metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.1C8, C8.31D28, C56.66D4, Dic14.1C8, (C2×C16)⋊2D7, C4.8(C8×D7), (C2×C112)⋊2C2, C7⋊1(D4.C8), C28.18(C2×C8), C4○D28.1C4, C28.C8⋊9C2, C2.9(D14⋊C8), (C2×C8).322D14, C8.45(C7⋊D4), C4.Dic7.1C4, C4.41(D14⋊C4), C14.8(C22⋊C8), (C2×C14).9M4(2), C28.56(C22⋊C4), (C2×C56).402C22, D28.2C4.3C2, C22.2(C8⋊D7), (C2×C4).97(C4×D7), (C2×C28).231(C2×C4), SmallGroup(448,65)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.C8
G = < a,b,c | a28=b2=1, c8=a14, bab=a-1, ac=ca, cbc-1=a7b >
Smallest permutation representation of D28.C8
►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)
(1 115)(2 114)(3 113)(4 140)(5 139)(6 138)(7 137)(8 136)(9 135)(10 134)(11 133)(12 132)(13 131)(14 130)(15 129)(16 128)(17 127)(18 126)(19 125)(20 124)(21 123)(22 122)(23 121)(24 120)(25 119)(26 118)(27 117)(28 116)(29 144)(30 143)(31 142)(32 141)(33 168)(34 167)(35 166)(36 165)(37 164)(38 163)(39 162)(40 161)(41 160)(42 159)(43 158)(44 157)(45 156)(46 155)(47 154)(48 153)(49 152)(50 151)(51 150)(52 149)(53 148)(54 147)(55 146)(56 145)(57 191)(58 190)(59 189)(60 188)(61 187)(62 186)(63 185)(64 184)(65 183)(66 182)(67 181)(68 180)(69 179)(70 178)(71 177)(72 176)(73 175)(74 174)(75 173)(76 172)(77 171)(78 170)(79 169)(80 196)(81 195)(82 194)(83 193)(84 192)(85 223)(86 222)(87 221)(88 220)(89 219)(90 218)(91 217)(92 216)(93 215)(94 214)(95 213)(96 212)(97 211)(98 210)(99 209)(100 208)(101 207)(102 206)(103 205)(104 204)(105 203)(106 202)(107 201)(108 200)(109 199)(110 198)(111 197)(112 224)
(1 110 30 74 8 89 37 81 15 96 44 60 22 103 51 67)(2 111 31 75 9 90 38 82 16 97 45 61 23 104 52 68)(3 112 32 76 10 91 39 83 17 98 46 62 24 105 53 69)(4 85 33 77 11 92 40 84 18 99 47 63 25 106 54 70)(5 86 34 78 12 93 41 57 19 100 48 64 26 107 55 71)(6 87 35 79 13 94 42 58 20 101 49 65 27 108 56 72)(7 88 36 80 14 95 43 59 21 102 50 66 28 109 29 73)(113 203 155 193 134 224 148 186 127 217 141 179 120 210 162 172)(114 204 156 194 135 197 149 187 128 218 142 180 121 211 163 173)(115 205 157 195 136 198 150 188 129 219 143 181 122 212 164 174)(116 206 158 196 137 199 151 189 130 220 144 182 123 213 165 175)(117 207 159 169 138 200 152 190 131 221 145 183 124 214 166 176)(118 208 160 170 139 201 153 191 132 222 146 184 125 215 167 177)(119 209 161 171 140 202 154 192 133 223 147 185 126 216 168 178)
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), (1,115)(2,114)(3,113)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,126)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,118)(27,117)(28,116)(29,144)(30,143)(31,142)(32,141)(33,168)(34,167)(35,166)(36,165)(37,164)(38,163)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,146)(56,145)(57,191)(58,190)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,196)(81,195)(82,194)(83,193)(84,192)(85,223)(86,222)(87,221)(88,220)(89,219)(90,218)(91,217)(92,216)(93,215)(94,214)(95,213)(96,212)(97,211)(98,210)(99,209)(100,208)(101,207)(102,206)(103,205)(104,204)(105,203)(106,202)(107,201)(108,200)(109,199)(110,198)(111,197)(112,224), (1,110,30,74,8,89,37,81,15,96,44,60,22,103,51,67)(2,111,31,75,9,90,38,82,16,97,45,61,23,104,52,68)(3,112,32,76,10,91,39,83,17,98,46,62,24,105,53,69)(4,85,33,77,11,92,40,84,18,99,47,63,25,106,54,70)(5,86,34,78,12,93,41,57,19,100,48,64,26,107,55,71)(6,87,35,79,13,94,42,58,20,101,49,65,27,108,56,72)(7,88,36,80,14,95,43,59,21,102,50,66,28,109,29,73)(113,203,155,193,134,224,148,186,127,217,141,179,120,210,162,172)(114,204,156,194,135,197,149,187,128,218,142,180,121,211,163,173)(115,205,157,195,136,198,150,188,129,219,143,181,122,212,164,174)(116,206,158,196,137,199,151,189,130,220,144,182,123,213,165,175)(117,207,159,169,138,200,152,190,131,221,145,183,124,214,166,176)(118,208,160,170,139,201,153,191,132,222,146,184,125,215,167,177)(119,209,161,171,140,202,154,192,133,223,147,185,126,216,168,178)>;
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), (1,115)(2,114)(3,113)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,126)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,118)(27,117)(28,116)(29,144)(30,143)(31,142)(32,141)(33,168)(34,167)(35,166)(36,165)(37,164)(38,163)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,146)(56,145)(57,191)(58,190)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,196)(81,195)(82,194)(83,193)(84,192)(85,223)(86,222)(87,221)(88,220)(89,219)(90,218)(91,217)(92,216)(93,215)(94,214)(95,213)(96,212)(97,211)(98,210)(99,209)(100,208)(101,207)(102,206)(103,205)(104,204)(105,203)(106,202)(107,201)(108,200)(109,199)(110,198)(111,197)(112,224), (1,110,30,74,8,89,37,81,15,96,44,60,22,103,51,67)(2,111,31,75,9,90,38,82,16,97,45,61,23,104,52,68)(3,112,32,76,10,91,39,83,17,98,46,62,24,105,53,69)(4,85,33,77,11,92,40,84,18,99,47,63,25,106,54,70)(5,86,34,78,12,93,41,57,19,100,48,64,26,107,55,71)(6,87,35,79,13,94,42,58,20,101,49,65,27,108,56,72)(7,88,36,80,14,95,43,59,21,102,50,66,28,109,29,73)(113,203,155,193,134,224,148,186,127,217,141,179,120,210,162,172)(114,204,156,194,135,197,149,187,128,218,142,180,121,211,163,173)(115,205,157,195,136,198,150,188,129,219,143,181,122,212,164,174)(116,206,158,196,137,199,151,189,130,220,144,182,123,213,165,175)(117,207,159,169,138,200,152,190,131,221,145,183,124,214,166,176)(118,208,160,170,139,201,153,191,132,222,146,184,125,215,167,177)(119,209,161,171,140,202,154,192,133,223,147,185,126,216,168,178) );
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)], [(1,115),(2,114),(3,113),(4,140),(5,139),(6,138),(7,137),(8,136),(9,135),(10,134),(11,133),(12,132),(13,131),(14,130),(15,129),(16,128),(17,127),(18,126),(19,125),(20,124),(21,123),(22,122),(23,121),(24,120),(25,119),(26,118),(27,117),(28,116),(29,144),(30,143),(31,142),(32,141),(33,168),(34,167),(35,166),(36,165),(37,164),(38,163),(39,162),(40,161),(41,160),(42,159),(43,158),(44,157),(45,156),(46,155),(47,154),(48,153),(49,152),(50,151),(51,150),(52,149),(53,148),(54,147),(55,146),(56,145),(57,191),(58,190),(59,189),(60,188),(61,187),(62,186),(63,185),(64,184),(65,183),(66,182),(67,181),(68,180),(69,179),(70,178),(71,177),(72,176),(73,175),(74,174),(75,173),(76,172),(77,171),(78,170),(79,169),(80,196),(81,195),(82,194),(83,193),(84,192),(85,223),(86,222),(87,221),(88,220),(89,219),(90,218),(91,217),(92,216),(93,215),(94,214),(95,213),(96,212),(97,211),(98,210),(99,209),(100,208),(101,207),(102,206),(103,205),(104,204),(105,203),(106,202),(107,201),(108,200),(109,199),(110,198),(111,197),(112,224)], [(1,110,30,74,8,89,37,81,15,96,44,60,22,103,51,67),(2,111,31,75,9,90,38,82,16,97,45,61,23,104,52,68),(3,112,32,76,10,91,39,83,17,98,46,62,24,105,53,69),(4,85,33,77,11,92,40,84,18,99,47,63,25,106,54,70),(5,86,34,78,12,93,41,57,19,100,48,64,26,107,55,71),(6,87,35,79,13,94,42,58,20,101,49,65,27,108,56,72),(7,88,36,80,14,95,43,59,21,102,50,66,28,109,29,73),(113,203,155,193,134,224,148,186,127,217,141,179,120,210,162,172),(114,204,156,194,135,197,149,187,128,218,142,180,121,211,163,173),(115,205,157,195,136,198,150,188,129,219,143,181,122,212,164,174),(116,206,158,196,137,199,151,189,130,220,144,182,123,213,165,175),(117,207,159,169,138,200,152,190,131,221,145,183,124,214,166,176),(118,208,160,170,139,201,153,191,132,222,146,184,125,215,167,177),(119,209,161,171,140,202,154,192,133,223,147,185,126,216,168,178)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 28 | 1 | 1 | 2 | 28 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D7 | M4(2) | D14 | D28 | C7⋊D4 | C4×D7 | D4.C8 | C8×D7 | C8⋊D7 | D28.C8 |
| kernel | D28.C8 | C28.C8 | C2×C112 | D28.2C4 | C4.Dic7 | C4○D28 | Dic14 | D28 | C56 | C2×C16 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 3 | 2 | 3 | 6 | 6 | 6 | 8 | 12 | 12 | 48 |
Matrix representation of D28.C8 ►in GL2(𝔽113) generated by
| 32 | 0 |
| 0 | 53 |
| 0 | 53 |
| 32 | 0 |
| 65 | 0 |
| 0 | 42 |
G:=sub<GL(2,GF(113))| [32,0,0,53],[0,32,53,0],[65,0,0,42] >;
D28.C8 in GAP, Magma, Sage, TeX
D_{28}.C_8 % in TeX
G:=Group("D28.C8"); // GroupNames label
G:=SmallGroup(448,65);
// by ID
G=gap.SmallGroup(448,65);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,100,1123,136,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^2=1,c^8=a^14,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations
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