metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊2C8, C28.54D8, C28.49SD16, C28.3M4(2), C42.192D14, C4⋊C8⋊1D7, C7⋊2(D4⋊C8), C4.1(C8×D7), C28.3(C2×C8), C14.10C4≀C2, (C4×D28).6C2, C4.27(D4⋊D7), C2.7(D14⋊C8), (C2×D28).11C4, (C2×C28).225D4, (C2×C4).109D28, C4.1(C8⋊D7), C4.15(Q8⋊D7), C4⋊Dic7.14C4, C14.5(C22⋊C8), (C4×C28).41C22, C14.3(D4⋊C4), C2.1(C14.D8), C2.1(D28⋊4C4), C22.34(D14⋊C4), (C4×C7⋊C8)⋊1C2, (C7×C4⋊C8)⋊1C2, (C2×C4).65(C4×D7), (C2×C28).48(C2×C4), (C2×C4).265(C7⋊D4), (C2×C14).45(C22⋊C4), SmallGroup(448,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊2C8
G = < a,b,c | a28=b2=c8=1, bab=a-1, cac-1=a15, cbc-1=a7b >
Subgroups: 484 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C4×C8, C4⋊C8, C4×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×C28, C22×D7, D4⋊C8, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C4×C7⋊C8, C7×C4⋊C8, C4×D28, D28⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), D8, SD16, D14, C22⋊C8, D4⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, D4⋊C8, C8×D7, C8⋊D7, D14⋊C4, D4⋊D7, Q8⋊D7, C14.D8, D14⋊C8, D28⋊4C4, D28⋊2C8
►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 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 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(57 77)(58 76)(59 75)(60 74)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(78 84)(79 83)(80 82)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(113 137)(114 136)(115 135)(116 134)(117 133)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)(138 140)(141 161)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)(162 168)(163 167)(164 166)(169 184)(170 183)(171 182)(172 181)(173 180)(174 179)(175 178)(176 177)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(197 222)(198 221)(199 220)(200 219)(201 218)(202 217)(203 216)(204 215)(205 214)(206 213)(207 212)(208 211)(209 210)(223 224)
(1 129 191 31 106 57 224 155)(2 116 192 46 107 72 197 142)(3 131 193 33 108 59 198 157)(4 118 194 48 109 74 199 144)(5 133 195 35 110 61 200 159)(6 120 196 50 111 76 201 146)(7 135 169 37 112 63 202 161)(8 122 170 52 85 78 203 148)(9 137 171 39 86 65 204 163)(10 124 172 54 87 80 205 150)(11 139 173 41 88 67 206 165)(12 126 174 56 89 82 207 152)(13 113 175 43 90 69 208 167)(14 128 176 30 91 84 209 154)(15 115 177 45 92 71 210 141)(16 130 178 32 93 58 211 156)(17 117 179 47 94 73 212 143)(18 132 180 34 95 60 213 158)(19 119 181 49 96 75 214 145)(20 134 182 36 97 62 215 160)(21 121 183 51 98 77 216 147)(22 136 184 38 99 64 217 162)(23 123 185 53 100 79 218 149)(24 138 186 40 101 66 219 164)(25 125 187 55 102 81 220 151)(26 140 188 42 103 68 221 166)(27 127 189 29 104 83 222 153)(28 114 190 44 105 70 223 168)
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), (1,28)(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)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,77)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(78,84)(79,83)(80,82)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)(138,140)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(162,168)(163,167)(164,166)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,177)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(197,222)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)(204,215)(205,214)(206,213)(207,212)(208,211)(209,210)(223,224), (1,129,191,31,106,57,224,155)(2,116,192,46,107,72,197,142)(3,131,193,33,108,59,198,157)(4,118,194,48,109,74,199,144)(5,133,195,35,110,61,200,159)(6,120,196,50,111,76,201,146)(7,135,169,37,112,63,202,161)(8,122,170,52,85,78,203,148)(9,137,171,39,86,65,204,163)(10,124,172,54,87,80,205,150)(11,139,173,41,88,67,206,165)(12,126,174,56,89,82,207,152)(13,113,175,43,90,69,208,167)(14,128,176,30,91,84,209,154)(15,115,177,45,92,71,210,141)(16,130,178,32,93,58,211,156)(17,117,179,47,94,73,212,143)(18,132,180,34,95,60,213,158)(19,119,181,49,96,75,214,145)(20,134,182,36,97,62,215,160)(21,121,183,51,98,77,216,147)(22,136,184,38,99,64,217,162)(23,123,185,53,100,79,218,149)(24,138,186,40,101,66,219,164)(25,125,187,55,102,81,220,151)(26,140,188,42,103,68,221,166)(27,127,189,29,104,83,222,153)(28,114,190,44,105,70,223,168)>;
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), (1,28)(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)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,77)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(78,84)(79,83)(80,82)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)(138,140)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(162,168)(163,167)(164,166)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,177)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(197,222)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)(204,215)(205,214)(206,213)(207,212)(208,211)(209,210)(223,224), (1,129,191,31,106,57,224,155)(2,116,192,46,107,72,197,142)(3,131,193,33,108,59,198,157)(4,118,194,48,109,74,199,144)(5,133,195,35,110,61,200,159)(6,120,196,50,111,76,201,146)(7,135,169,37,112,63,202,161)(8,122,170,52,85,78,203,148)(9,137,171,39,86,65,204,163)(10,124,172,54,87,80,205,150)(11,139,173,41,88,67,206,165)(12,126,174,56,89,82,207,152)(13,113,175,43,90,69,208,167)(14,128,176,30,91,84,209,154)(15,115,177,45,92,71,210,141)(16,130,178,32,93,58,211,156)(17,117,179,47,94,73,212,143)(18,132,180,34,95,60,213,158)(19,119,181,49,96,75,214,145)(20,134,182,36,97,62,215,160)(21,121,183,51,98,77,216,147)(22,136,184,38,99,64,217,162)(23,123,185,53,100,79,218,149)(24,138,186,40,101,66,219,164)(25,125,187,55,102,81,220,151)(26,140,188,42,103,68,221,166)(27,127,189,29,104,83,222,153)(28,114,190,44,105,70,223,168) );
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)], [(1,28),(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),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,77),(58,76),(59,75),(60,74),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(78,84),(79,83),(80,82),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(113,137),(114,136),(115,135),(116,134),(117,133),(118,132),(119,131),(120,130),(121,129),(122,128),(123,127),(124,126),(138,140),(141,161),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152),(162,168),(163,167),(164,166),(169,184),(170,183),(171,182),(172,181),(173,180),(174,179),(175,178),(176,177),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(197,222),(198,221),(199,220),(200,219),(201,218),(202,217),(203,216),(204,215),(205,214),(206,213),(207,212),(208,211),(209,210),(223,224)], [(1,129,191,31,106,57,224,155),(2,116,192,46,107,72,197,142),(3,131,193,33,108,59,198,157),(4,118,194,48,109,74,199,144),(5,133,195,35,110,61,200,159),(6,120,196,50,111,76,201,146),(7,135,169,37,112,63,202,161),(8,122,170,52,85,78,203,148),(9,137,171,39,86,65,204,163),(10,124,172,54,87,80,205,150),(11,139,173,41,88,67,206,165),(12,126,174,56,89,82,207,152),(13,113,175,43,90,69,208,167),(14,128,176,30,91,84,209,154),(15,115,177,45,92,71,210,141),(16,130,178,32,93,58,211,156),(17,117,179,47,94,73,212,143),(18,132,180,34,95,60,213,158),(19,119,181,49,96,75,214,145),(20,134,182,36,97,62,215,160),(21,121,183,51,98,77,216,147),(22,136,184,38,99,64,217,162),(23,123,185,53,100,79,218,149),(24,138,186,40,101,66,219,164),(25,125,187,55,102,81,220,151),(26,140,188,42,103,68,221,166),(27,127,189,29,104,83,222,153),(28,114,190,44,105,70,223,168)]])
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 | ··· | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | D8 | SD16 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | C8×D7 | C8⋊D7 | D4⋊D7 | Q8⋊D7 | D28⋊4C4 |
| kernel | D28⋊2C8 | C4×C7⋊C8 | C7×C4⋊C8 | C4×D28 | C4⋊Dic7 | C2×D28 | D28 | C2×C28 | C4⋊C8 | C28 | C28 | C28 | C42 | C14 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 3 | 3 | 6 |
Matrix representation of D28⋊2C8 ►in GL4(𝔽113) generated by
| 24 | 65 | 0 | 0 |
| 24 | 98 | 0 | 0 |
| 0 | 0 | 1 | 69 |
| 0 | 0 | 36 | 112 |
| 90 | 15 | 0 | 0 |
| 10 | 23 | 0 | 0 |
| 0 | 0 | 1 | 69 |
| 0 | 0 | 0 | 112 |
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 16 | 100 |
| 0 | 0 | 62 | 97 |
G:=sub<GL(4,GF(113))| [24,24,0,0,65,98,0,0,0,0,1,36,0,0,69,112],[90,10,0,0,15,23,0,0,0,0,1,0,0,0,69,112],[18,0,0,0,0,18,0,0,0,0,16,62,0,0,100,97] >;
D28⋊2C8 in GAP, Magma, Sage, TeX
D_{28}\rtimes_2C_8 % in TeX
G:=Group("D28:2C8"); // GroupNames label
G:=SmallGroup(448,40);
// by ID
G=gap.SmallGroup(448,40);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,100,1123,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^7*b>;
// generators/relations