metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.13D28, Q8.13D28, C28.63C24, C56.12C23, D56.14C22, D28.26C23, M4(2).28D14, Dic28.10C22, Dic14.26C23, C8○D4⋊5D7, C7⋊1(Q8○D8), (C7×D4).25D4, C28.75(C2×D4), C4.29(C2×D28), (C7×Q8).25D4, C4○D4.40D14, (C2×C8).102D14, D56⋊7C2⋊13C2, C8.54(C22×D7), C22.5(C2×D28), C4.60(C23×D7), (C2×Dic28)⋊15C2, C8.D14⋊12C2, (C2×C56).70C22, C56⋊C2.2C22, C2.32(C22×D28), C14.30(C22×D4), D4.10D14⋊4C2, (C2×C28).517C23, C4○D28.27C22, (C7×M4(2)).30C22, (C2×Dic14).194C22, (C7×C8○D4)⋊5C2, (C2×C14).10(C2×D4), (C7×C4○D4).47C22, (C2×C4).228(C22×D7), SmallGroup(448,1206)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.13D28
G = < a,b,c,d | a4=b2=d2=1, c28=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c27 >
Subgroups: 1180 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C56, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, Q8○D8, C56⋊C2, D56, Dic28, C2×C56, C7×M4(2), C2×Dic14, C4○D28, D4⋊2D7, Q8×D7, C7×C4○D4, D56⋊7C2, C2×Dic28, C8.D14, C7×C8○D4, D4.10D14, D4.13D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, D28, C22×D7, Q8○D8, C2×D28, C23×D7, C22×D28, D4.13D28
►On 224 points
Generators in S224
(1 60 29 88)(2 61 30 89)(3 62 31 90)(4 63 32 91)(5 64 33 92)(6 65 34 93)(7 66 35 94)(8 67 36 95)(9 68 37 96)(10 69 38 97)(11 70 39 98)(12 71 40 99)(13 72 41 100)(14 73 42 101)(15 74 43 102)(16 75 44 103)(17 76 45 104)(18 77 46 105)(19 78 47 106)(20 79 48 107)(21 80 49 108)(22 81 50 109)(23 82 51 110)(24 83 52 111)(25 84 53 112)(26 85 54 57)(27 86 55 58)(28 87 56 59)(113 206 141 178)(114 207 142 179)(115 208 143 180)(116 209 144 181)(117 210 145 182)(118 211 146 183)(119 212 147 184)(120 213 148 185)(121 214 149 186)(122 215 150 187)(123 216 151 188)(124 217 152 189)(125 218 153 190)(126 219 154 191)(127 220 155 192)(128 221 156 193)(129 222 157 194)(130 223 158 195)(131 224 159 196)(132 169 160 197)(133 170 161 198)(134 171 162 199)(135 172 163 200)(136 173 164 201)(137 174 165 202)(138 175 166 203)(139 176 167 204)(140 177 168 205)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(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)
(1 147)(2 146)(3 145)(4 144)(5 143)(6 142)(7 141)(8 140)(9 139)(10 138)(11 137)(12 136)(13 135)(14 134)(15 133)(16 132)(17 131)(18 130)(19 129)(20 128)(21 127)(22 126)(23 125)(24 124)(25 123)(26 122)(27 121)(28 120)(29 119)(30 118)(31 117)(32 116)(33 115)(34 114)(35 113)(36 168)(37 167)(38 166)(39 165)(40 164)(41 163)(42 162)(43 161)(44 160)(45 159)(46 158)(47 157)(48 156)(49 155)(50 154)(51 153)(52 152)(53 151)(54 150)(55 149)(56 148)(57 187)(58 186)(59 185)(60 184)(61 183)(62 182)(63 181)(64 180)(65 179)(66 178)(67 177)(68 176)(69 175)(70 174)(71 173)(72 172)(73 171)(74 170)(75 169)(76 224)(77 223)(78 222)(79 221)(80 220)(81 219)(82 218)(83 217)(84 216)(85 215)(86 214)(87 213)(88 212)(89 211)(90 210)(91 209)(92 208)(93 207)(94 206)(95 205)(96 204)(97 203)(98 202)(99 201)(100 200)(101 199)(102 198)(103 197)(104 196)(105 195)(106 194)(107 193)(108 192)(109 191)(110 190)(111 189)(112 188)
G:=sub<Sym(224)| (1,60,29,88)(2,61,30,89)(3,62,31,90)(4,63,32,91)(5,64,33,92)(6,65,34,93)(7,66,35,94)(8,67,36,95)(9,68,37,96)(10,69,38,97)(11,70,39,98)(12,71,40,99)(13,72,41,100)(14,73,42,101)(15,74,43,102)(16,75,44,103)(17,76,45,104)(18,77,46,105)(19,78,47,106)(20,79,48,107)(21,80,49,108)(22,81,50,109)(23,82,51,110)(24,83,52,111)(25,84,53,112)(26,85,54,57)(27,86,55,58)(28,87,56,59)(113,206,141,178)(114,207,142,179)(115,208,143,180)(116,209,144,181)(117,210,145,182)(118,211,146,183)(119,212,147,184)(120,213,148,185)(121,214,149,186)(122,215,150,187)(123,216,151,188)(124,217,152,189)(125,218,153,190)(126,219,154,191)(127,220,155,192)(128,221,156,193)(129,222,157,194)(130,223,158,195)(131,224,159,196)(132,169,160,197)(133,170,161,198)(134,171,162,199)(135,172,163,200)(136,173,164,201)(137,174,165,202)(138,175,166,203)(139,176,167,204)(140,177,168,205), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,168)(37,167)(38,166)(39,165)(40,164)(41,163)(42,162)(43,161)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,187)(58,186)(59,185)(60,184)(61,183)(62,182)(63,181)(64,180)(65,179)(66,178)(67,177)(68,176)(69,175)(70,174)(71,173)(72,172)(73,171)(74,170)(75,169)(76,224)(77,223)(78,222)(79,221)(80,220)(81,219)(82,218)(83,217)(84,216)(85,215)(86,214)(87,213)(88,212)(89,211)(90,210)(91,209)(92,208)(93,207)(94,206)(95,205)(96,204)(97,203)(98,202)(99,201)(100,200)(101,199)(102,198)(103,197)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188)>;
G:=Group( (1,60,29,88)(2,61,30,89)(3,62,31,90)(4,63,32,91)(5,64,33,92)(6,65,34,93)(7,66,35,94)(8,67,36,95)(9,68,37,96)(10,69,38,97)(11,70,39,98)(12,71,40,99)(13,72,41,100)(14,73,42,101)(15,74,43,102)(16,75,44,103)(17,76,45,104)(18,77,46,105)(19,78,47,106)(20,79,48,107)(21,80,49,108)(22,81,50,109)(23,82,51,110)(24,83,52,111)(25,84,53,112)(26,85,54,57)(27,86,55,58)(28,87,56,59)(113,206,141,178)(114,207,142,179)(115,208,143,180)(116,209,144,181)(117,210,145,182)(118,211,146,183)(119,212,147,184)(120,213,148,185)(121,214,149,186)(122,215,150,187)(123,216,151,188)(124,217,152,189)(125,218,153,190)(126,219,154,191)(127,220,155,192)(128,221,156,193)(129,222,157,194)(130,223,158,195)(131,224,159,196)(132,169,160,197)(133,170,161,198)(134,171,162,199)(135,172,163,200)(136,173,164,201)(137,174,165,202)(138,175,166,203)(139,176,167,204)(140,177,168,205), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,168)(37,167)(38,166)(39,165)(40,164)(41,163)(42,162)(43,161)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,187)(58,186)(59,185)(60,184)(61,183)(62,182)(63,181)(64,180)(65,179)(66,178)(67,177)(68,176)(69,175)(70,174)(71,173)(72,172)(73,171)(74,170)(75,169)(76,224)(77,223)(78,222)(79,221)(80,220)(81,219)(82,218)(83,217)(84,216)(85,215)(86,214)(87,213)(88,212)(89,211)(90,210)(91,209)(92,208)(93,207)(94,206)(95,205)(96,204)(97,203)(98,202)(99,201)(100,200)(101,199)(102,198)(103,197)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188) );
G=PermutationGroup([[(1,60,29,88),(2,61,30,89),(3,62,31,90),(4,63,32,91),(5,64,33,92),(6,65,34,93),(7,66,35,94),(8,67,36,95),(9,68,37,96),(10,69,38,97),(11,70,39,98),(12,71,40,99),(13,72,41,100),(14,73,42,101),(15,74,43,102),(16,75,44,103),(17,76,45,104),(18,77,46,105),(19,78,47,106),(20,79,48,107),(21,80,49,108),(22,81,50,109),(23,82,51,110),(24,83,52,111),(25,84,53,112),(26,85,54,57),(27,86,55,58),(28,87,56,59),(113,206,141,178),(114,207,142,179),(115,208,143,180),(116,209,144,181),(117,210,145,182),(118,211,146,183),(119,212,147,184),(120,213,148,185),(121,214,149,186),(122,215,150,187),(123,216,151,188),(124,217,152,189),(125,218,153,190),(126,219,154,191),(127,220,155,192),(128,221,156,193),(129,222,157,194),(130,223,158,195),(131,224,159,196),(132,169,160,197),(133,170,161,198),(134,171,162,199),(135,172,163,200),(136,173,164,201),(137,174,165,202),(138,175,166,203),(139,176,167,204),(140,177,168,205)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(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)], [(1,147),(2,146),(3,145),(4,144),(5,143),(6,142),(7,141),(8,140),(9,139),(10,138),(11,137),(12,136),(13,135),(14,134),(15,133),(16,132),(17,131),(18,130),(19,129),(20,128),(21,127),(22,126),(23,125),(24,124),(25,123),(26,122),(27,121),(28,120),(29,119),(30,118),(31,117),(32,116),(33,115),(34,114),(35,113),(36,168),(37,167),(38,166),(39,165),(40,164),(41,163),(42,162),(43,161),(44,160),(45,159),(46,158),(47,157),(48,156),(49,155),(50,154),(51,153),(52,152),(53,151),(54,150),(55,149),(56,148),(57,187),(58,186),(59,185),(60,184),(61,183),(62,182),(63,181),(64,180),(65,179),(66,178),(67,177),(68,176),(69,175),(70,174),(71,173),(72,172),(73,171),(74,170),(75,169),(76,224),(77,223),(78,222),(79,221),(80,220),(81,219),(82,218),(83,217),(84,216),(85,215),(86,214),(87,213),(88,212),(89,211),(90,210),(91,209),(92,208),(93,207),(94,206),(95,205),(96,204),(97,203),(98,202),(99,201),(100,200),(101,199),(102,198),(103,197),(104,196),(105,195),(106,194),(107,193),(108,192),(109,191),(110,190),(111,189),(112,188)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | Q8○D8 | D4.13D28 |
| kernel | D4.13D28 | D56⋊7C2 | C2×Dic28 | C8.D14 | C7×C8○D4 | D4.10D14 | C7×D4 | C7×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 3 | 1 | 3 | 9 | 9 | 3 | 18 | 6 | 2 | 12 |
Matrix representation of D4.13D28 ►in GL4(𝔽113) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 96 | 35 | 0 | 0 |
| 78 | 72 | 0 | 0 |
| 0 | 0 | 96 | 35 |
| 0 | 0 | 78 | 72 |
| 0 | 0 | 3 | 55 |
| 0 | 0 | 82 | 110 |
| 110 | 58 | 0 | 0 |
| 31 | 3 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,0,112,0,0,0,0,112,1,0,0,0,0,1,0,0],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[96,78,0,0,35,72,0,0,0,0,96,78,0,0,35,72],[0,0,110,31,0,0,58,3,3,82,0,0,55,110,0,0] >;
D4.13D28 in GAP, Magma, Sage, TeX
D_4._{13}D_{28} % in TeX
G:=Group("D4.13D28"); // GroupNames label
G:=SmallGroup(448,1206);
// by ID
G=gap.SmallGroup(448,1206);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^28=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^27>;
// generators/relations