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