metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊5D28, C56⋊23D4, C28⋊7SD16, C42.260D14, (C4×C8)⋊12D7, (C4×C56)⋊17C2, C7⋊1(C8⋊5D4), C4⋊1(C56⋊C2), C28⋊2Q8⋊1C2, C4.30(C2×D28), (C2×C4).79D28, (C2×C28).376D4, C28⋊4D4.1C2, C28.273(C2×D4), (C2×C8).317D14, C14.4(C2×SD16), C2.5(C28⋊4D4), C14.3(C4⋊1D4), (C2×D28).1C22, C22.89(C2×D28), (C2×C56).389C22, (C2×C28).722C23, (C4×C28).306C22, (C2×Dic14).2C22, (C2×C56⋊C2)⋊7C2, C2.7(C2×C56⋊C2), (C2×C14).105(C2×D4), (C2×C4).665(C22×D7), SmallGroup(448,227)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊5D28
G = < a,b,c | a8=b28=c2=1, ab=ba, cac=a3, cbc=b-1 >
Subgroups: 1028 in 142 conjugacy classes, 55 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C56, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C8⋊5D4, C56⋊C2, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C2×D28, C2×D28, C4×C56, C28⋊2Q8, C28⋊4D4, C2×C56⋊C2, C8⋊5D28
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C4⋊1D4, C2×SD16, D28, C22×D7, C8⋊5D4, C56⋊C2, C2×D28, C28⋊4D4, C2×C56⋊C2, C8⋊5D28
►On 224 points
Generators in S224
(1 221 141 86 46 186 58 125)(2 222 142 87 47 187 59 126)(3 223 143 88 48 188 60 127)(4 224 144 89 49 189 61 128)(5 197 145 90 50 190 62 129)(6 198 146 91 51 191 63 130)(7 199 147 92 52 192 64 131)(8 200 148 93 53 193 65 132)(9 201 149 94 54 194 66 133)(10 202 150 95 55 195 67 134)(11 203 151 96 56 196 68 135)(12 204 152 97 29 169 69 136)(13 205 153 98 30 170 70 137)(14 206 154 99 31 171 71 138)(15 207 155 100 32 172 72 139)(16 208 156 101 33 173 73 140)(17 209 157 102 34 174 74 113)(18 210 158 103 35 175 75 114)(19 211 159 104 36 176 76 115)(20 212 160 105 37 177 77 116)(21 213 161 106 38 178 78 117)(22 214 162 107 39 179 79 118)(23 215 163 108 40 180 80 119)(24 216 164 109 41 181 81 120)(25 217 165 110 42 182 82 121)(26 218 166 111 43 183 83 122)(27 219 167 112 44 184 84 123)(28 220 168 85 45 185 57 124)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(57 162)(58 161)(59 160)(60 159)(61 158)(62 157)(63 156)(64 155)(65 154)(66 153)(67 152)(68 151)(69 150)(70 149)(71 148)(72 147)(73 146)(74 145)(75 144)(76 143)(77 142)(78 141)(79 168)(80 167)(81 166)(82 165)(83 164)(84 163)(85 214)(86 213)(87 212)(88 211)(89 210)(90 209)(91 208)(92 207)(93 206)(94 205)(95 204)(96 203)(97 202)(98 201)(99 200)(100 199)(101 198)(102 197)(103 224)(104 223)(105 222)(106 221)(107 220)(108 219)(109 218)(110 217)(111 216)(112 215)(113 190)(114 189)(115 188)(116 187)(117 186)(118 185)(119 184)(120 183)(121 182)(122 181)(123 180)(124 179)(125 178)(126 177)(127 176)(128 175)(129 174)(130 173)(131 172)(132 171)(133 170)(134 169)(135 196)(136 195)(137 194)(138 193)(139 192)(140 191)
G:=sub<Sym(224)| (1,221,141,86,46,186,58,125)(2,222,142,87,47,187,59,126)(3,223,143,88,48,188,60,127)(4,224,144,89,49,189,61,128)(5,197,145,90,50,190,62,129)(6,198,146,91,51,191,63,130)(7,199,147,92,52,192,64,131)(8,200,148,93,53,193,65,132)(9,201,149,94,54,194,66,133)(10,202,150,95,55,195,67,134)(11,203,151,96,56,196,68,135)(12,204,152,97,29,169,69,136)(13,205,153,98,30,170,70,137)(14,206,154,99,31,171,71,138)(15,207,155,100,32,172,72,139)(16,208,156,101,33,173,73,140)(17,209,157,102,34,174,74,113)(18,210,158,103,35,175,75,114)(19,211,159,104,36,176,76,115)(20,212,160,105,37,177,77,116)(21,213,161,106,38,178,78,117)(22,214,162,107,39,179,79,118)(23,215,163,108,40,180,80,119)(24,216,164,109,41,181,81,120)(25,217,165,110,42,182,82,121)(26,218,166,111,43,183,83,122)(27,219,167,112,44,184,84,123)(28,220,168,85,45,185,57,124), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,162)(58,161)(59,160)(60,159)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,214)(86,213)(87,212)(88,211)(89,210)(90,209)(91,208)(92,207)(93,206)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,224)(104,223)(105,222)(106,221)(107,220)(108,219)(109,218)(110,217)(111,216)(112,215)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,196)(136,195)(137,194)(138,193)(139,192)(140,191)>;
G:=Group( (1,221,141,86,46,186,58,125)(2,222,142,87,47,187,59,126)(3,223,143,88,48,188,60,127)(4,224,144,89,49,189,61,128)(5,197,145,90,50,190,62,129)(6,198,146,91,51,191,63,130)(7,199,147,92,52,192,64,131)(8,200,148,93,53,193,65,132)(9,201,149,94,54,194,66,133)(10,202,150,95,55,195,67,134)(11,203,151,96,56,196,68,135)(12,204,152,97,29,169,69,136)(13,205,153,98,30,170,70,137)(14,206,154,99,31,171,71,138)(15,207,155,100,32,172,72,139)(16,208,156,101,33,173,73,140)(17,209,157,102,34,174,74,113)(18,210,158,103,35,175,75,114)(19,211,159,104,36,176,76,115)(20,212,160,105,37,177,77,116)(21,213,161,106,38,178,78,117)(22,214,162,107,39,179,79,118)(23,215,163,108,40,180,80,119)(24,216,164,109,41,181,81,120)(25,217,165,110,42,182,82,121)(26,218,166,111,43,183,83,122)(27,219,167,112,44,184,84,123)(28,220,168,85,45,185,57,124), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,162)(58,161)(59,160)(60,159)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,214)(86,213)(87,212)(88,211)(89,210)(90,209)(91,208)(92,207)(93,206)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,224)(104,223)(105,222)(106,221)(107,220)(108,219)(109,218)(110,217)(111,216)(112,215)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,196)(136,195)(137,194)(138,193)(139,192)(140,191) );
G=PermutationGroup([[(1,221,141,86,46,186,58,125),(2,222,142,87,47,187,59,126),(3,223,143,88,48,188,60,127),(4,224,144,89,49,189,61,128),(5,197,145,90,50,190,62,129),(6,198,146,91,51,191,63,130),(7,199,147,92,52,192,64,131),(8,200,148,93,53,193,65,132),(9,201,149,94,54,194,66,133),(10,202,150,95,55,195,67,134),(11,203,151,96,56,196,68,135),(12,204,152,97,29,169,69,136),(13,205,153,98,30,170,70,137),(14,206,154,99,31,171,71,138),(15,207,155,100,32,172,72,139),(16,208,156,101,33,173,73,140),(17,209,157,102,34,174,74,113),(18,210,158,103,35,175,75,114),(19,211,159,104,36,176,76,115),(20,212,160,105,37,177,77,116),(21,213,161,106,38,178,78,117),(22,214,162,107,39,179,79,118),(23,215,163,108,40,180,80,119),(24,216,164,109,41,181,81,120),(25,217,165,110,42,182,82,121),(26,218,166,111,43,183,83,122),(27,219,167,112,44,184,84,123),(28,220,168,85,45,185,57,124)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(57,162),(58,161),(59,160),(60,159),(61,158),(62,157),(63,156),(64,155),(65,154),(66,153),(67,152),(68,151),(69,150),(70,149),(71,148),(72,147),(73,146),(74,145),(75,144),(76,143),(77,142),(78,141),(79,168),(80,167),(81,166),(82,165),(83,164),(84,163),(85,214),(86,213),(87,212),(88,211),(89,210),(90,209),(91,208),(92,207),(93,206),(94,205),(95,204),(96,203),(97,202),(98,201),(99,200),(100,199),(101,198),(102,197),(103,224),(104,223),(105,222),(106,221),(107,220),(108,219),(109,218),(110,217),(111,216),(112,215),(113,190),(114,189),(115,188),(116,187),(117,186),(118,185),(119,184),(120,183),(121,182),(122,181),(123,180),(124,179),(125,178),(126,177),(127,176),(128,175),(129,174),(130,173),(131,172),(132,171),(133,170),(134,169),(135,196),(136,195),(137,194),(138,193),(139,192),(140,191)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 56 | 2 | ··· | 2 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | SD16 | D14 | D14 | D28 | D28 | C56⋊C2 |
| kernel | C8⋊5D28 | C4×C56 | C28⋊2Q8 | C28⋊4D4 | C2×C56⋊C2 | C56 | C2×C28 | C4×C8 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 3 | 8 | 3 | 6 | 24 | 12 | 48 |
Matrix representation of C8⋊5D28 ►in GL4(𝔽113) generated by
| 17 | 80 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 0 | 0 | 96 | 42 |
| 0 | 0 | 79 | 104 |
| 94 | 13 | 0 | 0 |
| 100 | 104 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 25 | 34 |
| 25 | 34 | 0 | 0 |
| 88 | 88 | 0 | 0 |
| 0 | 0 | 112 | 80 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [17,33,0,0,80,9,0,0,0,0,96,79,0,0,42,104],[94,100,0,0,13,104,0,0,0,0,0,25,0,0,9,34],[25,88,0,0,34,88,0,0,0,0,112,0,0,0,80,1] >;
C8⋊5D28 in GAP, Magma, Sage, TeX
C_8\rtimes_5D_{28} % in TeX
G:=Group("C8:5D28"); // GroupNames label
G:=SmallGroup(448,227);
// by ID
G=gap.SmallGroup(448,227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations