metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊3C4, C42.17D14, C8⋊2(C4×D7), C56⋊2(C2×C4), (C4×D28)⋊2C2, D28⋊9(C2×C4), C8⋊C4⋊2D7, C8⋊Dic7⋊2C2, C7⋊1(D8⋊C4), (C2×D56).7C2, C14.12(C4×D4), (C2×C8).54D14, C2.15(C4×D28), (C2×C28).237D4, (C2×C4).115D28, C2.D56⋊37C2, C2.2(C8⋊D14), C14.3(C8⋊C22), (C4×C28).15C22, (C2×C56).55C22, C22.31(C2×D28), C4.107(C4○D28), C28.223(C4○D4), (C2×C28).732C23, C28.106(C22×C4), (C2×D28).190C22, C4⋊Dic7.266C22, C4.64(C2×C4×D7), (C7×C8⋊C4)⋊3C2, (C2×C14).115(C2×D4), (C2×C4).676(C22×D7), SmallGroup(448,245)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊C4
G = < a,b,c | a56=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a28b >
Subgroups: 868 in 132 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C56, C56, C4×D7, D28, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, D8⋊C4, D56, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C8⋊Dic7, C2.D56, C7×C8⋊C4, C4×D28, C2×D56, D56⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8⋊C22, C4×D7, D28, C22×D7, D8⋊C4, C2×C4×D7, C2×D28, C4○D28, C4×D28, C8⋊D14, D56⋊C4
►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 7)(2 6)(3 5)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(57 59)(60 112)(61 111)(62 110)(63 109)(64 108)(65 107)(66 106)(67 105)(68 104)(69 103)(70 102)(71 101)(72 100)(73 99)(74 98)(75 97)(76 96)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)(114 168)(115 167)(116 166)(117 165)(118 164)(119 163)(120 162)(121 161)(122 160)(123 159)(124 158)(125 157)(126 156)(127 155)(128 154)(129 153)(130 152)(131 151)(132 150)(133 149)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)(169 221)(170 220)(171 219)(172 218)(173 217)(174 216)(175 215)(176 214)(177 213)(178 212)(179 211)(180 210)(181 209)(182 208)(183 207)(184 206)(185 205)(186 204)(187 203)(188 202)(189 201)(190 200)(191 199)(192 198)(193 197)(194 196)(222 224)
(1 206 83 124)(2 179 84 153)(3 208 85 126)(4 181 86 155)(5 210 87 128)(6 183 88 157)(7 212 89 130)(8 185 90 159)(9 214 91 132)(10 187 92 161)(11 216 93 134)(12 189 94 163)(13 218 95 136)(14 191 96 165)(15 220 97 138)(16 193 98 167)(17 222 99 140)(18 195 100 113)(19 224 101 142)(20 197 102 115)(21 170 103 144)(22 199 104 117)(23 172 105 146)(24 201 106 119)(25 174 107 148)(26 203 108 121)(27 176 109 150)(28 205 110 123)(29 178 111 152)(30 207 112 125)(31 180 57 154)(32 209 58 127)(33 182 59 156)(34 211 60 129)(35 184 61 158)(36 213 62 131)(37 186 63 160)(38 215 64 133)(39 188 65 162)(40 217 66 135)(41 190 67 164)(42 219 68 137)(43 192 69 166)(44 221 70 139)(45 194 71 168)(46 223 72 141)(47 196 73 114)(48 169 74 143)(49 198 75 116)(50 171 76 145)(51 200 77 118)(52 173 78 147)(53 202 79 120)(54 175 80 149)(55 204 81 122)(56 177 82 151)
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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,59)(60,112)(61,111)(62,110)(63,109)(64,108)(65,107)(66,106)(67,105)(68,104)(69,103)(70,102)(71,101)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(114,168)(115,167)(116,166)(117,165)(118,164)(119,163)(120,162)(121,161)(122,160)(123,159)(124,158)(125,157)(126,156)(127,155)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(169,221)(170,220)(171,219)(172,218)(173,217)(174,216)(175,215)(176,214)(177,213)(178,212)(179,211)(180,210)(181,209)(182,208)(183,207)(184,206)(185,205)(186,204)(187,203)(188,202)(189,201)(190,200)(191,199)(192,198)(193,197)(194,196)(222,224), (1,206,83,124)(2,179,84,153)(3,208,85,126)(4,181,86,155)(5,210,87,128)(6,183,88,157)(7,212,89,130)(8,185,90,159)(9,214,91,132)(10,187,92,161)(11,216,93,134)(12,189,94,163)(13,218,95,136)(14,191,96,165)(15,220,97,138)(16,193,98,167)(17,222,99,140)(18,195,100,113)(19,224,101,142)(20,197,102,115)(21,170,103,144)(22,199,104,117)(23,172,105,146)(24,201,106,119)(25,174,107,148)(26,203,108,121)(27,176,109,150)(28,205,110,123)(29,178,111,152)(30,207,112,125)(31,180,57,154)(32,209,58,127)(33,182,59,156)(34,211,60,129)(35,184,61,158)(36,213,62,131)(37,186,63,160)(38,215,64,133)(39,188,65,162)(40,217,66,135)(41,190,67,164)(42,219,68,137)(43,192,69,166)(44,221,70,139)(45,194,71,168)(46,223,72,141)(47,196,73,114)(48,169,74,143)(49,198,75,116)(50,171,76,145)(51,200,77,118)(52,173,78,147)(53,202,79,120)(54,175,80,149)(55,204,81,122)(56,177,82,151)>;
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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,59)(60,112)(61,111)(62,110)(63,109)(64,108)(65,107)(66,106)(67,105)(68,104)(69,103)(70,102)(71,101)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(114,168)(115,167)(116,166)(117,165)(118,164)(119,163)(120,162)(121,161)(122,160)(123,159)(124,158)(125,157)(126,156)(127,155)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(169,221)(170,220)(171,219)(172,218)(173,217)(174,216)(175,215)(176,214)(177,213)(178,212)(179,211)(180,210)(181,209)(182,208)(183,207)(184,206)(185,205)(186,204)(187,203)(188,202)(189,201)(190,200)(191,199)(192,198)(193,197)(194,196)(222,224), (1,206,83,124)(2,179,84,153)(3,208,85,126)(4,181,86,155)(5,210,87,128)(6,183,88,157)(7,212,89,130)(8,185,90,159)(9,214,91,132)(10,187,92,161)(11,216,93,134)(12,189,94,163)(13,218,95,136)(14,191,96,165)(15,220,97,138)(16,193,98,167)(17,222,99,140)(18,195,100,113)(19,224,101,142)(20,197,102,115)(21,170,103,144)(22,199,104,117)(23,172,105,146)(24,201,106,119)(25,174,107,148)(26,203,108,121)(27,176,109,150)(28,205,110,123)(29,178,111,152)(30,207,112,125)(31,180,57,154)(32,209,58,127)(33,182,59,156)(34,211,60,129)(35,184,61,158)(36,213,62,131)(37,186,63,160)(38,215,64,133)(39,188,65,162)(40,217,66,135)(41,190,67,164)(42,219,68,137)(43,192,69,166)(44,221,70,139)(45,194,71,168)(46,223,72,141)(47,196,73,114)(48,169,74,143)(49,198,75,116)(50,171,76,145)(51,200,77,118)(52,173,78,147)(53,202,79,120)(54,175,80,149)(55,204,81,122)(56,177,82,151) );
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)], [(1,7),(2,6),(3,5),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(57,59),(60,112),(61,111),(62,110),(63,109),(64,108),(65,107),(66,106),(67,105),(68,104),(69,103),(70,102),(71,101),(72,100),(73,99),(74,98),(75,97),(76,96),(77,95),(78,94),(79,93),(80,92),(81,91),(82,90),(83,89),(84,88),(85,87),(114,168),(115,167),(116,166),(117,165),(118,164),(119,163),(120,162),(121,161),(122,160),(123,159),(124,158),(125,157),(126,156),(127,155),(128,154),(129,153),(130,152),(131,151),(132,150),(133,149),(134,148),(135,147),(136,146),(137,145),(138,144),(139,143),(140,142),(169,221),(170,220),(171,219),(172,218),(173,217),(174,216),(175,215),(176,214),(177,213),(178,212),(179,211),(180,210),(181,209),(182,208),(183,207),(184,206),(185,205),(186,204),(187,203),(188,202),(189,201),(190,200),(191,199),(192,198),(193,197),(194,196),(222,224)], [(1,206,83,124),(2,179,84,153),(3,208,85,126),(4,181,86,155),(5,210,87,128),(6,183,88,157),(7,212,89,130),(8,185,90,159),(9,214,91,132),(10,187,92,161),(11,216,93,134),(12,189,94,163),(13,218,95,136),(14,191,96,165),(15,220,97,138),(16,193,98,167),(17,222,99,140),(18,195,100,113),(19,224,101,142),(20,197,102,115),(21,170,103,144),(22,199,104,117),(23,172,105,146),(24,201,106,119),(25,174,107,148),(26,203,108,121),(27,176,109,150),(28,205,110,123),(29,178,111,152),(30,207,112,125),(31,180,57,154),(32,209,58,127),(33,182,59,156),(34,211,60,129),(35,184,61,158),(36,213,62,131),(37,186,63,160),(38,215,64,133),(39,188,65,162),(40,217,66,135),(41,190,67,164),(42,219,68,137),(43,192,69,166),(44,221,70,139),(45,194,71,168),(46,223,72,141),(47,196,73,114),(48,169,74,143),(49,198,75,116),(50,171,76,145),(51,200,77,118),(52,173,78,147),(53,202,79,120),(54,175,80,149),(55,204,81,122),(56,177,82,151)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 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 | 28 | 28 | 28 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D28 | C4○D28 | C8⋊C22 | C8⋊D14 |
| kernel | D56⋊C4 | C8⋊Dic7 | C2.D56 | C7×C8⋊C4 | C4×D28 | C2×D56 | D56 | C2×C28 | C8⋊C4 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 3 | 2 | 3 | 6 | 12 | 12 | 12 | 2 | 12 |
Matrix representation of D56⋊C4 ►in GL6(𝔽113)
| 29 | 112 | 0 | 0 | 0 | 0 |
| 51 | 84 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 12 | 56 | 95 |
| 0 | 0 | 14 | 73 | 1 | 27 |
| 0 | 0 | 112 | 77 | 12 | 92 |
| 0 | 0 | 8 | 16 | 85 | 25 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 55 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 17 | 0 | 0 |
| 0 | 0 | 90 | 77 | 0 | 0 |
| 0 | 0 | 41 | 15 | 4 | 90 |
| 0 | 0 | 17 | 34 | 94 | 109 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 103 | 86 | 83 |
| 0 | 0 | 108 | 104 | 30 | 0 |
| 0 | 0 | 4 | 86 | 61 | 108 |
| 0 | 0 | 57 | 5 | 41 | 28 |
G:=sub<GL(6,GF(113))| [29,51,0,0,0,0,112,84,0,0,0,0,0,0,3,14,112,8,0,0,12,73,77,16,0,0,56,1,12,85,0,0,95,27,92,25],[112,55,0,0,0,0,0,1,0,0,0,0,0,0,36,90,41,17,0,0,17,77,15,34,0,0,0,0,4,94,0,0,0,0,90,109],[15,0,0,0,0,0,0,15,0,0,0,0,0,0,33,108,4,57,0,0,103,104,86,5,0,0,86,30,61,41,0,0,83,0,108,28] >;
D56⋊C4 in GAP, Magma, Sage, TeX
D_{56}\rtimes C_4 % in TeX
G:=Group("D56:C4"); // GroupNames label
G:=SmallGroup(448,245);
// by ID
G=gap.SmallGroup(448,245);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,387,58,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^28*b>;
// generators/relations