metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.2Q8, C14.4SD16, C4.2Dic14, C7⋊C8⋊2C4, C4⋊C4.2D7, C7⋊1(C4.Q8), C28.2(C2×C4), C4.12(C4×D7), C14.3(C4⋊C4), (C2×C14).29D4, (C2×C4).34D14, C2.1(Q8⋊D7), C4⋊Dic7.9C2, (C2×C28).9C22, C2.1(D4.D7), C2.4(Dic7⋊C4), C22.13(C7⋊D4), (C2×C7⋊C8).2C2, (C7×C4⋊C4).2C2, SmallGroup(224,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.SD16
G = < a,b,c | a14=b8=1, c2=a7, bab-1=a-1, ac=ca, cbc-1=b3 >
Smallest permutation representation of C14.SD16
►Regular action on 224 points
Generators in S224
(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 15 118 70 213 135 75 51)(2 28 119 69 214 134 76 50)(3 27 120 68 215 133 77 49)(4 26 121 67 216 132 78 48)(5 25 122 66 217 131 79 47)(6 24 123 65 218 130 80 46)(7 23 124 64 219 129 81 45)(8 22 125 63 220 128 82 44)(9 21 126 62 221 127 83 43)(10 20 113 61 222 140 84 56)(11 19 114 60 223 139 71 55)(12 18 115 59 224 138 72 54)(13 17 116 58 211 137 73 53)(14 16 117 57 212 136 74 52)(29 109 191 197 169 90 143 165)(30 108 192 210 170 89 144 164)(31 107 193 209 171 88 145 163)(32 106 194 208 172 87 146 162)(33 105 195 207 173 86 147 161)(34 104 196 206 174 85 148 160)(35 103 183 205 175 98 149 159)(36 102 184 204 176 97 150 158)(37 101 185 203 177 96 151 157)(38 100 186 202 178 95 152 156)(39 99 187 201 179 94 153 155)(40 112 188 200 180 93 154 168)(41 111 189 199 181 92 141 167)(42 110 190 198 182 91 142 166)
(1 101 8 108)(2 102 9 109)(3 103 10 110)(4 104 11 111)(5 105 12 112)(6 106 13 99)(7 107 14 100)(15 177 22 170)(16 178 23 171)(17 179 24 172)(18 180 25 173)(19 181 26 174)(20 182 27 175)(21 169 28 176)(29 134 36 127)(30 135 37 128)(31 136 38 129)(32 137 39 130)(33 138 40 131)(34 139 41 132)(35 140 42 133)(43 143 50 150)(44 144 51 151)(45 145 52 152)(46 146 53 153)(47 147 54 154)(48 148 55 141)(49 149 56 142)(57 186 64 193)(58 187 65 194)(59 188 66 195)(60 189 67 196)(61 190 68 183)(62 191 69 184)(63 192 70 185)(71 199 78 206)(72 200 79 207)(73 201 80 208)(74 202 81 209)(75 203 82 210)(76 204 83 197)(77 205 84 198)(85 223 92 216)(86 224 93 217)(87 211 94 218)(88 212 95 219)(89 213 96 220)(90 214 97 221)(91 215 98 222)(113 166 120 159)(114 167 121 160)(115 168 122 161)(116 155 123 162)(117 156 124 163)(118 157 125 164)(119 158 126 165)
G:=sub<Sym(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), (1,15,118,70,213,135,75,51)(2,28,119,69,214,134,76,50)(3,27,120,68,215,133,77,49)(4,26,121,67,216,132,78,48)(5,25,122,66,217,131,79,47)(6,24,123,65,218,130,80,46)(7,23,124,64,219,129,81,45)(8,22,125,63,220,128,82,44)(9,21,126,62,221,127,83,43)(10,20,113,61,222,140,84,56)(11,19,114,60,223,139,71,55)(12,18,115,59,224,138,72,54)(13,17,116,58,211,137,73,53)(14,16,117,57,212,136,74,52)(29,109,191,197,169,90,143,165)(30,108,192,210,170,89,144,164)(31,107,193,209,171,88,145,163)(32,106,194,208,172,87,146,162)(33,105,195,207,173,86,147,161)(34,104,196,206,174,85,148,160)(35,103,183,205,175,98,149,159)(36,102,184,204,176,97,150,158)(37,101,185,203,177,96,151,157)(38,100,186,202,178,95,152,156)(39,99,187,201,179,94,153,155)(40,112,188,200,180,93,154,168)(41,111,189,199,181,92,141,167)(42,110,190,198,182,91,142,166), (1,101,8,108)(2,102,9,109)(3,103,10,110)(4,104,11,111)(5,105,12,112)(6,106,13,99)(7,107,14,100)(15,177,22,170)(16,178,23,171)(17,179,24,172)(18,180,25,173)(19,181,26,174)(20,182,27,175)(21,169,28,176)(29,134,36,127)(30,135,37,128)(31,136,38,129)(32,137,39,130)(33,138,40,131)(34,139,41,132)(35,140,42,133)(43,143,50,150)(44,144,51,151)(45,145,52,152)(46,146,53,153)(47,147,54,154)(48,148,55,141)(49,149,56,142)(57,186,64,193)(58,187,65,194)(59,188,66,195)(60,189,67,196)(61,190,68,183)(62,191,69,184)(63,192,70,185)(71,199,78,206)(72,200,79,207)(73,201,80,208)(74,202,81,209)(75,203,82,210)(76,204,83,197)(77,205,84,198)(85,223,92,216)(86,224,93,217)(87,211,94,218)(88,212,95,219)(89,213,96,220)(90,214,97,221)(91,215,98,222)(113,166,120,159)(114,167,121,160)(115,168,122,161)(116,155,123,162)(117,156,124,163)(118,157,125,164)(119,158,126,165)>;
G:=Group( (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), (1,15,118,70,213,135,75,51)(2,28,119,69,214,134,76,50)(3,27,120,68,215,133,77,49)(4,26,121,67,216,132,78,48)(5,25,122,66,217,131,79,47)(6,24,123,65,218,130,80,46)(7,23,124,64,219,129,81,45)(8,22,125,63,220,128,82,44)(9,21,126,62,221,127,83,43)(10,20,113,61,222,140,84,56)(11,19,114,60,223,139,71,55)(12,18,115,59,224,138,72,54)(13,17,116,58,211,137,73,53)(14,16,117,57,212,136,74,52)(29,109,191,197,169,90,143,165)(30,108,192,210,170,89,144,164)(31,107,193,209,171,88,145,163)(32,106,194,208,172,87,146,162)(33,105,195,207,173,86,147,161)(34,104,196,206,174,85,148,160)(35,103,183,205,175,98,149,159)(36,102,184,204,176,97,150,158)(37,101,185,203,177,96,151,157)(38,100,186,202,178,95,152,156)(39,99,187,201,179,94,153,155)(40,112,188,200,180,93,154,168)(41,111,189,199,181,92,141,167)(42,110,190,198,182,91,142,166), (1,101,8,108)(2,102,9,109)(3,103,10,110)(4,104,11,111)(5,105,12,112)(6,106,13,99)(7,107,14,100)(15,177,22,170)(16,178,23,171)(17,179,24,172)(18,180,25,173)(19,181,26,174)(20,182,27,175)(21,169,28,176)(29,134,36,127)(30,135,37,128)(31,136,38,129)(32,137,39,130)(33,138,40,131)(34,139,41,132)(35,140,42,133)(43,143,50,150)(44,144,51,151)(45,145,52,152)(46,146,53,153)(47,147,54,154)(48,148,55,141)(49,149,56,142)(57,186,64,193)(58,187,65,194)(59,188,66,195)(60,189,67,196)(61,190,68,183)(62,191,69,184)(63,192,70,185)(71,199,78,206)(72,200,79,207)(73,201,80,208)(74,202,81,209)(75,203,82,210)(76,204,83,197)(77,205,84,198)(85,223,92,216)(86,224,93,217)(87,211,94,218)(88,212,95,219)(89,213,96,220)(90,214,97,221)(91,215,98,222)(113,166,120,159)(114,167,121,160)(115,168,122,161)(116,155,123,162)(117,156,124,163)(118,157,125,164)(119,158,126,165) );
G=PermutationGroup([(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)], [(1,15,118,70,213,135,75,51),(2,28,119,69,214,134,76,50),(3,27,120,68,215,133,77,49),(4,26,121,67,216,132,78,48),(5,25,122,66,217,131,79,47),(6,24,123,65,218,130,80,46),(7,23,124,64,219,129,81,45),(8,22,125,63,220,128,82,44),(9,21,126,62,221,127,83,43),(10,20,113,61,222,140,84,56),(11,19,114,60,223,139,71,55),(12,18,115,59,224,138,72,54),(13,17,116,58,211,137,73,53),(14,16,117,57,212,136,74,52),(29,109,191,197,169,90,143,165),(30,108,192,210,170,89,144,164),(31,107,193,209,171,88,145,163),(32,106,194,208,172,87,146,162),(33,105,195,207,173,86,147,161),(34,104,196,206,174,85,148,160),(35,103,183,205,175,98,149,159),(36,102,184,204,176,97,150,158),(37,101,185,203,177,96,151,157),(38,100,186,202,178,95,152,156),(39,99,187,201,179,94,153,155),(40,112,188,200,180,93,154,168),(41,111,189,199,181,92,141,167),(42,110,190,198,182,91,142,166)], [(1,101,8,108),(2,102,9,109),(3,103,10,110),(4,104,11,111),(5,105,12,112),(6,106,13,99),(7,107,14,100),(15,177,22,170),(16,178,23,171),(17,179,24,172),(18,180,25,173),(19,181,26,174),(20,182,27,175),(21,169,28,176),(29,134,36,127),(30,135,37,128),(31,136,38,129),(32,137,39,130),(33,138,40,131),(34,139,41,132),(35,140,42,133),(43,143,50,150),(44,144,51,151),(45,145,52,152),(46,146,53,153),(47,147,54,154),(48,148,55,141),(49,149,56,142),(57,186,64,193),(58,187,65,194),(59,188,66,195),(60,189,67,196),(61,190,68,183),(62,191,69,184),(63,192,70,185),(71,199,78,206),(72,200,79,207),(73,201,80,208),(74,202,81,209),(75,203,82,210),(76,204,83,197),(77,205,84,198),(85,223,92,216),(86,224,93,217),(87,211,94,218),(88,212,95,219),(89,213,96,220),(90,214,97,221),(91,215,98,222),(113,166,120,159),(114,167,121,160),(115,168,122,161),(116,155,123,162),(117,156,124,163),(118,157,125,164),(119,158,126,165)])
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 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | SD16 | D14 | Dic14 | C4×D7 | C7⋊D4 | D4.D7 | Q8⋊D7 |
| kernel | C14.SD16 | C2×C7⋊C8 | C4⋊Dic7 | C7×C4⋊C4 | C7⋊C8 | C28 | C2×C14 | C4⋊C4 | C14 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 4 | 3 | 6 | 6 | 6 | 3 | 3 |
Matrix representation of C14.SD16 ►in GL4(𝔽113) generated by
| 80 | 80 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 81 | 38 | 0 | 0 |
| 89 | 32 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 87 |
| 55 | 38 | 0 | 0 |
| 75 | 58 | 0 | 0 |
| 0 | 0 | 68 | 97 |
| 0 | 0 | 56 | 45 |
G:=sub<GL(4,GF(113))| [80,33,0,0,80,9,0,0,0,0,112,0,0,0,0,112],[81,89,0,0,38,32,0,0,0,0,0,1,0,0,1,87],[55,75,0,0,38,58,0,0,0,0,68,56,0,0,97,45] >;
C14.SD16 in GAP, Magma, Sage, TeX
C_{14}.{\rm SD}_{16} % in TeX
G:=Group("C14.SD16"); // GroupNames label
G:=SmallGroup(224,14);
// by ID
G=gap.SmallGroup(224,14);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,313,31,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^14=b^8=1,c^2=a^7,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations