direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×Q16, C28.43D4, C28.46C23, C56.27C22, C4.8(C7×D4), C8.5(C2×C14), (C2×C8).4C14, (C2×C56).14C2, C14.76(C2×D4), C2.13(D4×C14), (C2×C14).54D4, (Q8×C14).9C2, Q8.1(C2×C14), (C2×Q8).4C14, C4.3(C22×C14), C22.16(C7×D4), (C7×Q8).12C22, (C2×C28).131C22, (C2×C4).27(C2×C14), SmallGroup(224,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×Q16
G = < a,b,c | a14=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 76 in 60 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C14, C2×C8, Q16, C2×Q8, C28, C28, C2×C14, C2×Q16, C56, C2×C28, C2×C28, C7×Q8, C7×Q8, C2×C56, C7×Q16, Q8×C14, C14×Q16
Quotients: C1, C2, C22, C7, D4, C23, C14, Q16, C2×D4, C2×C14, C2×Q16, C7×D4, C22×C14, C7×Q16, D4×C14, C14×Q16
►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)
(1 168 178 53 221 139 210 90)(2 155 179 54 222 140 197 91)(3 156 180 55 223 127 198 92)(4 157 181 56 224 128 199 93)(5 158 182 43 211 129 200 94)(6 159 169 44 212 130 201 95)(7 160 170 45 213 131 202 96)(8 161 171 46 214 132 203 97)(9 162 172 47 215 133 204 98)(10 163 173 48 216 134 205 85)(11 164 174 49 217 135 206 86)(12 165 175 50 218 136 207 87)(13 166 176 51 219 137 208 88)(14 167 177 52 220 138 209 89)(15 30 188 107 141 77 116 59)(16 31 189 108 142 78 117 60)(17 32 190 109 143 79 118 61)(18 33 191 110 144 80 119 62)(19 34 192 111 145 81 120 63)(20 35 193 112 146 82 121 64)(21 36 194 99 147 83 122 65)(22 37 195 100 148 84 123 66)(23 38 196 101 149 71 124 67)(24 39 183 102 150 72 125 68)(25 40 184 103 151 73 126 69)(26 41 185 104 152 74 113 70)(27 42 186 105 153 75 114 57)(28 29 187 106 154 76 115 58)
(1 26 221 152)(2 27 222 153)(3 28 223 154)(4 15 224 141)(5 16 211 142)(6 17 212 143)(7 18 213 144)(8 19 214 145)(9 20 215 146)(10 21 216 147)(11 22 217 148)(12 23 218 149)(13 24 219 150)(14 25 220 151)(29 55 76 92)(30 56 77 93)(31 43 78 94)(32 44 79 95)(33 45 80 96)(34 46 81 97)(35 47 82 98)(36 48 83 85)(37 49 84 86)(38 50 71 87)(39 51 72 88)(40 52 73 89)(41 53 74 90)(42 54 75 91)(57 140 105 155)(58 127 106 156)(59 128 107 157)(60 129 108 158)(61 130 109 159)(62 131 110 160)(63 132 111 161)(64 133 112 162)(65 134 99 163)(66 135 100 164)(67 136 101 165)(68 137 102 166)(69 138 103 167)(70 139 104 168)(113 210 185 178)(114 197 186 179)(115 198 187 180)(116 199 188 181)(117 200 189 182)(118 201 190 169)(119 202 191 170)(120 203 192 171)(121 204 193 172)(122 205 194 173)(123 206 195 174)(124 207 196 175)(125 208 183 176)(126 209 184 177)
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), (1,168,178,53,221,139,210,90)(2,155,179,54,222,140,197,91)(3,156,180,55,223,127,198,92)(4,157,181,56,224,128,199,93)(5,158,182,43,211,129,200,94)(6,159,169,44,212,130,201,95)(7,160,170,45,213,131,202,96)(8,161,171,46,214,132,203,97)(9,162,172,47,215,133,204,98)(10,163,173,48,216,134,205,85)(11,164,174,49,217,135,206,86)(12,165,175,50,218,136,207,87)(13,166,176,51,219,137,208,88)(14,167,177,52,220,138,209,89)(15,30,188,107,141,77,116,59)(16,31,189,108,142,78,117,60)(17,32,190,109,143,79,118,61)(18,33,191,110,144,80,119,62)(19,34,192,111,145,81,120,63)(20,35,193,112,146,82,121,64)(21,36,194,99,147,83,122,65)(22,37,195,100,148,84,123,66)(23,38,196,101,149,71,124,67)(24,39,183,102,150,72,125,68)(25,40,184,103,151,73,126,69)(26,41,185,104,152,74,113,70)(27,42,186,105,153,75,114,57)(28,29,187,106,154,76,115,58), (1,26,221,152)(2,27,222,153)(3,28,223,154)(4,15,224,141)(5,16,211,142)(6,17,212,143)(7,18,213,144)(8,19,214,145)(9,20,215,146)(10,21,216,147)(11,22,217,148)(12,23,218,149)(13,24,219,150)(14,25,220,151)(29,55,76,92)(30,56,77,93)(31,43,78,94)(32,44,79,95)(33,45,80,96)(34,46,81,97)(35,47,82,98)(36,48,83,85)(37,49,84,86)(38,50,71,87)(39,51,72,88)(40,52,73,89)(41,53,74,90)(42,54,75,91)(57,140,105,155)(58,127,106,156)(59,128,107,157)(60,129,108,158)(61,130,109,159)(62,131,110,160)(63,132,111,161)(64,133,112,162)(65,134,99,163)(66,135,100,164)(67,136,101,165)(68,137,102,166)(69,138,103,167)(70,139,104,168)(113,210,185,178)(114,197,186,179)(115,198,187,180)(116,199,188,181)(117,200,189,182)(118,201,190,169)(119,202,191,170)(120,203,192,171)(121,204,193,172)(122,205,194,173)(123,206,195,174)(124,207,196,175)(125,208,183,176)(126,209,184,177)>;
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), (1,168,178,53,221,139,210,90)(2,155,179,54,222,140,197,91)(3,156,180,55,223,127,198,92)(4,157,181,56,224,128,199,93)(5,158,182,43,211,129,200,94)(6,159,169,44,212,130,201,95)(7,160,170,45,213,131,202,96)(8,161,171,46,214,132,203,97)(9,162,172,47,215,133,204,98)(10,163,173,48,216,134,205,85)(11,164,174,49,217,135,206,86)(12,165,175,50,218,136,207,87)(13,166,176,51,219,137,208,88)(14,167,177,52,220,138,209,89)(15,30,188,107,141,77,116,59)(16,31,189,108,142,78,117,60)(17,32,190,109,143,79,118,61)(18,33,191,110,144,80,119,62)(19,34,192,111,145,81,120,63)(20,35,193,112,146,82,121,64)(21,36,194,99,147,83,122,65)(22,37,195,100,148,84,123,66)(23,38,196,101,149,71,124,67)(24,39,183,102,150,72,125,68)(25,40,184,103,151,73,126,69)(26,41,185,104,152,74,113,70)(27,42,186,105,153,75,114,57)(28,29,187,106,154,76,115,58), (1,26,221,152)(2,27,222,153)(3,28,223,154)(4,15,224,141)(5,16,211,142)(6,17,212,143)(7,18,213,144)(8,19,214,145)(9,20,215,146)(10,21,216,147)(11,22,217,148)(12,23,218,149)(13,24,219,150)(14,25,220,151)(29,55,76,92)(30,56,77,93)(31,43,78,94)(32,44,79,95)(33,45,80,96)(34,46,81,97)(35,47,82,98)(36,48,83,85)(37,49,84,86)(38,50,71,87)(39,51,72,88)(40,52,73,89)(41,53,74,90)(42,54,75,91)(57,140,105,155)(58,127,106,156)(59,128,107,157)(60,129,108,158)(61,130,109,159)(62,131,110,160)(63,132,111,161)(64,133,112,162)(65,134,99,163)(66,135,100,164)(67,136,101,165)(68,137,102,166)(69,138,103,167)(70,139,104,168)(113,210,185,178)(114,197,186,179)(115,198,187,180)(116,199,188,181)(117,200,189,182)(118,201,190,169)(119,202,191,170)(120,203,192,171)(121,204,193,172)(122,205,194,173)(123,206,195,174)(124,207,196,175)(125,208,183,176)(126,209,184,177) );
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)], [(1,168,178,53,221,139,210,90),(2,155,179,54,222,140,197,91),(3,156,180,55,223,127,198,92),(4,157,181,56,224,128,199,93),(5,158,182,43,211,129,200,94),(6,159,169,44,212,130,201,95),(7,160,170,45,213,131,202,96),(8,161,171,46,214,132,203,97),(9,162,172,47,215,133,204,98),(10,163,173,48,216,134,205,85),(11,164,174,49,217,135,206,86),(12,165,175,50,218,136,207,87),(13,166,176,51,219,137,208,88),(14,167,177,52,220,138,209,89),(15,30,188,107,141,77,116,59),(16,31,189,108,142,78,117,60),(17,32,190,109,143,79,118,61),(18,33,191,110,144,80,119,62),(19,34,192,111,145,81,120,63),(20,35,193,112,146,82,121,64),(21,36,194,99,147,83,122,65),(22,37,195,100,148,84,123,66),(23,38,196,101,149,71,124,67),(24,39,183,102,150,72,125,68),(25,40,184,103,151,73,126,69),(26,41,185,104,152,74,113,70),(27,42,186,105,153,75,114,57),(28,29,187,106,154,76,115,58)], [(1,26,221,152),(2,27,222,153),(3,28,223,154),(4,15,224,141),(5,16,211,142),(6,17,212,143),(7,18,213,144),(8,19,214,145),(9,20,215,146),(10,21,216,147),(11,22,217,148),(12,23,218,149),(13,24,219,150),(14,25,220,151),(29,55,76,92),(30,56,77,93),(31,43,78,94),(32,44,79,95),(33,45,80,96),(34,46,81,97),(35,47,82,98),(36,48,83,85),(37,49,84,86),(38,50,71,87),(39,51,72,88),(40,52,73,89),(41,53,74,90),(42,54,75,91),(57,140,105,155),(58,127,106,156),(59,128,107,157),(60,129,108,158),(61,130,109,159),(62,131,110,160),(63,132,111,161),(64,133,112,162),(65,134,99,163),(66,135,100,164),(67,136,101,165),(68,137,102,166),(69,138,103,167),(70,139,104,168),(113,210,185,178),(114,197,186,179),(115,198,187,180),(116,199,188,181),(117,200,189,182),(118,201,190,169),(119,202,191,170),(120,203,192,171),(121,204,193,172),(122,205,194,173),(123,206,195,174),(124,207,196,175),(125,208,183,176),(126,209,184,177)]])
C14×Q16 is a maximal subgroup of
C14.Q32 Q16.Dic7 Q16.D14 C56.26D4 Dic7⋊3Q16 Q16⋊Dic7 (C2×Q16)⋊D7 D14⋊5Q16 D28.17D4 D14⋊3Q16 C56.36D4 C56.37D4 C56.28D4 C56.29D4 D28.30D4
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 28A | ··· | 28L | 28M | ··· | 28AJ | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | Q16 | C7×D4 | C7×D4 | C7×Q16 |
| kernel | C14×Q16 | C2×C56 | C7×Q16 | Q8×C14 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C28 | C2×C14 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 1 | 1 | 4 | 6 | 6 | 24 |
Matrix representation of C14×Q16 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 83 | 0 |
| 0 | 0 | 83 |
| 112 | 0 | 0 |
| 0 | 0 | 51 |
| 0 | 31 | 51 |
| 112 | 0 | 0 |
| 0 | 29 | 104 |
| 0 | 81 | 84 |
G:=sub<GL(3,GF(113))| [112,0,0,0,83,0,0,0,83],[112,0,0,0,0,31,0,51,51],[112,0,0,0,29,81,0,104,84] >;
C14×Q16 in GAP, Magma, Sage, TeX
C_{14}\times Q_{16} % in TeX
G:=Group("C14xQ16"); // GroupNames label
G:=SmallGroup(224,169);
// by ID
G=gap.SmallGroup(224,169);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,679,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c|a^14=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations