Copied to
clipboard

## G = Q8×Dic7order 224 = 25·7

### Direct product of Q8 and Dic7

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×Dic7
G = < a,b,c,d | a4=c14=1, b2=a2, d2=c7, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 190 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×Q8, Dic7, Dic7, C28, C2×C14, C4×Q8, C2×Dic7, C2×Dic7, C2×C28, C7×Q8, C4×Dic7, C4⋊Dic7, Q8×C14, Q8×Dic7
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, C22×C4, C2×Q8, C4○D4, Dic7, D14, C4×Q8, C2×Dic7, C22×D7, Q8×D7, Q82D7, C22×Dic7, Q8×Dic7

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 7A 7B 7C 14A ··· 14I 28A ··· 28R order 1 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 7 7 7 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + - + + - - + image C1 C2 C2 C2 C4 Q8 D7 C4○D4 D14 Dic7 Q8×D7 Q8⋊2D7 kernel Q8×Dic7 C4×Dic7 C4⋊Dic7 Q8×C14 C7×Q8 Dic7 C2×Q8 C14 C2×C4 Q8 C2 C2 # reps 1 3 3 1 8 2 3 2 9 12 3 3

Matrix representation of Q8×Dic7 in GL4(𝔽29) generated by

 0 28 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 18 20 0 0 20 11 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 1 28 0 0 9 21
,
 1 0 0 0 0 1 0 0 0 0 22 3 0 0 22 7
G:=sub<GL(4,GF(29))| [0,1,0,0,28,0,0,0,0,0,1,0,0,0,0,1],[18,20,0,0,20,11,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,1,9,0,0,28,21],[1,0,0,0,0,1,0,0,0,0,22,22,0,0,3,7] >;

Q8×Dic7 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xDic7");
// GroupNames label

G:=SmallGroup(224,140);
// by ID

G=gap.SmallGroup(224,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,103,188,86,6917]);
// Polycyclic

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

׿
×
𝔽