Copied to
clipboard

## G = Q8⋊Dic7order 224 = 25·7

### 1st semidirect product of Q8 and Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Q8⋊Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4⋊Dic7 — Q8⋊Dic7
 Lower central C7 — C14 — C28 — Q8⋊Dic7
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for Q8⋊Dic7
G = < a,b,c,d | a4=c14=1, b2=a2, d2=c7, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Smallest permutation representation of Q8⋊Dic7
Regular action on 224 points
Generators in S224
(1 67 88 42)(2 68 89 29)(3 69 90 30)(4 70 91 31)(5 57 92 32)(6 58 93 33)(7 59 94 34)(8 60 95 35)(9 61 96 36)(10 62 97 37)(11 63 98 38)(12 64 85 39)(13 65 86 40)(14 66 87 41)(15 135 103 77)(16 136 104 78)(17 137 105 79)(18 138 106 80)(19 139 107 81)(20 140 108 82)(21 127 109 83)(22 128 110 84)(23 129 111 71)(24 130 112 72)(25 131 99 73)(26 132 100 74)(27 133 101 75)(28 134 102 76)(43 214 206 145)(44 215 207 146)(45 216 208 147)(46 217 209 148)(47 218 210 149)(48 219 197 150)(49 220 198 151)(50 221 199 152)(51 222 200 153)(52 223 201 154)(53 224 202 141)(54 211 203 142)(55 212 204 143)(56 213 205 144)(113 155 169 192)(114 156 170 193)(115 157 171 194)(116 158 172 195)(117 159 173 196)(118 160 174 183)(119 161 175 184)(120 162 176 185)(121 163 177 186)(122 164 178 187)(123 165 179 188)(124 166 180 189)(125 167 181 190)(126 168 182 191)
(1 109 88 21)(2 110 89 22)(3 111 90 23)(4 112 91 24)(5 99 92 25)(6 100 93 26)(7 101 94 27)(8 102 95 28)(9 103 96 15)(10 104 97 16)(11 105 98 17)(12 106 85 18)(13 107 86 19)(14 108 87 20)(29 84 68 128)(30 71 69 129)(31 72 70 130)(32 73 57 131)(33 74 58 132)(34 75 59 133)(35 76 60 134)(36 77 61 135)(37 78 62 136)(38 79 63 137)(39 80 64 138)(40 81 65 139)(41 82 66 140)(42 83 67 127)(43 181 206 125)(44 182 207 126)(45 169 208 113)(46 170 209 114)(47 171 210 115)(48 172 197 116)(49 173 198 117)(50 174 199 118)(51 175 200 119)(52 176 201 120)(53 177 202 121)(54 178 203 122)(55 179 204 123)(56 180 205 124)(141 186 224 163)(142 187 211 164)(143 188 212 165)(144 189 213 166)(145 190 214 167)(146 191 215 168)(147 192 216 155)(148 193 217 156)(149 194 218 157)(150 195 219 158)(151 196 220 159)(152 183 221 160)(153 184 222 161)(154 185 223 162)
(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)
(1 224 8 217)(2 223 9 216)(3 222 10 215)(4 221 11 214)(5 220 12 213)(6 219 13 212)(7 218 14 211)(15 113 22 120)(16 126 23 119)(17 125 24 118)(18 124 25 117)(19 123 26 116)(20 122 27 115)(21 121 28 114)(29 201 36 208)(30 200 37 207)(31 199 38 206)(32 198 39 205)(33 197 40 204)(34 210 41 203)(35 209 42 202)(43 70 50 63)(44 69 51 62)(45 68 52 61)(46 67 53 60)(47 66 54 59)(48 65 55 58)(49 64 56 57)(71 161 78 168)(72 160 79 167)(73 159 80 166)(74 158 81 165)(75 157 82 164)(76 156 83 163)(77 155 84 162)(85 144 92 151)(86 143 93 150)(87 142 94 149)(88 141 95 148)(89 154 96 147)(90 153 97 146)(91 152 98 145)(99 173 106 180)(100 172 107 179)(101 171 108 178)(102 170 109 177)(103 169 110 176)(104 182 111 175)(105 181 112 174)(127 186 134 193)(128 185 135 192)(129 184 136 191)(130 183 137 190)(131 196 138 189)(132 195 139 188)(133 194 140 187)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 28A ··· 28R order 1 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 4 4 28 28 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - + - image C1 C2 C2 C2 C4 D4 D4 D7 SD16 Q16 D14 Dic7 C7⋊D4 C7⋊D4 Q8⋊D7 C7⋊Q16 kernel Q8⋊Dic7 C2×C7⋊C8 C4⋊Dic7 Q8×C14 C7×Q8 C28 C2×C14 C2×Q8 C14 C14 C2×C4 Q8 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 3 2 2 3 6 6 6 3 3

Matrix representation of Q8⋊Dic7 in GL6(𝔽113)

 1 111 0 0 0 0 1 112 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 92 18 0 0 0 0 101 21 0 0 0 0 0 0 95 50 0 0 0 0 50 18 0 0 0 0 0 0 29 5 0 0 0 0 58 84
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 1 0 0 0 0 102 10
,
 7 40 0 0 0 0 27 106 0 0 0 0 0 0 77 93 0 0 0 0 93 36 0 0 0 0 0 0 50 24 0 0 0 0 23 63

G:=sub<GL(6,GF(113))| [1,1,0,0,0,0,111,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[92,101,0,0,0,0,18,21,0,0,0,0,0,0,95,50,0,0,0,0,50,18,0,0,0,0,0,0,29,58,0,0,0,0,5,84],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,102,0,0,0,0,1,10],[7,27,0,0,0,0,40,106,0,0,0,0,0,0,77,93,0,0,0,0,93,36,0,0,0,0,0,0,50,23,0,0,0,0,24,63] >;

Q8⋊Dic7 in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm Dic}_7
% in TeX

G:=Group("Q8:Dic7");
// GroupNames label

G:=SmallGroup(224,41);
// by ID

G=gap.SmallGroup(224,41);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,103,579,297,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^14=1,b^2=a^2,d^2=c^7,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=c^-1>;
// generators/relations

Export

׿
×
𝔽