metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.16D4, C28.12D8, Q16.2Dic7, C56.26(C2×C4), (C2×C8).50D14, (C2×Q16).5D7, (C7×Q16).2C4, C8.2(C2×Dic7), C4.15(D4⋊D7), (C2×C28).117D4, C8.26(C7⋊D4), C7⋊3(C8.17D4), (C14×Q16).1C2, C28.C8.1C2, C56.C4.2C2, (C2×C56).30C22, (C2×C14).31SD16, C4.4(C23.D7), C28.16(C22⋊C4), C2.9(D4⋊Dic7), C22.7(D4.D7), C14.29(D4⋊C4), (C2×C4).25(C7⋊D4), SmallGroup(448,122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16.Dic7
G = < a,b,c,d | a8=1, b2=c14=a4, d2=a4c7, bab-1=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a3b, dcd-1=c13 >
Smallest permutation representation of Q16.Dic7
►On 224 points
Generators in S224
(1 46 8 53 15 32 22 39)(2 47 9 54 16 33 23 40)(3 48 10 55 17 34 24 41)(4 49 11 56 18 35 25 42)(5 50 12 29 19 36 26 43)(6 51 13 30 20 37 27 44)(7 52 14 31 21 38 28 45)(57 88 64 95 71 102 78 109)(58 89 65 96 72 103 79 110)(59 90 66 97 73 104 80 111)(60 91 67 98 74 105 81 112)(61 92 68 99 75 106 82 85)(62 93 69 100 76 107 83 86)(63 94 70 101 77 108 84 87)(113 148 134 141 127 162 120 155)(114 149 135 142 128 163 121 156)(115 150 136 143 129 164 122 157)(116 151 137 144 130 165 123 158)(117 152 138 145 131 166 124 159)(118 153 139 146 132 167 125 160)(119 154 140 147 133 168 126 161)(169 197 190 218 183 211 176 204)(170 198 191 219 184 212 177 205)(171 199 192 220 185 213 178 206)(172 200 193 221 186 214 179 207)(173 201 194 222 187 215 180 208)(174 202 195 223 188 216 181 209)(175 203 196 224 189 217 182 210)
(1 77 15 63)(2 64 16 78)(3 79 17 65)(4 66 18 80)(5 81 19 67)(6 68 20 82)(7 83 21 69)(8 70 22 84)(9 57 23 71)(10 72 24 58)(11 59 25 73)(12 74 26 60)(13 61 27 75)(14 76 28 62)(29 98 43 112)(30 85 44 99)(31 100 45 86)(32 87 46 101)(33 102 47 88)(34 89 48 103)(35 104 49 90)(36 91 50 105)(37 106 51 92)(38 93 52 107)(39 108 53 94)(40 95 54 109)(41 110 55 96)(42 97 56 111)(113 204 127 218)(114 219 128 205)(115 206 129 220)(116 221 130 207)(117 208 131 222)(118 223 132 209)(119 210 133 224)(120 197 134 211)(121 212 135 198)(122 199 136 213)(123 214 137 200)(124 201 138 215)(125 216 139 202)(126 203 140 217)(141 183 155 169)(142 170 156 184)(143 185 157 171)(144 172 158 186)(145 187 159 173)(146 174 160 188)(147 189 161 175)(148 176 162 190)(149 191 163 177)(150 178 164 192)(151 193 165 179)(152 180 166 194)(153 195 167 181)(154 182 168 196)
(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)
(1 127 22 120 15 113 8 134)(2 140 23 133 16 126 9 119)(3 125 24 118 17 139 10 132)(4 138 25 131 18 124 11 117)(5 123 26 116 19 137 12 130)(6 136 27 129 20 122 13 115)(7 121 28 114 21 135 14 128)(29 158 50 151 43 144 36 165)(30 143 51 164 44 157 37 150)(31 156 52 149 45 142 38 163)(32 141 53 162 46 155 39 148)(33 154 54 147 47 168 40 161)(34 167 55 160 48 153 41 146)(35 152 56 145 49 166 42 159)(57 175 78 196 71 189 64 182)(58 188 79 181 72 174 65 195)(59 173 80 194 73 187 66 180)(60 186 81 179 74 172 67 193)(61 171 82 192 75 185 68 178)(62 184 83 177 76 170 69 191)(63 169 84 190 77 183 70 176)(85 213 106 206 99 199 92 220)(86 198 107 219 100 212 93 205)(87 211 108 204 101 197 94 218)(88 224 109 217 102 210 95 203)(89 209 110 202 103 223 96 216)(90 222 111 215 104 208 97 201)(91 207 112 200 105 221 98 214)
G:=sub<Sym(224)| (1,46,8,53,15,32,22,39)(2,47,9,54,16,33,23,40)(3,48,10,55,17,34,24,41)(4,49,11,56,18,35,25,42)(5,50,12,29,19,36,26,43)(6,51,13,30,20,37,27,44)(7,52,14,31,21,38,28,45)(57,88,64,95,71,102,78,109)(58,89,65,96,72,103,79,110)(59,90,66,97,73,104,80,111)(60,91,67,98,74,105,81,112)(61,92,68,99,75,106,82,85)(62,93,69,100,76,107,83,86)(63,94,70,101,77,108,84,87)(113,148,134,141,127,162,120,155)(114,149,135,142,128,163,121,156)(115,150,136,143,129,164,122,157)(116,151,137,144,130,165,123,158)(117,152,138,145,131,166,124,159)(118,153,139,146,132,167,125,160)(119,154,140,147,133,168,126,161)(169,197,190,218,183,211,176,204)(170,198,191,219,184,212,177,205)(171,199,192,220,185,213,178,206)(172,200,193,221,186,214,179,207)(173,201,194,222,187,215,180,208)(174,202,195,223,188,216,181,209)(175,203,196,224,189,217,182,210), (1,77,15,63)(2,64,16,78)(3,79,17,65)(4,66,18,80)(5,81,19,67)(6,68,20,82)(7,83,21,69)(8,70,22,84)(9,57,23,71)(10,72,24,58)(11,59,25,73)(12,74,26,60)(13,61,27,75)(14,76,28,62)(29,98,43,112)(30,85,44,99)(31,100,45,86)(32,87,46,101)(33,102,47,88)(34,89,48,103)(35,104,49,90)(36,91,50,105)(37,106,51,92)(38,93,52,107)(39,108,53,94)(40,95,54,109)(41,110,55,96)(42,97,56,111)(113,204,127,218)(114,219,128,205)(115,206,129,220)(116,221,130,207)(117,208,131,222)(118,223,132,209)(119,210,133,224)(120,197,134,211)(121,212,135,198)(122,199,136,213)(123,214,137,200)(124,201,138,215)(125,216,139,202)(126,203,140,217)(141,183,155,169)(142,170,156,184)(143,185,157,171)(144,172,158,186)(145,187,159,173)(146,174,160,188)(147,189,161,175)(148,176,162,190)(149,191,163,177)(150,178,164,192)(151,193,165,179)(152,180,166,194)(153,195,167,181)(154,182,168,196), (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), (1,127,22,120,15,113,8,134)(2,140,23,133,16,126,9,119)(3,125,24,118,17,139,10,132)(4,138,25,131,18,124,11,117)(5,123,26,116,19,137,12,130)(6,136,27,129,20,122,13,115)(7,121,28,114,21,135,14,128)(29,158,50,151,43,144,36,165)(30,143,51,164,44,157,37,150)(31,156,52,149,45,142,38,163)(32,141,53,162,46,155,39,148)(33,154,54,147,47,168,40,161)(34,167,55,160,48,153,41,146)(35,152,56,145,49,166,42,159)(57,175,78,196,71,189,64,182)(58,188,79,181,72,174,65,195)(59,173,80,194,73,187,66,180)(60,186,81,179,74,172,67,193)(61,171,82,192,75,185,68,178)(62,184,83,177,76,170,69,191)(63,169,84,190,77,183,70,176)(85,213,106,206,99,199,92,220)(86,198,107,219,100,212,93,205)(87,211,108,204,101,197,94,218)(88,224,109,217,102,210,95,203)(89,209,110,202,103,223,96,216)(90,222,111,215,104,208,97,201)(91,207,112,200,105,221,98,214)>;
G:=Group( (1,46,8,53,15,32,22,39)(2,47,9,54,16,33,23,40)(3,48,10,55,17,34,24,41)(4,49,11,56,18,35,25,42)(5,50,12,29,19,36,26,43)(6,51,13,30,20,37,27,44)(7,52,14,31,21,38,28,45)(57,88,64,95,71,102,78,109)(58,89,65,96,72,103,79,110)(59,90,66,97,73,104,80,111)(60,91,67,98,74,105,81,112)(61,92,68,99,75,106,82,85)(62,93,69,100,76,107,83,86)(63,94,70,101,77,108,84,87)(113,148,134,141,127,162,120,155)(114,149,135,142,128,163,121,156)(115,150,136,143,129,164,122,157)(116,151,137,144,130,165,123,158)(117,152,138,145,131,166,124,159)(118,153,139,146,132,167,125,160)(119,154,140,147,133,168,126,161)(169,197,190,218,183,211,176,204)(170,198,191,219,184,212,177,205)(171,199,192,220,185,213,178,206)(172,200,193,221,186,214,179,207)(173,201,194,222,187,215,180,208)(174,202,195,223,188,216,181,209)(175,203,196,224,189,217,182,210), (1,77,15,63)(2,64,16,78)(3,79,17,65)(4,66,18,80)(5,81,19,67)(6,68,20,82)(7,83,21,69)(8,70,22,84)(9,57,23,71)(10,72,24,58)(11,59,25,73)(12,74,26,60)(13,61,27,75)(14,76,28,62)(29,98,43,112)(30,85,44,99)(31,100,45,86)(32,87,46,101)(33,102,47,88)(34,89,48,103)(35,104,49,90)(36,91,50,105)(37,106,51,92)(38,93,52,107)(39,108,53,94)(40,95,54,109)(41,110,55,96)(42,97,56,111)(113,204,127,218)(114,219,128,205)(115,206,129,220)(116,221,130,207)(117,208,131,222)(118,223,132,209)(119,210,133,224)(120,197,134,211)(121,212,135,198)(122,199,136,213)(123,214,137,200)(124,201,138,215)(125,216,139,202)(126,203,140,217)(141,183,155,169)(142,170,156,184)(143,185,157,171)(144,172,158,186)(145,187,159,173)(146,174,160,188)(147,189,161,175)(148,176,162,190)(149,191,163,177)(150,178,164,192)(151,193,165,179)(152,180,166,194)(153,195,167,181)(154,182,168,196), (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), (1,127,22,120,15,113,8,134)(2,140,23,133,16,126,9,119)(3,125,24,118,17,139,10,132)(4,138,25,131,18,124,11,117)(5,123,26,116,19,137,12,130)(6,136,27,129,20,122,13,115)(7,121,28,114,21,135,14,128)(29,158,50,151,43,144,36,165)(30,143,51,164,44,157,37,150)(31,156,52,149,45,142,38,163)(32,141,53,162,46,155,39,148)(33,154,54,147,47,168,40,161)(34,167,55,160,48,153,41,146)(35,152,56,145,49,166,42,159)(57,175,78,196,71,189,64,182)(58,188,79,181,72,174,65,195)(59,173,80,194,73,187,66,180)(60,186,81,179,74,172,67,193)(61,171,82,192,75,185,68,178)(62,184,83,177,76,170,69,191)(63,169,84,190,77,183,70,176)(85,213,106,206,99,199,92,220)(86,198,107,219,100,212,93,205)(87,211,108,204,101,197,94,218)(88,224,109,217,102,210,95,203)(89,209,110,202,103,223,96,216)(90,222,111,215,104,208,97,201)(91,207,112,200,105,221,98,214) );
G=PermutationGroup([[(1,46,8,53,15,32,22,39),(2,47,9,54,16,33,23,40),(3,48,10,55,17,34,24,41),(4,49,11,56,18,35,25,42),(5,50,12,29,19,36,26,43),(6,51,13,30,20,37,27,44),(7,52,14,31,21,38,28,45),(57,88,64,95,71,102,78,109),(58,89,65,96,72,103,79,110),(59,90,66,97,73,104,80,111),(60,91,67,98,74,105,81,112),(61,92,68,99,75,106,82,85),(62,93,69,100,76,107,83,86),(63,94,70,101,77,108,84,87),(113,148,134,141,127,162,120,155),(114,149,135,142,128,163,121,156),(115,150,136,143,129,164,122,157),(116,151,137,144,130,165,123,158),(117,152,138,145,131,166,124,159),(118,153,139,146,132,167,125,160),(119,154,140,147,133,168,126,161),(169,197,190,218,183,211,176,204),(170,198,191,219,184,212,177,205),(171,199,192,220,185,213,178,206),(172,200,193,221,186,214,179,207),(173,201,194,222,187,215,180,208),(174,202,195,223,188,216,181,209),(175,203,196,224,189,217,182,210)], [(1,77,15,63),(2,64,16,78),(3,79,17,65),(4,66,18,80),(5,81,19,67),(6,68,20,82),(7,83,21,69),(8,70,22,84),(9,57,23,71),(10,72,24,58),(11,59,25,73),(12,74,26,60),(13,61,27,75),(14,76,28,62),(29,98,43,112),(30,85,44,99),(31,100,45,86),(32,87,46,101),(33,102,47,88),(34,89,48,103),(35,104,49,90),(36,91,50,105),(37,106,51,92),(38,93,52,107),(39,108,53,94),(40,95,54,109),(41,110,55,96),(42,97,56,111),(113,204,127,218),(114,219,128,205),(115,206,129,220),(116,221,130,207),(117,208,131,222),(118,223,132,209),(119,210,133,224),(120,197,134,211),(121,212,135,198),(122,199,136,213),(123,214,137,200),(124,201,138,215),(125,216,139,202),(126,203,140,217),(141,183,155,169),(142,170,156,184),(143,185,157,171),(144,172,158,186),(145,187,159,173),(146,174,160,188),(147,189,161,175),(148,176,162,190),(149,191,163,177),(150,178,164,192),(151,193,165,179),(152,180,166,194),(153,195,167,181),(154,182,168,196)], [(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)], [(1,127,22,120,15,113,8,134),(2,140,23,133,16,126,9,119),(3,125,24,118,17,139,10,132),(4,138,25,131,18,124,11,117),(5,123,26,116,19,137,12,130),(6,136,27,129,20,122,13,115),(7,121,28,114,21,135,14,128),(29,158,50,151,43,144,36,165),(30,143,51,164,44,157,37,150),(31,156,52,149,45,142,38,163),(32,141,53,162,46,155,39,148),(33,154,54,147,47,168,40,161),(34,167,55,160,48,153,41,146),(35,152,56,145,49,166,42,159),(57,175,78,196,71,189,64,182),(58,188,79,181,72,174,65,195),(59,173,80,194,73,187,66,180),(60,186,81,179,74,172,67,193),(61,171,82,192,75,185,68,178),(62,184,83,177,76,170,69,191),(63,169,84,190,77,183,70,176),(85,213,106,206,99,199,92,220),(86,198,107,219,100,212,93,205),(87,211,108,204,101,197,94,218),(88,224,109,217,102,210,95,203),(89,209,110,202,103,223,96,216),(90,222,111,215,104,208,97,201),(91,207,112,200,105,221,98,214)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14I | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | Dic7 | C7⋊D4 | C7⋊D4 | C8.17D4 | D4⋊D7 | D4.D7 | Q16.Dic7 |
| kernel | Q16.Dic7 | C28.C8 | C56.C4 | C14×Q16 | C7×Q16 | C56 | C2×C28 | C2×Q16 | C28 | C2×C14 | C2×C8 | Q16 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of Q16.Dic7 ►in GL4(𝔽113) generated by
| 82 | 82 | 0 | 0 |
| 31 | 82 | 0 | 0 |
| 0 | 0 | 31 | 82 |
| 0 | 0 | 31 | 31 |
| 60 | 69 | 0 | 0 |
| 69 | 53 | 0 | 0 |
| 0 | 0 | 69 | 60 |
| 0 | 0 | 60 | 44 |
| 0 | 85 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(113))| [82,31,0,0,82,82,0,0,0,0,31,31,0,0,82,31],[60,69,0,0,69,53,0,0,0,0,69,60,0,0,60,44],[0,28,0,0,85,0,0,0,0,0,0,109,0,0,4,0],[0,0,0,1,0,0,112,0,1,0,0,0,0,1,0,0] >;
Q16.Dic7 in GAP, Magma, Sage, TeX
Q_{16}.{\rm Dic}_7 % in TeX
G:=Group("Q16.Dic7"); // GroupNames label
G:=SmallGroup(448,122);
// by ID
G=gap.SmallGroup(448,122);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,387,184,675,794,80,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=1,b^2=c^14=a^4,d^2=a^4*c^7,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^13>;
// generators/relations
Export