metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊C16, C8.30D28, C56.89D4, C14.3M5(2), C28.17M4(2), (C2×C16)⋊1D7, (C2×C112)⋊1C2, C2.5(D7×C16), C7⋊1(C22⋊C16), C14.5(C2×C16), C2.1(D14⋊C8), (C2×C8).335D14, C8.49(C7⋊D4), (C2×Dic7).4C8, (C22×D7).2C8, C22.11(C8×D7), C4.40(D14⋊C4), C4.15(C8⋊D7), C2.3(C16⋊D7), C14.7(C22⋊C8), C28.55(C22⋊C4), (C2×C56).401C22, (C2×C7⋊C16)⋊9C2, (D7×C2×C8).9C2, (C2×C7⋊C8).14C4, (C2×C4×D7).11C4, (C2×C14).12(C2×C8), (C2×C4).169(C4×D7), (C2×C28).243(C2×C4), SmallGroup(448,64)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊C16
G = < a,b,c | a14=b2=c16=1, bab=a-1, ac=ca, cbc-1=a7b >
Subgroups: 276 in 66 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C16, C2×C8, C2×C8, C22×C4, Dic7, C28, D14, D14, C2×C14, C2×C16, C2×C16, C22×C8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C22⋊C16, C7⋊C16, C112, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C2×C7⋊C16, C2×C112, D7×C2×C8, D14⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C16, C22⋊C4, C2×C8, M4(2), D14, C22⋊C8, C2×C16, M5(2), C4×D7, D28, C7⋊D4, C22⋊C16, C8×D7, C8⋊D7, D14⋊C4, D7×C16, C16⋊D7, D14⋊C8, D14⋊C16
►On 224 points
Generators in S224
(1 167 111 44 218 159 122 63 139 178 27 198 68 82)(2 168 112 45 219 160 123 64 140 179 28 199 69 83)(3 169 97 46 220 145 124 49 141 180 29 200 70 84)(4 170 98 47 221 146 125 50 142 181 30 201 71 85)(5 171 99 48 222 147 126 51 143 182 31 202 72 86)(6 172 100 33 223 148 127 52 144 183 32 203 73 87)(7 173 101 34 224 149 128 53 129 184 17 204 74 88)(8 174 102 35 209 150 113 54 130 185 18 205 75 89)(9 175 103 36 210 151 114 55 131 186 19 206 76 90)(10 176 104 37 211 152 115 56 132 187 20 207 77 91)(11 161 105 38 212 153 116 57 133 188 21 208 78 92)(12 162 106 39 213 154 117 58 134 189 22 193 79 93)(13 163 107 40 214 155 118 59 135 190 23 194 80 94)(14 164 108 41 215 156 119 60 136 191 24 195 65 95)(15 165 109 42 216 157 120 61 137 192 25 196 66 96)(16 166 110 43 217 158 121 62 138 177 26 197 67 81)
(1 90)(2 115)(3 92)(4 117)(5 94)(6 119)(7 96)(8 121)(9 82)(10 123)(11 84)(12 125)(13 86)(14 127)(15 88)(16 113)(17 42)(18 26)(19 44)(20 28)(21 46)(22 30)(23 48)(24 32)(25 34)(27 36)(29 38)(31 40)(33 41)(35 43)(37 45)(39 47)(49 116)(50 93)(51 118)(52 95)(53 120)(54 81)(55 122)(56 83)(57 124)(58 85)(59 126)(60 87)(61 128)(62 89)(63 114)(64 91)(65 144)(66 173)(67 130)(68 175)(69 132)(70 161)(71 134)(72 163)(73 136)(74 165)(75 138)(76 167)(77 140)(78 169)(79 142)(80 171)(97 208)(98 213)(99 194)(100 215)(101 196)(102 217)(103 198)(104 219)(105 200)(106 221)(107 202)(108 223)(109 204)(110 209)(111 206)(112 211)(129 157)(131 159)(133 145)(135 147)(137 149)(139 151)(141 153)(143 155)(146 162)(148 164)(150 166)(152 168)(154 170)(156 172)(158 174)(160 176)(177 205)(178 210)(179 207)(180 212)(181 193)(182 214)(183 195)(184 216)(185 197)(186 218)(187 199)(188 220)(189 201)(190 222)(191 203)(192 224)
(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)
G:=sub<Sym(224)| (1,167,111,44,218,159,122,63,139,178,27,198,68,82)(2,168,112,45,219,160,123,64,140,179,28,199,69,83)(3,169,97,46,220,145,124,49,141,180,29,200,70,84)(4,170,98,47,221,146,125,50,142,181,30,201,71,85)(5,171,99,48,222,147,126,51,143,182,31,202,72,86)(6,172,100,33,223,148,127,52,144,183,32,203,73,87)(7,173,101,34,224,149,128,53,129,184,17,204,74,88)(8,174,102,35,209,150,113,54,130,185,18,205,75,89)(9,175,103,36,210,151,114,55,131,186,19,206,76,90)(10,176,104,37,211,152,115,56,132,187,20,207,77,91)(11,161,105,38,212,153,116,57,133,188,21,208,78,92)(12,162,106,39,213,154,117,58,134,189,22,193,79,93)(13,163,107,40,214,155,118,59,135,190,23,194,80,94)(14,164,108,41,215,156,119,60,136,191,24,195,65,95)(15,165,109,42,216,157,120,61,137,192,25,196,66,96)(16,166,110,43,217,158,121,62,138,177,26,197,67,81), (1,90)(2,115)(3,92)(4,117)(5,94)(6,119)(7,96)(8,121)(9,82)(10,123)(11,84)(12,125)(13,86)(14,127)(15,88)(16,113)(17,42)(18,26)(19,44)(20,28)(21,46)(22,30)(23,48)(24,32)(25,34)(27,36)(29,38)(31,40)(33,41)(35,43)(37,45)(39,47)(49,116)(50,93)(51,118)(52,95)(53,120)(54,81)(55,122)(56,83)(57,124)(58,85)(59,126)(60,87)(61,128)(62,89)(63,114)(64,91)(65,144)(66,173)(67,130)(68,175)(69,132)(70,161)(71,134)(72,163)(73,136)(74,165)(75,138)(76,167)(77,140)(78,169)(79,142)(80,171)(97,208)(98,213)(99,194)(100,215)(101,196)(102,217)(103,198)(104,219)(105,200)(106,221)(107,202)(108,223)(109,204)(110,209)(111,206)(112,211)(129,157)(131,159)(133,145)(135,147)(137,149)(139,151)(141,153)(143,155)(146,162)(148,164)(150,166)(152,168)(154,170)(156,172)(158,174)(160,176)(177,205)(178,210)(179,207)(180,212)(181,193)(182,214)(183,195)(184,216)(185,197)(186,218)(187,199)(188,220)(189,201)(190,222)(191,203)(192,224), (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)>;
G:=Group( (1,167,111,44,218,159,122,63,139,178,27,198,68,82)(2,168,112,45,219,160,123,64,140,179,28,199,69,83)(3,169,97,46,220,145,124,49,141,180,29,200,70,84)(4,170,98,47,221,146,125,50,142,181,30,201,71,85)(5,171,99,48,222,147,126,51,143,182,31,202,72,86)(6,172,100,33,223,148,127,52,144,183,32,203,73,87)(7,173,101,34,224,149,128,53,129,184,17,204,74,88)(8,174,102,35,209,150,113,54,130,185,18,205,75,89)(9,175,103,36,210,151,114,55,131,186,19,206,76,90)(10,176,104,37,211,152,115,56,132,187,20,207,77,91)(11,161,105,38,212,153,116,57,133,188,21,208,78,92)(12,162,106,39,213,154,117,58,134,189,22,193,79,93)(13,163,107,40,214,155,118,59,135,190,23,194,80,94)(14,164,108,41,215,156,119,60,136,191,24,195,65,95)(15,165,109,42,216,157,120,61,137,192,25,196,66,96)(16,166,110,43,217,158,121,62,138,177,26,197,67,81), (1,90)(2,115)(3,92)(4,117)(5,94)(6,119)(7,96)(8,121)(9,82)(10,123)(11,84)(12,125)(13,86)(14,127)(15,88)(16,113)(17,42)(18,26)(19,44)(20,28)(21,46)(22,30)(23,48)(24,32)(25,34)(27,36)(29,38)(31,40)(33,41)(35,43)(37,45)(39,47)(49,116)(50,93)(51,118)(52,95)(53,120)(54,81)(55,122)(56,83)(57,124)(58,85)(59,126)(60,87)(61,128)(62,89)(63,114)(64,91)(65,144)(66,173)(67,130)(68,175)(69,132)(70,161)(71,134)(72,163)(73,136)(74,165)(75,138)(76,167)(77,140)(78,169)(79,142)(80,171)(97,208)(98,213)(99,194)(100,215)(101,196)(102,217)(103,198)(104,219)(105,200)(106,221)(107,202)(108,223)(109,204)(110,209)(111,206)(112,211)(129,157)(131,159)(133,145)(135,147)(137,149)(139,151)(141,153)(143,155)(146,162)(148,164)(150,166)(152,168)(154,170)(156,172)(158,174)(160,176)(177,205)(178,210)(179,207)(180,212)(181,193)(182,214)(183,195)(184,216)(185,197)(186,218)(187,199)(188,220)(189,201)(190,222)(191,203)(192,224), (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) );
G=PermutationGroup([[(1,167,111,44,218,159,122,63,139,178,27,198,68,82),(2,168,112,45,219,160,123,64,140,179,28,199,69,83),(3,169,97,46,220,145,124,49,141,180,29,200,70,84),(4,170,98,47,221,146,125,50,142,181,30,201,71,85),(5,171,99,48,222,147,126,51,143,182,31,202,72,86),(6,172,100,33,223,148,127,52,144,183,32,203,73,87),(7,173,101,34,224,149,128,53,129,184,17,204,74,88),(8,174,102,35,209,150,113,54,130,185,18,205,75,89),(9,175,103,36,210,151,114,55,131,186,19,206,76,90),(10,176,104,37,211,152,115,56,132,187,20,207,77,91),(11,161,105,38,212,153,116,57,133,188,21,208,78,92),(12,162,106,39,213,154,117,58,134,189,22,193,79,93),(13,163,107,40,214,155,118,59,135,190,23,194,80,94),(14,164,108,41,215,156,119,60,136,191,24,195,65,95),(15,165,109,42,216,157,120,61,137,192,25,196,66,96),(16,166,110,43,217,158,121,62,138,177,26,197,67,81)], [(1,90),(2,115),(3,92),(4,117),(5,94),(6,119),(7,96),(8,121),(9,82),(10,123),(11,84),(12,125),(13,86),(14,127),(15,88),(16,113),(17,42),(18,26),(19,44),(20,28),(21,46),(22,30),(23,48),(24,32),(25,34),(27,36),(29,38),(31,40),(33,41),(35,43),(37,45),(39,47),(49,116),(50,93),(51,118),(52,95),(53,120),(54,81),(55,122),(56,83),(57,124),(58,85),(59,126),(60,87),(61,128),(62,89),(63,114),(64,91),(65,144),(66,173),(67,130),(68,175),(69,132),(70,161),(71,134),(72,163),(73,136),(74,165),(75,138),(76,167),(77,140),(78,169),(79,142),(80,171),(97,208),(98,213),(99,194),(100,215),(101,196),(102,217),(103,198),(104,219),(105,200),(106,221),(107,202),(108,223),(109,204),(110,209),(111,206),(112,211),(129,157),(131,159),(133,145),(135,147),(137,149),(139,151),(141,153),(143,155),(146,162),(148,164),(150,166),(152,168),(154,170),(156,172),(158,174),(160,176),(177,205),(178,210),(179,207),(180,212),(181,193),(182,214),(183,195),(184,216),(185,197),(186,218),(187,199),(188,220),(189,201),(190,222),(191,203),(192,224)], [(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)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 16A | ··· | 16H | 16I | ··· | 16P | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 1 | 1 | 1 | 1 | 14 | 14 | 2 | 2 | 2 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | D7 | M4(2) | D14 | M5(2) | D28 | C7⋊D4 | C4×D7 | C8⋊D7 | C8×D7 | D7×C16 | C16⋊D7 |
| kernel | D14⋊C16 | C2×C7⋊C16 | C2×C112 | D7×C2×C8 | C2×C7⋊C8 | C2×C4×D7 | C2×Dic7 | C22×D7 | D14 | C56 | C2×C16 | C28 | C2×C8 | C14 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 3 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 24 | 24 |
Matrix representation of D14⋊C16 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 79 | 25 |
| 0 | 88 | 25 |
| 1 | 0 | 0 |
| 0 | 25 | 79 |
| 0 | 25 | 88 |
| 71 | 0 | 0 |
| 0 | 56 | 10 |
| 0 | 103 | 57 |
G:=sub<GL(3,GF(113))| [1,0,0,0,79,88,0,25,25],[1,0,0,0,25,25,0,79,88],[71,0,0,0,56,103,0,10,57] >;
D14⋊C16 in GAP, Magma, Sage, TeX
D_{14}\rtimes C_{16} % in TeX
G:=Group("D14:C16"); // GroupNames label
G:=SmallGroup(448,64);
// by ID
G=gap.SmallGroup(448,64);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,100,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^14=b^2=c^16=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations