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## G = D4.1D28order 448 = 26·7

### 1st non-split extension by D4 of D28 acting via D28/C28=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D4.1D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C4.D28 — D4.1D28
 Lower central C7 — C14 — C2×C28 — D4.1D28
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.1D28
G = < a,b,c,d | a4=b2=c28=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 628 in 124 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C7⋊C8, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, D4.2D4, C2×C7⋊C8, D14⋊C4, D4⋊D7, D4.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×D28, C22×C28, D4×C14, C28⋊C8, C14.D8, C14.Q16, C4.D28, C2×D4⋊D7, C2×D4.D7, D4×C28, D4.1D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8⋊C22, D28, C7⋊D4, C22×D7, D4.2D4, C2×D28, C4○D28, C2×C7⋊D4, C287D4, D4.D14, D4.8D14, D4.1D28

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

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

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

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

79 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14U 28A ··· 28L 28M ··· 28AJ order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 56 2 2 2 2 4 4 4 56 2 2 2 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 D14 C4○D8 C7⋊D4 D28 C4○D28 C8⋊C22 D4.D14 D4.8D14 kernel D4.1D28 C28⋊C8 C14.D8 C14.Q16 C4.D28 C2×D4⋊D7 C2×D4.D7 D4×C28 C2×C28 C7×D4 C4×D4 C28 C42 C4⋊C4 C2×D4 C14 C2×C4 D4 C4 C14 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 3 2 3 3 3 4 12 12 12 1 6 6

Matrix representation of D4.1D28 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 1 111 0 0 1 112
,
 1 0 0 0 0 1 0 0 0 0 1 111 0 0 0 112
,
 12 36 0 0 89 107 0 0 0 0 15 0 0 0 0 15
,
 103 17 0 0 54 10 0 0 0 0 0 26 0 0 13 0
`G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,111,112],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,111,112],[12,89,0,0,36,107,0,0,0,0,15,0,0,0,0,15],[103,54,0,0,17,10,0,0,0,0,0,13,0,0,26,0] >;`

D4.1D28 in GAP, Magma, Sage, TeX

`D_4._1D_{28}`
`% in TeX`

`G:=Group("D4.1D28");`
`// GroupNames label`

`G:=SmallGroup(448,550);`
`// by ID`

`G=gap.SmallGroup(448,550);`
`# by ID`

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

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

׿
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