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## G = D4.5D28order 448 = 26·7

### 5th non-split extension by D4 of D28 acting via D28/C28=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D4.5D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×Dic14 — C2×Dic28 — D4.5D28
 Lower central C7 — C14 — C2×C28 — D4.5D28
 Upper central C1 — C2 — C2×C4 — C8○D4

Generators and relations for D4.5D28
G = < a,b,c,d | a4=b2=1, c28=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c27 >

Subgroups: 444 in 100 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C14, C14, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C7⋊C8, C56, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, D4.5D4, Dic28, C4.Dic7, D4.D7, C7⋊Q16, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C2×Dic14, C7×C4○D4, C56.C4, C4.12D28, C2×Dic28, D4.9D14, C7×C8○D4, D4.5D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C7⋊D4, C22×D7, D4.5D4, C2×D28, C4○D28, C2×C7⋊D4, C287D4, D4.5D28

Smallest permutation representation of D4.5D28
On 224 points
Generators in S224
```(1 43 29 15)(2 44 30 16)(3 45 31 17)(4 46 32 18)(5 47 33 19)(6 48 34 20)(7 49 35 21)(8 50 36 22)(9 51 37 23)(10 52 38 24)(11 53 39 25)(12 54 40 26)(13 55 41 27)(14 56 42 28)(57 99 85 71)(58 100 86 72)(59 101 87 73)(60 102 88 74)(61 103 89 75)(62 104 90 76)(63 105 91 77)(64 106 92 78)(65 107 93 79)(66 108 94 80)(67 109 95 81)(68 110 96 82)(69 111 97 83)(70 112 98 84)(113 127 141 155)(114 128 142 156)(115 129 143 157)(116 130 144 158)(117 131 145 159)(118 132 146 160)(119 133 147 161)(120 134 148 162)(121 135 149 163)(122 136 150 164)(123 137 151 165)(124 138 152 166)(125 139 153 167)(126 140 154 168)(169 183 197 211)(170 184 198 212)(171 185 199 213)(172 186 200 214)(173 187 201 215)(174 188 202 216)(175 189 203 217)(176 190 204 218)(177 191 205 219)(178 192 206 220)(179 193 207 221)(180 194 208 222)(181 195 209 223)(182 196 210 224)
(1 197)(2 198)(3 199)(4 200)(5 201)(6 202)(7 203)(8 204)(9 205)(10 206)(11 207)(12 208)(13 209)(14 210)(15 211)(16 212)(17 213)(18 214)(19 215)(20 216)(21 217)(22 218)(23 219)(24 220)(25 221)(26 222)(27 223)(28 224)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 168)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)(101 156)(102 157)(103 158)(104 159)(105 160)(106 161)(107 162)(108 163)(109 164)(110 165)(111 166)(112 167)
(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 63 29 91)(2 62 30 90)(3 61 31 89)(4 60 32 88)(5 59 33 87)(6 58 34 86)(7 57 35 85)(8 112 36 84)(9 111 37 83)(10 110 38 82)(11 109 39 81)(12 108 40 80)(13 107 41 79)(14 106 42 78)(15 105 43 77)(16 104 44 76)(17 103 45 75)(18 102 46 74)(19 101 47 73)(20 100 48 72)(21 99 49 71)(22 98 50 70)(23 97 51 69)(24 96 52 68)(25 95 53 67)(26 94 54 66)(27 93 55 65)(28 92 56 64)(113 216 141 188)(114 215 142 187)(115 214 143 186)(116 213 144 185)(117 212 145 184)(118 211 146 183)(119 210 147 182)(120 209 148 181)(121 208 149 180)(122 207 150 179)(123 206 151 178)(124 205 152 177)(125 204 153 176)(126 203 154 175)(127 202 155 174)(128 201 156 173)(129 200 157 172)(130 199 158 171)(131 198 159 170)(132 197 160 169)(133 196 161 224)(134 195 162 223)(135 194 163 222)(136 193 164 221)(137 192 165 220)(138 191 166 219)(139 190 167 218)(140 189 168 217)```

`G:=sub<Sym(224)| (1,43,29,15)(2,44,30,16)(3,45,31,17)(4,46,32,18)(5,47,33,19)(6,48,34,20)(7,49,35,21)(8,50,36,22)(9,51,37,23)(10,52,38,24)(11,53,39,25)(12,54,40,26)(13,55,41,27)(14,56,42,28)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84)(113,127,141,155)(114,128,142,156)(115,129,143,157)(116,130,144,158)(117,131,145,159)(118,132,146,160)(119,133,147,161)(120,134,148,162)(121,135,149,163)(122,136,150,164)(123,137,151,165)(124,138,152,166)(125,139,153,167)(126,140,154,168)(169,183,197,211)(170,184,198,212)(171,185,199,213)(172,186,200,214)(173,187,201,215)(174,188,202,216)(175,189,203,217)(176,190,204,218)(177,191,205,219)(178,192,206,220)(179,193,207,221)(180,194,208,222)(181,195,209,223)(182,196,210,224), (1,197)(2,198)(3,199)(4,200)(5,201)(6,202)(7,203)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,218)(23,219)(24,220)(25,221)(26,222)(27,223)(28,224)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,168)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167), (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), (1,63,29,91)(2,62,30,90)(3,61,31,89)(4,60,32,88)(5,59,33,87)(6,58,34,86)(7,57,35,85)(8,112,36,84)(9,111,37,83)(10,110,38,82)(11,109,39,81)(12,108,40,80)(13,107,41,79)(14,106,42,78)(15,105,43,77)(16,104,44,76)(17,103,45,75)(18,102,46,74)(19,101,47,73)(20,100,48,72)(21,99,49,71)(22,98,50,70)(23,97,51,69)(24,96,52,68)(25,95,53,67)(26,94,54,66)(27,93,55,65)(28,92,56,64)(113,216,141,188)(114,215,142,187)(115,214,143,186)(116,213,144,185)(117,212,145,184)(118,211,146,183)(119,210,147,182)(120,209,148,181)(121,208,149,180)(122,207,150,179)(123,206,151,178)(124,205,152,177)(125,204,153,176)(126,203,154,175)(127,202,155,174)(128,201,156,173)(129,200,157,172)(130,199,158,171)(131,198,159,170)(132,197,160,169)(133,196,161,224)(134,195,162,223)(135,194,163,222)(136,193,164,221)(137,192,165,220)(138,191,166,219)(139,190,167,218)(140,189,168,217)>;`

`G:=Group( (1,43,29,15)(2,44,30,16)(3,45,31,17)(4,46,32,18)(5,47,33,19)(6,48,34,20)(7,49,35,21)(8,50,36,22)(9,51,37,23)(10,52,38,24)(11,53,39,25)(12,54,40,26)(13,55,41,27)(14,56,42,28)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84)(113,127,141,155)(114,128,142,156)(115,129,143,157)(116,130,144,158)(117,131,145,159)(118,132,146,160)(119,133,147,161)(120,134,148,162)(121,135,149,163)(122,136,150,164)(123,137,151,165)(124,138,152,166)(125,139,153,167)(126,140,154,168)(169,183,197,211)(170,184,198,212)(171,185,199,213)(172,186,200,214)(173,187,201,215)(174,188,202,216)(175,189,203,217)(176,190,204,218)(177,191,205,219)(178,192,206,220)(179,193,207,221)(180,194,208,222)(181,195,209,223)(182,196,210,224), (1,197)(2,198)(3,199)(4,200)(5,201)(6,202)(7,203)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,218)(23,219)(24,220)(25,221)(26,222)(27,223)(28,224)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,168)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167), (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), (1,63,29,91)(2,62,30,90)(3,61,31,89)(4,60,32,88)(5,59,33,87)(6,58,34,86)(7,57,35,85)(8,112,36,84)(9,111,37,83)(10,110,38,82)(11,109,39,81)(12,108,40,80)(13,107,41,79)(14,106,42,78)(15,105,43,77)(16,104,44,76)(17,103,45,75)(18,102,46,74)(19,101,47,73)(20,100,48,72)(21,99,49,71)(22,98,50,70)(23,97,51,69)(24,96,52,68)(25,95,53,67)(26,94,54,66)(27,93,55,65)(28,92,56,64)(113,216,141,188)(114,215,142,187)(115,214,143,186)(116,213,144,185)(117,212,145,184)(118,211,146,183)(119,210,147,182)(120,209,148,181)(121,208,149,180)(122,207,150,179)(123,206,151,178)(124,205,152,177)(125,204,153,176)(126,203,154,175)(127,202,155,174)(128,201,156,173)(129,200,157,172)(130,199,158,171)(131,198,159,170)(132,197,160,169)(133,196,161,224)(134,195,162,223)(135,194,163,222)(136,193,164,221)(137,192,165,220)(138,191,166,219)(139,190,167,218)(140,189,168,217) );`

`G=PermutationGroup([[(1,43,29,15),(2,44,30,16),(3,45,31,17),(4,46,32,18),(5,47,33,19),(6,48,34,20),(7,49,35,21),(8,50,36,22),(9,51,37,23),(10,52,38,24),(11,53,39,25),(12,54,40,26),(13,55,41,27),(14,56,42,28),(57,99,85,71),(58,100,86,72),(59,101,87,73),(60,102,88,74),(61,103,89,75),(62,104,90,76),(63,105,91,77),(64,106,92,78),(65,107,93,79),(66,108,94,80),(67,109,95,81),(68,110,96,82),(69,111,97,83),(70,112,98,84),(113,127,141,155),(114,128,142,156),(115,129,143,157),(116,130,144,158),(117,131,145,159),(118,132,146,160),(119,133,147,161),(120,134,148,162),(121,135,149,163),(122,136,150,164),(123,137,151,165),(124,138,152,166),(125,139,153,167),(126,140,154,168),(169,183,197,211),(170,184,198,212),(171,185,199,213),(172,186,200,214),(173,187,201,215),(174,188,202,216),(175,189,203,217),(176,190,204,218),(177,191,205,219),(178,192,206,220),(179,193,207,221),(180,194,208,222),(181,195,209,223),(182,196,210,224)], [(1,197),(2,198),(3,199),(4,200),(5,201),(6,202),(7,203),(8,204),(9,205),(10,206),(11,207),(12,208),(13,209),(14,210),(15,211),(16,212),(17,213),(18,214),(19,215),(20,216),(21,217),(22,218),(23,219),(24,220),(25,221),(26,222),(27,223),(28,224),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,193),(54,194),(55,195),(56,196),(57,168),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,119),(65,120),(66,121),(67,122),(68,123),(69,124),(70,125),(71,126),(72,127),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155),(101,156),(102,157),(103,158),(104,159),(105,160),(106,161),(107,162),(108,163),(109,164),(110,165),(111,166),(112,167)], [(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)], [(1,63,29,91),(2,62,30,90),(3,61,31,89),(4,60,32,88),(5,59,33,87),(6,58,34,86),(7,57,35,85),(8,112,36,84),(9,111,37,83),(10,110,38,82),(11,109,39,81),(12,108,40,80),(13,107,41,79),(14,106,42,78),(15,105,43,77),(16,104,44,76),(17,103,45,75),(18,102,46,74),(19,101,47,73),(20,100,48,72),(21,99,49,71),(22,98,50,70),(23,97,51,69),(24,96,52,68),(25,95,53,67),(26,94,54,66),(27,93,55,65),(28,92,56,64),(113,216,141,188),(114,215,142,187),(115,214,143,186),(116,213,144,185),(117,212,145,184),(118,211,146,183),(119,210,147,182),(120,209,148,181),(121,208,149,180),(122,207,150,179),(123,206,151,178),(124,205,152,177),(125,204,153,176),(126,203,154,175),(127,202,155,174),(128,201,156,173),(129,200,157,172),(130,199,158,171),(131,198,159,170),(132,197,160,169),(133,196,161,224),(134,195,162,223),(135,194,163,222),(136,193,164,221),(137,192,165,220),(138,191,166,219),(139,190,167,218),(140,189,168,217)]])`

76 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 14A 14B 14C 14D ··· 14L 28A ··· 28F 28G ··· 28O 56A ··· 56L 56M ··· 56AD order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 14 14 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 4 2 2 4 56 56 2 2 2 2 2 4 4 4 56 56 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

76 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 C4○D4 D14 D14 D14 C7⋊D4 D28 D28 C4○D28 D4.5D4 D4.5D28 kernel D4.5D28 C56.C4 C4.12D28 C2×Dic28 D4.9D14 C7×C8○D4 C56 C7×D4 C7×Q8 C8○D4 C2×C14 C2×C8 M4(2) C4○D4 C8 D4 Q8 C22 C7 C1 # reps 1 1 2 1 2 1 2 1 1 3 2 3 3 3 12 6 6 12 2 12

Matrix representation of D4.5D28 in GL4(𝔽113) generated by

 17 105 0 0 8 96 0 0 81 45 17 8 91 81 105 96
,
 54 94 107 85 29 109 59 107 77 107 99 48 76 40 20 77
,
 44 61 0 0 52 49 0 0 99 70 49 61 70 112 52 44
,
 40 86 0 0 30 73 0 0 63 92 72 28 53 0 61 41
`G:=sub<GL(4,GF(113))| [17,8,81,91,105,96,45,81,0,0,17,105,0,0,8,96],[54,29,77,76,94,109,107,40,107,59,99,20,85,107,48,77],[44,52,99,70,61,49,70,112,0,0,49,52,0,0,61,44],[40,30,63,53,86,73,92,0,0,0,72,61,0,0,28,41] >;`

D4.5D28 in GAP, Magma, Sage, TeX

`D_4._5D_{28}`
`% in TeX`

`G:=Group("D4.5D28");`
`// GroupNames label`

`G:=SmallGroup(448,677);`
`// by ID`

`G=gap.SmallGroup(448,677);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,344,254,1123,297,136,1684,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^28=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^27>;`
`// generators/relations`

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