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## G = Q8.1D28order 448 = 26·7

### 1st non-split extension by Q8 of D28 acting via D28/C28=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — Q8.1D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C4.D28 — Q8.1D28
 Lower central C7 — C14 — C2×C28 — Q8.1D28
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for Q8.1D28
G = < a,b,c,d | a4=c28=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 580 in 112 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, D14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C7⋊C8, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, Q8.D4, C2×C7⋊C8, D14⋊C4, Q8⋊D7, C7⋊Q16, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×D28, Q8×C14, C28⋊C8, C14.D8, C14.Q16, C4.D28, C2×Q8⋊D7, C2×C7⋊Q16, Q8×C28, Q8.1D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8.C22, D28, C7⋊D4, C22×D7, Q8.D4, C2×D28, C4○D28, C2×C7⋊D4, C287D4, C28.C23, D4.8D14, Q8.1D28

Smallest permutation representation of Q8.1D28
On 224 points
Generators in S224
```(1 92 150 51)(2 93 151 52)(3 94 152 53)(4 95 153 54)(5 96 154 55)(6 97 155 56)(7 98 156 29)(8 99 157 30)(9 100 158 31)(10 101 159 32)(11 102 160 33)(12 103 161 34)(13 104 162 35)(14 105 163 36)(15 106 164 37)(16 107 165 38)(17 108 166 39)(18 109 167 40)(19 110 168 41)(20 111 141 42)(21 112 142 43)(22 85 143 44)(23 86 144 45)(24 87 145 46)(25 88 146 47)(26 89 147 48)(27 90 148 49)(28 91 149 50)(57 180 208 135)(58 181 209 136)(59 182 210 137)(60 183 211 138)(61 184 212 139)(62 185 213 140)(63 186 214 113)(64 187 215 114)(65 188 216 115)(66 189 217 116)(67 190 218 117)(68 191 219 118)(69 192 220 119)(70 193 221 120)(71 194 222 121)(72 195 223 122)(73 196 224 123)(74 169 197 124)(75 170 198 125)(76 171 199 126)(77 172 200 127)(78 173 201 128)(79 174 202 129)(80 175 203 130)(81 176 204 131)(82 177 205 132)(83 178 206 133)(84 179 207 134)
(1 205 150 82)(2 206 151 83)(3 207 152 84)(4 208 153 57)(5 209 154 58)(6 210 155 59)(7 211 156 60)(8 212 157 61)(9 213 158 62)(10 214 159 63)(11 215 160 64)(12 216 161 65)(13 217 162 66)(14 218 163 67)(15 219 164 68)(16 220 165 69)(17 221 166 70)(18 222 167 71)(19 223 168 72)(20 224 141 73)(21 197 142 74)(22 198 143 75)(23 199 144 76)(24 200 145 77)(25 201 146 78)(26 202 147 79)(27 203 148 80)(28 204 149 81)(29 138 98 183)(30 139 99 184)(31 140 100 185)(32 113 101 186)(33 114 102 187)(34 115 103 188)(35 116 104 189)(36 117 105 190)(37 118 106 191)(38 119 107 192)(39 120 108 193)(40 121 109 194)(41 122 110 195)(42 123 111 196)(43 124 112 169)(44 125 85 170)(45 126 86 171)(46 127 87 172)(47 128 88 173)(48 129 89 174)(49 130 90 175)(50 131 91 176)(51 132 92 177)(52 133 93 178)(53 134 94 179)(54 135 95 180)(55 136 96 181)(56 137 97 182)
(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)
(1 149 150 28)(2 27 151 148)(3 147 152 26)(4 25 153 146)(5 145 154 24)(6 23 155 144)(7 143 156 22)(8 21 157 142)(9 141 158 20)(10 19 159 168)(11 167 160 18)(12 17 161 166)(13 165 162 16)(14 15 163 164)(29 44 98 85)(30 112 99 43)(31 42 100 111)(32 110 101 41)(33 40 102 109)(34 108 103 39)(35 38 104 107)(36 106 105 37)(45 56 86 97)(46 96 87 55)(47 54 88 95)(48 94 89 53)(49 52 90 93)(50 92 91 51)(57 173 208 128)(58 127 209 172)(59 171 210 126)(60 125 211 170)(61 169 212 124)(62 123 213 196)(63 195 214 122)(64 121 215 194)(65 193 216 120)(66 119 217 192)(67 191 218 118)(68 117 219 190)(69 189 220 116)(70 115 221 188)(71 187 222 114)(72 113 223 186)(73 185 224 140)(74 139 197 184)(75 183 198 138)(76 137 199 182)(77 181 200 136)(78 135 201 180)(79 179 202 134)(80 133 203 178)(81 177 204 132)(82 131 205 176)(83 175 206 130)(84 129 207 174)```

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

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

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

79 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E ··· 4I 4J 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 28A ··· 28L 28M ··· 28AV order 1 2 2 2 2 4 4 4 4 4 ··· 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 56 2 2 2 2 4 ··· 4 56 2 2 2 28 28 28 28 2 ··· 2 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 C28.C23 D4.8D14 kernel Q8.1D28 C28⋊C8 C14.D8 C14.Q16 C4.D28 C2×Q8⋊D7 C2×C7⋊Q16 Q8×C28 C2×C28 C7×Q8 C4×Q8 C28 C42 C4⋊C4 C2×Q8 C14 C2×C4 Q8 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 Q8.1D28 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 0 1 0 0 112 0
,
 79 106 0 0 52 34 0 0 0 0 13 13 0 0 13 100
,
 32 77 0 0 106 75 0 0 0 0 98 0 0 0 0 98
,
 75 36 0 0 7 38 0 0 0 0 98 0 0 0 0 15
`G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,0,112,0,0,1,0],[79,52,0,0,106,34,0,0,0,0,13,13,0,0,13,100],[32,106,0,0,77,75,0,0,0,0,98,0,0,0,0,98],[75,7,0,0,36,38,0,0,0,0,98,0,0,0,0,15] >;`

Q8.1D28 in GAP, Magma, Sage, TeX

`Q_8._1D_{28}`
`% in TeX`

`G:=Group("Q8.1D28");`
`// GroupNames label`

`G:=SmallGroup(448,562);`
`// by ID`

`G=gap.SmallGroup(448,562);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,184,1123,297,136,18822]);`
`// Polycyclic`

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

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