,97,181,150)(27,96,182,149)(28,95,183,148)(29,94,184,147)(30,93,185,146)(31,92,186,145)(32,91,187,144)(33,90,188,143)(34,89,189,142)(35,88,190,141)(36,87,191,140)(37,86,192,139)(38,85,193,138)(39,84,194,137)(40,83,195,136)(41,82,196,135)(42,81,197,134)(43,80,198,133)(44,79,199,132)(45,78,200,131)(46,77,201,130)(47,76,202,129)(48,75,203,128)(49,74,204,127)(50,73,205,126)(51,72,206,125)(52,71,207,124)(53,70,208,123)(54,69,209,122)(55,68,210,121)(56,67,211,120), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,142)(58,127)(59,168)(60,153)(61,138)(62,123)(63,164)(64,149)(65,134)(66,119)(67,160)(68,145)(69,130)(70,115)(71,156)(72,141)(73,126)(74,167)(75,152)(76,137)(77,122)(78,163)(79,148)(80,133)(81,118)(82,159)(83,144)(84,129)(85,114)(86,155)(87,140)(88,125)(89,166)(90,151)(91,136)(92,121)(93,162)(94,147)(95,132)(96,117)(97,158)(98,143)(99,128)(100,113)(101,154)(102,139)(103,124)(104,165)(105,150)(106,135)(107,120)(108,161)(109,146)(110,131)(111,116)(112,157)(169,185)(171,211)(172,196)(173,181)(174,222)(175,207)(176,192)(178,218)(179,203)(180,188)(182,214)(183,199)(186,210)(187,195)(189,221)(190,206)(193,217)(194,202)(197,213)(200,224)(201,209)(204,220)(208,216)(215,223)>;
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), (1,66,212,119)(2,65,213,118)(3,64,214,117)(4,63,215,116)(5,62,216,115)(6,61,217,114)(7,60,218,113)(8,59,219,168)(9,58,220,167)(10,57,221,166)(11,112,222,165)(12,111,223,164)(13,110,224,163)(14,109,169,162)(15,108,170,161)(16,107,171,160)(17,106,172,159)(18,105,173,158)(19,104,174,157)(20,103,175,156)(21,102,176,155)(22,101,177,154)(23,100,178,153)(24,99,179,152)(25,98,180,151)(26,97,181,150)(27,96,182,149)(28,95,183,148)(29,94,184,147)(30,93,185,146)(31,92,186,145)(32,91,187,144)(33,90,188,143)(34,89,189,142)(35,88,190,141)(36,87,191,140)(37,86,192,139)(38,85,193,138)(39,84,194,137)(40,83,195,136)(41,82,196,135)(42,81,197,134)(43,80,198,133)(44,79,199,132)(45,78,200,131)(46,77,201,130)(47,76,202,129)(48,75,203,128)(49,74,204,127)(50,73,205,126)(51,72,206,125)(52,71,207,124)(53,70,208,123)(54,69,209,122)(55,68,210,121)(56,67,211,120), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,142)(58,127)(59,168)(60,153)(61,138)(62,123)(63,164)(64,149)(65,134)(66,119)(67,160)(68,145)(69,130)(70,115)(71,156)(72,141)(73,126)(74,167)(75,152)(76,137)(77,122)(78,163)(79,148)(80,133)(81,118)(82,159)(83,144)(84,129)(85,114)(86,155)(87,140)(88,125)(89,166)(90,151)(91,136)(92,121)(93,162)(94,147)(95,132)(96,117)(97,158)(98,143)(99,128)(100,113)(101,154)(102,139)(103,124)(104,165)(105,150)(106,135)(107,120)(108,161)(109,146)(110,131)(111,116)(112,157)(169,185)(171,211)(172,196)(173,181)(174,222)(175,207)(176,192)(178,218)(179,203)(180,188)(182,214)(183,199)(186,210)(187,195)(189,221)(190,206)(193,217)(194,202)(197,213)(200,224)(201,209)(204,220)(208,216)(215,223) );
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)], [(1,66,212,119),(2,65,213,118),(3,64,214,117),(4,63,215,116),(5,62,216,115),(6,61,217,114),(7,60,218,113),(8,59,219,168),(9,58,220,167),(10,57,221,166),(11,112,222,165),(12,111,223,164),(13,110,224,163),(14,109,169,162),(15,108,170,161),(16,107,171,160),(17,106,172,159),(18,105,173,158),(19,104,174,157),(20,103,175,156),(21,102,176,155),(22,101,177,154),(23,100,178,153),(24,99,179,152),(25,98,180,151),(26,97,181,150),(27,96,182,149),(28,95,183,148),(29,94,184,147),(30,93,185,146),(31,92,186,145),(32,91,187,144),(33,90,188,143),(34,89,189,142),(35,88,190,141),(36,87,191,140),(37,86,192,139),(38,85,193,138),(39,84,194,137),(40,83,195,136),(41,82,196,135),(42,81,197,134),(43,80,198,133),(44,79,199,132),(45,78,200,131),(46,77,201,130),(47,76,202,129),(48,75,203,128),(49,74,204,127),(50,73,205,126),(51,72,206,125),(52,71,207,124),(53,70,208,123),(54,69,209,122),(55,68,210,121),(56,67,211,120)], [(2,42),(3,27),(4,12),(5,53),(6,38),(7,23),(9,49),(10,34),(11,19),(13,45),(14,30),(16,56),(17,41),(18,26),(20,52),(21,37),(24,48),(25,33),(28,44),(31,55),(32,40),(35,51),(39,47),(46,54),(57,142),(58,127),(59,168),(60,153),(61,138),(62,123),(63,164),(64,149),(65,134),(66,119),(67,160),(68,145),(69,130),(70,115),(71,156),(72,141),(73,126),(74,167),(75,152),(76,137),(77,122),(78,163),(79,148),(80,133),(81,118),(82,159),(83,144),(84,129),(85,114),(86,155),(87,140),(88,125),(89,166),(90,151),(91,136),(92,121),(93,162),(94,147),(95,132),(96,117),(97,158),(98,143),(99,128),(100,113),(101,154),(102,139),(103,124),(104,165),(105,150),(106,135),(107,120),(108,161),(109,146),(110,131),(111,116),(112,157),(169,185),(171,211),(172,196),(173,181),(174,222),(175,207),(176,192),(178,218),(179,203),(180,188),(182,214),(183,199),(186,210),(187,195),(189,221),(190,206),(193,217),(194,202),(197,213),(200,224),(201,209),(204,220),(208,216),(215,223)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 14 | 14 | 2 | 2 | 14 | 14 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D8 | D14 | D14 | C4○D8 | C7⋊D4 | D4⋊2D7 | D4×D7 | D7×D8 | D8⋊3D7 |
| kernel | C56⋊6D4 | C56⋊1C4 | D4⋊Dic7 | D7×C2×C8 | C28⋊2D4 | C14×D8 | C56 | C2×Dic7 | C22×D7 | C2×D8 | C28 | D14 | C2×C8 | C2×D4 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 3 | 6 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊6D4 ►in GL4(𝔽113) generated by
| 44 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 34 | 60 |
| 0 | 0 | 112 | 88 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 91 | 74 |
| 0 | 0 | 53 | 22 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 80 | 112 |
G:=sub<GL(4,GF(113))| [44,0,0,0,0,18,0,0,0,0,34,112,0,0,60,88],[0,1,0,0,112,0,0,0,0,0,91,53,0,0,74,22],[1,0,0,0,0,112,0,0,0,0,1,80,0,0,0,112] >;
C56⋊6D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_6D_4 % in TeX
G:=Group("C56:6D4"); // GroupNames label
G:=SmallGroup(448,691);
// by ID
G=gap.SmallGroup(448,691);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^41,c*b*c=b^-1>;
// generators/relations