metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D28, C56⋊1D4, C42.18D14, C8⋊C4⋊3D7, C7⋊1(C8⋊3D4), (C2×D56)⋊10C2, C28⋊4D4⋊2C2, (C2×C28).36D4, (C2×C8).55D14, C4.34(C2×D28), (C2×C4).25D28, C28.277(C2×D4), C4.D28⋊2C2, (C4×C28).3C22, C2.9(C28⋊4D4), C14.7(C4⋊1D4), C2.7(C8⋊D14), C14.4(C8⋊C22), (C2×C56).56C22, (C2×D28).6C22, C22.97(C2×D28), (C2×C28).733C23, (C2×Dic14).7C22, (C7×C8⋊C4)⋊4C2, (C2×C56⋊C2)⋊1C2, (C2×C14).116(C2×D4), (C2×C4).677(C22×D7), SmallGroup(448,246)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D28
G = < a,b,c | a8=b28=c2=1, bab-1=a5, cac=a3, cbc=b-1 >
Subgroups: 1156 in 144 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C56, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C8⋊3D4, C56⋊C2, D56, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×D28, C2×D28, C2×D28, C7×C8⋊C4, C28⋊4D4, C4.D28, C2×C56⋊C2, C2×D56, C8⋊D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C8⋊C22, D28, C22×D7, C8⋊3D4, C2×D28, C28⋊4D4, C8⋊D14, C8⋊D28
►On 224 points
Generators in S224
(1 65 206 107 38 114 179 151)(2 115 207 152 39 66 180 108)(3 67 208 109 40 116 181 153)(4 117 209 154 41 68 182 110)(5 69 210 111 42 118 183 155)(6 119 211 156 43 70 184 112)(7 71 212 85 44 120 185 157)(8 121 213 158 45 72 186 86)(9 73 214 87 46 122 187 159)(10 123 215 160 47 74 188 88)(11 75 216 89 48 124 189 161)(12 125 217 162 49 76 190 90)(13 77 218 91 50 126 191 163)(14 127 219 164 51 78 192 92)(15 79 220 93 52 128 193 165)(16 129 221 166 53 80 194 94)(17 81 222 95 54 130 195 167)(18 131 223 168 55 82 196 96)(19 83 224 97 56 132 169 141)(20 133 197 142 29 84 170 98)(21 57 198 99 30 134 171 143)(22 135 199 144 31 58 172 100)(23 59 200 101 32 136 173 145)(24 137 201 146 33 60 174 102)(25 61 202 103 34 138 175 147)(26 139 203 148 35 62 176 104)(27 63 204 105 36 140 177 149)(28 113 205 150 37 64 178 106)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(47 56)(48 55)(49 54)(50 53)(51 52)(57 158)(58 157)(59 156)(60 155)(61 154)(62 153)(63 152)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 144)(72 143)(73 142)(74 141)(75 168)(76 167)(77 166)(78 165)(79 164)(80 163)(81 162)(82 161)(83 160)(84 159)(85 135)(86 134)(87 133)(88 132)(89 131)(90 130)(91 129)(92 128)(93 127)(94 126)(95 125)(96 124)(97 123)(98 122)(99 121)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 140)(109 139)(110 138)(111 137)(112 136)(169 215)(170 214)(171 213)(172 212)(173 211)(174 210)(175 209)(176 208)(177 207)(178 206)(179 205)(180 204)(181 203)(182 202)(183 201)(184 200)(185 199)(186 198)(187 197)(188 224)(189 223)(190 222)(191 221)(192 220)(193 219)(194 218)(195 217)(196 216)
G:=sub<Sym(224)| (1,65,206,107,38,114,179,151)(2,115,207,152,39,66,180,108)(3,67,208,109,40,116,181,153)(4,117,209,154,41,68,182,110)(5,69,210,111,42,118,183,155)(6,119,211,156,43,70,184,112)(7,71,212,85,44,120,185,157)(8,121,213,158,45,72,186,86)(9,73,214,87,46,122,187,159)(10,123,215,160,47,74,188,88)(11,75,216,89,48,124,189,161)(12,125,217,162,49,76,190,90)(13,77,218,91,50,126,191,163)(14,127,219,164,51,78,192,92)(15,79,220,93,52,128,193,165)(16,129,221,166,53,80,194,94)(17,81,222,95,54,130,195,167)(18,131,223,168,55,82,196,96)(19,83,224,97,56,132,169,141)(20,133,197,142,29,84,170,98)(21,57,198,99,30,134,171,143)(22,135,199,144,31,58,172,100)(23,59,200,101,32,136,173,145)(24,137,201,146,33,60,174,102)(25,61,202,103,34,138,175,147)(26,139,203,148,35,62,176,104)(27,63,204,105,36,140,177,149)(28,113,205,150,37,64,178,106), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(47,56)(48,55)(49,54)(50,53)(51,52)(57,158)(58,157)(59,156)(60,155)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,168)(76,167)(77,166)(78,165)(79,164)(80,163)(81,162)(82,161)(83,160)(84,159)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,140)(109,139)(110,138)(111,137)(112,136)(169,215)(170,214)(171,213)(172,212)(173,211)(174,210)(175,209)(176,208)(177,207)(178,206)(179,205)(180,204)(181,203)(182,202)(183,201)(184,200)(185,199)(186,198)(187,197)(188,224)(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)(196,216)>;
G:=Group( (1,65,206,107,38,114,179,151)(2,115,207,152,39,66,180,108)(3,67,208,109,40,116,181,153)(4,117,209,154,41,68,182,110)(5,69,210,111,42,118,183,155)(6,119,211,156,43,70,184,112)(7,71,212,85,44,120,185,157)(8,121,213,158,45,72,186,86)(9,73,214,87,46,122,187,159)(10,123,215,160,47,74,188,88)(11,75,216,89,48,124,189,161)(12,125,217,162,49,76,190,90)(13,77,218,91,50,126,191,163)(14,127,219,164,51,78,192,92)(15,79,220,93,52,128,193,165)(16,129,221,166,53,80,194,94)(17,81,222,95,54,130,195,167)(18,131,223,168,55,82,196,96)(19,83,224,97,56,132,169,141)(20,133,197,142,29,84,170,98)(21,57,198,99,30,134,171,143)(22,135,199,144,31,58,172,100)(23,59,200,101,32,136,173,145)(24,137,201,146,33,60,174,102)(25,61,202,103,34,138,175,147)(26,139,203,148,35,62,176,104)(27,63,204,105,36,140,177,149)(28,113,205,150,37,64,178,106), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(47,56)(48,55)(49,54)(50,53)(51,52)(57,158)(58,157)(59,156)(60,155)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,168)(76,167)(77,166)(78,165)(79,164)(80,163)(81,162)(82,161)(83,160)(84,159)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,140)(109,139)(110,138)(111,137)(112,136)(169,215)(170,214)(171,213)(172,212)(173,211)(174,210)(175,209)(176,208)(177,207)(178,206)(179,205)(180,204)(181,203)(182,202)(183,201)(184,200)(185,199)(186,198)(187,197)(188,224)(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)(196,216) );
G=PermutationGroup([[(1,65,206,107,38,114,179,151),(2,115,207,152,39,66,180,108),(3,67,208,109,40,116,181,153),(4,117,209,154,41,68,182,110),(5,69,210,111,42,118,183,155),(6,119,211,156,43,70,184,112),(7,71,212,85,44,120,185,157),(8,121,213,158,45,72,186,86),(9,73,214,87,46,122,187,159),(10,123,215,160,47,74,188,88),(11,75,216,89,48,124,189,161),(12,125,217,162,49,76,190,90),(13,77,218,91,50,126,191,163),(14,127,219,164,51,78,192,92),(15,79,220,93,52,128,193,165),(16,129,221,166,53,80,194,94),(17,81,222,95,54,130,195,167),(18,131,223,168,55,82,196,96),(19,83,224,97,56,132,169,141),(20,133,197,142,29,84,170,98),(21,57,198,99,30,134,171,143),(22,135,199,144,31,58,172,100),(23,59,200,101,32,136,173,145),(24,137,201,146,33,60,174,102),(25,61,202,103,34,138,175,147),(26,139,203,148,35,62,176,104),(27,63,204,105,36,140,177,149),(28,113,205,150,37,64,178,106)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39),(37,38),(47,56),(48,55),(49,54),(50,53),(51,52),(57,158),(58,157),(59,156),(60,155),(61,154),(62,153),(63,152),(64,151),(65,150),(66,149),(67,148),(68,147),(69,146),(70,145),(71,144),(72,143),(73,142),(74,141),(75,168),(76,167),(77,166),(78,165),(79,164),(80,163),(81,162),(82,161),(83,160),(84,159),(85,135),(86,134),(87,133),(88,132),(89,131),(90,130),(91,129),(92,128),(93,127),(94,126),(95,125),(96,124),(97,123),(98,122),(99,121),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,140),(109,139),(110,138),(111,137),(112,136),(169,215),(170,214),(171,213),(172,212),(173,211),(174,210),(175,209),(176,208),(177,207),(178,206),(179,205),(180,204),(181,203),(182,202),(183,201),(184,200),(185,199),(186,198),(187,197),(188,224),(189,223),(190,222),(191,221),(192,220),(193,219),(194,218),(195,217),(196,216)]])
76 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 | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| 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 | 56 | 56 | 56 | 2 | 2 | 4 | 4 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D28 | D28 | C8⋊C22 | C8⋊D14 |
| kernel | C8⋊D28 | C7×C8⋊C4 | C28⋊4D4 | C4.D28 | C2×C56⋊C2 | C2×D56 | C56 | C2×C28 | C8⋊C4 | C42 | C2×C8 | C8 | C2×C4 | C14 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 3 | 3 | 6 | 24 | 12 | 2 | 12 |
Matrix representation of C8⋊D28 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 105 | 9 | 51 | 0 |
| 0 | 0 | 104 | 95 | 0 | 51 |
| 0 | 0 | 76 | 69 | 8 | 104 |
| 0 | 0 | 44 | 37 | 9 | 18 |
| 44 | 31 | 0 | 0 | 0 | 0 |
| 25 | 69 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 94 | 56 | 102 |
| 0 | 0 | 19 | 51 | 11 | 18 |
| 0 | 0 | 23 | 70 | 58 | 19 |
| 0 | 0 | 43 | 8 | 94 | 62 |
| 69 | 82 | 0 | 0 | 0 | 0 |
| 77 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 94 | 56 | 102 |
| 0 | 0 | 58 | 58 | 90 | 57 |
| 0 | 0 | 56 | 102 | 70 | 85 |
| 0 | 0 | 90 | 57 | 70 | 43 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,105,104,76,44,0,0,9,95,69,37,0,0,51,0,8,9,0,0,0,51,104,18],[44,25,0,0,0,0,31,69,0,0,0,0,0,0,55,19,23,43,0,0,94,51,70,8,0,0,56,11,58,94,0,0,102,18,19,62],[69,77,0,0,0,0,82,44,0,0,0,0,0,0,55,58,56,90,0,0,94,58,102,57,0,0,56,90,70,70,0,0,102,57,85,43] >;
C8⋊D28 in GAP, Magma, Sage, TeX
C_8\rtimes D_{28} % in TeX
G:=Group("C8:D28"); // GroupNames label
G:=SmallGroup(448,246);
// by ID
G=gap.SmallGroup(448,246);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,387,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations