direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C14×D16, C28.44D8, C56.72D4, C112⋊10C22, C56.72C23, C4.6(C7×D8), C8.9(C7×D4), C16⋊2(C2×C14), (C2×C16)⋊5C14, D8⋊1(C2×C14), (C2×D8)⋊6C14, C4.7(D4×C14), (C2×C112)⋊12C2, (C14×D8)⋊20C2, (C2×C14).55D8, C14.84(C2×D8), C2.12(C14×D8), (C2×C28).426D4, C28.314(C2×D4), (C7×D8)⋊17C22, C8.3(C22×C14), C22.14(C7×D8), (C2×C56).426C22, (C2×C4).82(C7×D4), (C2×C8).84(C2×C14), SmallGroup(448,913)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×D16
G = < a,b,c | a14=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 274 in 98 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, C23, C14, C14, C14, C16, C2×C8, D8, D8, C2×D4, C28, C2×C14, C2×C14, C2×C16, D16, C2×D8, C56, C2×C28, C7×D4, C22×C14, C2×D16, C112, C2×C56, C7×D8, C7×D8, D4×C14, C2×C112, C7×D16, C14×D8, C14×D16
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, D16, C2×D8, C7×D4, C22×C14, C2×D16, C7×D8, D4×C14, C7×D16, C14×D8, C14×D16
►On 224 points
Generators in S224
(1 165 192 95 42 144 73 223 27 52 149 116 205 97)(2 166 177 96 43 129 74 224 28 53 150 117 206 98)(3 167 178 81 44 130 75 209 29 54 151 118 207 99)(4 168 179 82 45 131 76 210 30 55 152 119 208 100)(5 169 180 83 46 132 77 211 31 56 153 120 193 101)(6 170 181 84 47 133 78 212 32 57 154 121 194 102)(7 171 182 85 48 134 79 213 17 58 155 122 195 103)(8 172 183 86 33 135 80 214 18 59 156 123 196 104)(9 173 184 87 34 136 65 215 19 60 157 124 197 105)(10 174 185 88 35 137 66 216 20 61 158 125 198 106)(11 175 186 89 36 138 67 217 21 62 159 126 199 107)(12 176 187 90 37 139 68 218 22 63 160 127 200 108)(13 161 188 91 38 140 69 219 23 64 145 128 201 109)(14 162 189 92 39 141 70 220 24 49 146 113 202 110)(15 163 190 93 40 142 71 221 25 50 147 114 203 111)(16 164 191 94 41 143 72 222 26 51 148 115 204 112)
(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)
(1 215)(2 214)(3 213)(4 212)(5 211)(6 210)(7 209)(8 224)(9 223)(10 222)(11 221)(12 220)(13 219)(14 218)(15 217)(16 216)(17 167)(18 166)(19 165)(20 164)(21 163)(22 162)(23 161)(24 176)(25 175)(26 174)(27 173)(28 172)(29 171)(30 170)(31 169)(32 168)(33 117)(34 116)(35 115)(36 114)(37 113)(38 128)(39 127)(40 126)(41 125)(42 124)(43 123)(44 122)(45 121)(46 120)(47 119)(48 118)(49 187)(50 186)(51 185)(52 184)(53 183)(54 182)(55 181)(56 180)(57 179)(58 178)(59 177)(60 192)(61 191)(62 190)(63 189)(64 188)(65 97)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 155)(82 154)(83 153)(84 152)(85 151)(86 150)(87 149)(88 148)(89 147)(90 146)(91 145)(92 160)(93 159)(94 158)(95 157)(96 156)(129 196)(130 195)(131 194)(132 193)(133 208)(134 207)(135 206)(136 205)(137 204)(138 203)(139 202)(140 201)(141 200)(142 199)(143 198)(144 197)
G:=sub<Sym(224)| (1,165,192,95,42,144,73,223,27,52,149,116,205,97)(2,166,177,96,43,129,74,224,28,53,150,117,206,98)(3,167,178,81,44,130,75,209,29,54,151,118,207,99)(4,168,179,82,45,131,76,210,30,55,152,119,208,100)(5,169,180,83,46,132,77,211,31,56,153,120,193,101)(6,170,181,84,47,133,78,212,32,57,154,121,194,102)(7,171,182,85,48,134,79,213,17,58,155,122,195,103)(8,172,183,86,33,135,80,214,18,59,156,123,196,104)(9,173,184,87,34,136,65,215,19,60,157,124,197,105)(10,174,185,88,35,137,66,216,20,61,158,125,198,106)(11,175,186,89,36,138,67,217,21,62,159,126,199,107)(12,176,187,90,37,139,68,218,22,63,160,127,200,108)(13,161,188,91,38,140,69,219,23,64,145,128,201,109)(14,162,189,92,39,141,70,220,24,49,146,113,202,110)(15,163,190,93,40,142,71,221,25,50,147,114,203,111)(16,164,191,94,41,143,72,222,26,51,148,115,204,112), (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), (1,215)(2,214)(3,213)(4,212)(5,211)(6,210)(7,209)(8,224)(9,223)(10,222)(11,221)(12,220)(13,219)(14,218)(15,217)(16,216)(17,167)(18,166)(19,165)(20,164)(21,163)(22,162)(23,161)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,117)(34,116)(35,115)(36,114)(37,113)(38,128)(39,127)(40,126)(41,125)(42,124)(43,123)(44,122)(45,121)(46,120)(47,119)(48,118)(49,187)(50,186)(51,185)(52,184)(53,183)(54,182)(55,181)(56,180)(57,179)(58,178)(59,177)(60,192)(61,191)(62,190)(63,189)(64,188)(65,97)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,160)(93,159)(94,158)(95,157)(96,156)(129,196)(130,195)(131,194)(132,193)(133,208)(134,207)(135,206)(136,205)(137,204)(138,203)(139,202)(140,201)(141,200)(142,199)(143,198)(144,197)>;
G:=Group( (1,165,192,95,42,144,73,223,27,52,149,116,205,97)(2,166,177,96,43,129,74,224,28,53,150,117,206,98)(3,167,178,81,44,130,75,209,29,54,151,118,207,99)(4,168,179,82,45,131,76,210,30,55,152,119,208,100)(5,169,180,83,46,132,77,211,31,56,153,120,193,101)(6,170,181,84,47,133,78,212,32,57,154,121,194,102)(7,171,182,85,48,134,79,213,17,58,155,122,195,103)(8,172,183,86,33,135,80,214,18,59,156,123,196,104)(9,173,184,87,34,136,65,215,19,60,157,124,197,105)(10,174,185,88,35,137,66,216,20,61,158,125,198,106)(11,175,186,89,36,138,67,217,21,62,159,126,199,107)(12,176,187,90,37,139,68,218,22,63,160,127,200,108)(13,161,188,91,38,140,69,219,23,64,145,128,201,109)(14,162,189,92,39,141,70,220,24,49,146,113,202,110)(15,163,190,93,40,142,71,221,25,50,147,114,203,111)(16,164,191,94,41,143,72,222,26,51,148,115,204,112), (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), (1,215)(2,214)(3,213)(4,212)(5,211)(6,210)(7,209)(8,224)(9,223)(10,222)(11,221)(12,220)(13,219)(14,218)(15,217)(16,216)(17,167)(18,166)(19,165)(20,164)(21,163)(22,162)(23,161)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,117)(34,116)(35,115)(36,114)(37,113)(38,128)(39,127)(40,126)(41,125)(42,124)(43,123)(44,122)(45,121)(46,120)(47,119)(48,118)(49,187)(50,186)(51,185)(52,184)(53,183)(54,182)(55,181)(56,180)(57,179)(58,178)(59,177)(60,192)(61,191)(62,190)(63,189)(64,188)(65,97)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,160)(93,159)(94,158)(95,157)(96,156)(129,196)(130,195)(131,194)(132,193)(133,208)(134,207)(135,206)(136,205)(137,204)(138,203)(139,202)(140,201)(141,200)(142,199)(143,198)(144,197) );
G=PermutationGroup([[(1,165,192,95,42,144,73,223,27,52,149,116,205,97),(2,166,177,96,43,129,74,224,28,53,150,117,206,98),(3,167,178,81,44,130,75,209,29,54,151,118,207,99),(4,168,179,82,45,131,76,210,30,55,152,119,208,100),(5,169,180,83,46,132,77,211,31,56,153,120,193,101),(6,170,181,84,47,133,78,212,32,57,154,121,194,102),(7,171,182,85,48,134,79,213,17,58,155,122,195,103),(8,172,183,86,33,135,80,214,18,59,156,123,196,104),(9,173,184,87,34,136,65,215,19,60,157,124,197,105),(10,174,185,88,35,137,66,216,20,61,158,125,198,106),(11,175,186,89,36,138,67,217,21,62,159,126,199,107),(12,176,187,90,37,139,68,218,22,63,160,127,200,108),(13,161,188,91,38,140,69,219,23,64,145,128,201,109),(14,162,189,92,39,141,70,220,24,49,146,113,202,110),(15,163,190,93,40,142,71,221,25,50,147,114,203,111),(16,164,191,94,41,143,72,222,26,51,148,115,204,112)], [(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)], [(1,215),(2,214),(3,213),(4,212),(5,211),(6,210),(7,209),(8,224),(9,223),(10,222),(11,221),(12,220),(13,219),(14,218),(15,217),(16,216),(17,167),(18,166),(19,165),(20,164),(21,163),(22,162),(23,161),(24,176),(25,175),(26,174),(27,173),(28,172),(29,171),(30,170),(31,169),(32,168),(33,117),(34,116),(35,115),(36,114),(37,113),(38,128),(39,127),(40,126),(41,125),(42,124),(43,123),(44,122),(45,121),(46,120),(47,119),(48,118),(49,187),(50,186),(51,185),(52,184),(53,183),(54,182),(55,181),(56,180),(57,179),(58,178),(59,177),(60,192),(61,191),(62,190),(63,189),(64,188),(65,97),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,155),(82,154),(83,153),(84,152),(85,151),(86,150),(87,149),(88,148),(89,147),(90,146),(91,145),(92,160),(93,159),(94,158),(95,157),(96,156),(129,196),(130,195),(131,194),(132,193),(133,208),(134,207),(135,206),(136,205),(137,204),(138,203),(139,202),(140,201),(141,200),(142,199),(143,198),(144,197)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AP | 16A | ··· | 16H | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | D8 | D8 | D16 | C7×D4 | C7×D4 | C7×D8 | C7×D8 | C7×D16 |
| kernel | C14×D16 | C2×C112 | C7×D16 | C14×D8 | C2×D16 | C2×C16 | D16 | C2×D8 | C56 | C2×C28 | C28 | C2×C14 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 1 | 1 | 2 | 2 | 8 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C14×D16 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 106 | 0 |
| 0 | 0 | 106 |
| 1 | 0 | 0 |
| 0 | 4 | 18 |
| 0 | 95 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 112 |
| 0 | 112 | 0 |
G:=sub<GL(3,GF(113))| [112,0,0,0,106,0,0,0,106],[1,0,0,0,4,95,0,18,4],[1,0,0,0,0,112,0,112,0] >;
C14×D16 in GAP, Magma, Sage, TeX
C_{14}\times D_{16} % in TeX
G:=Group("C14xD16"); // GroupNames label
G:=SmallGroup(448,913);
// by ID
G=gap.SmallGroup(448,913);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,5884,2951,242,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c|a^14=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations