metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.3C8, C16.12D14, M5(2)⋊7D7, C56.67C23, Dic14.3C8, C112.13C22, C7⋊D4.C8, C4.5(C8×D7), (D7×C16)⋊8C2, C7⋊2(D4○C16), C8.17(C4×D7), C16⋊D7⋊6C2, C56.39(C2×C4), C28.13(C2×C8), C4○D28.3C4, D14.2(C2×C8), C8⋊D7.2C4, C22.1(C8×D7), (C2×C8).273D14, C7⋊C16.13C22, (C7×M5(2))⋊6C2, Dic7.4(C2×C8), C8.61(C22×D7), C4.Dic7.5C4, C14.16(C22×C8), (C8×D7).19C22, C28.132(C22×C4), (C2×C56).231C22, D28.2C4.5C2, (C2×C7⋊C16)⋊15C2, C2.17(D7×C2×C8), C7⋊C8.14(C2×C4), C4.106(C2×C4×D7), (C2×C14).6(C2×C8), (C2×C4).74(C4×D7), (C2×C28).73(C2×C4), (C4×D7).22(C2×C4), SmallGroup(448,441)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16.12D14
G = < a,b,c | a16=b14=1, c2=a8, bab-1=a9, ac=ca, cbc-1=a8b-1 >
Subgroups: 276 in 84 conjugacy classes, 51 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C16, C16, C2×C8, C2×C8, M4(2), C4○D4, Dic7, C28, D14, C2×C14, C2×C16, M5(2), M5(2), C8○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, D4○C16, C7⋊C16, C112, C8×D7, C8⋊D7, C4.Dic7, C2×C56, C4○D28, D7×C16, C16⋊D7, C2×C7⋊C16, C7×M5(2), D28.2C4, C16.12D14
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, D14, C22×C8, C4×D7, C22×D7, D4○C16, C8×D7, C2×C4×D7, D7×C2×C8, C16.12D14
►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)
(1 129 113 97 173 35 85 209 184 193 78 50 32 160)(2 138 114 106 174 44 86 218 185 202 79 59 17 153)(3 131 115 99 175 37 87 211 186 195 80 52 18 146)(4 140 116 108 176 46 88 220 187 204 65 61 19 155)(5 133 117 101 161 39 89 213 188 197 66 54 20 148)(6 142 118 110 162 48 90 222 189 206 67 63 21 157)(7 135 119 103 163 41 91 215 190 199 68 56 22 150)(8 144 120 112 164 34 92 224 191 208 69 49 23 159)(9 137 121 105 165 43 93 217 192 201 70 58 24 152)(10 130 122 98 166 36 94 210 177 194 71 51 25 145)(11 139 123 107 167 45 95 219 178 203 72 60 26 154)(12 132 124 100 168 38 96 212 179 196 73 53 27 147)(13 141 125 109 169 47 81 221 180 205 74 62 28 156)(14 134 126 102 170 40 82 214 181 198 75 55 29 149)(15 143 127 111 171 33 83 223 182 207 76 64 30 158)(16 136 128 104 172 42 84 216 183 200 77 57 31 151)
(1 160 9 152)(2 145 10 153)(3 146 11 154)(4 147 12 155)(5 148 13 156)(6 149 14 157)(7 150 15 158)(8 151 16 159)(17 130 25 138)(18 131 26 139)(19 132 27 140)(20 133 28 141)(21 134 29 142)(22 135 30 143)(23 136 31 144)(24 137 32 129)(33 190 41 182)(34 191 42 183)(35 192 43 184)(36 177 44 185)(37 178 45 186)(38 179 46 187)(39 180 47 188)(40 181 48 189)(49 120 57 128)(50 121 58 113)(51 122 59 114)(52 123 60 115)(53 124 61 116)(54 125 62 117)(55 126 63 118)(56 127 64 119)(65 100 73 108)(66 101 74 109)(67 102 75 110)(68 103 76 111)(69 104 77 112)(70 105 78 97)(71 106 79 98)(72 107 80 99)(81 221 89 213)(82 222 90 214)(83 223 91 215)(84 224 92 216)(85 209 93 217)(86 210 94 218)(87 211 95 219)(88 212 96 220)(161 197 169 205)(162 198 170 206)(163 199 171 207)(164 200 172 208)(165 201 173 193)(166 202 174 194)(167 203 175 195)(168 204 176 196)
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), (1,129,113,97,173,35,85,209,184,193,78,50,32,160)(2,138,114,106,174,44,86,218,185,202,79,59,17,153)(3,131,115,99,175,37,87,211,186,195,80,52,18,146)(4,140,116,108,176,46,88,220,187,204,65,61,19,155)(5,133,117,101,161,39,89,213,188,197,66,54,20,148)(6,142,118,110,162,48,90,222,189,206,67,63,21,157)(7,135,119,103,163,41,91,215,190,199,68,56,22,150)(8,144,120,112,164,34,92,224,191,208,69,49,23,159)(9,137,121,105,165,43,93,217,192,201,70,58,24,152)(10,130,122,98,166,36,94,210,177,194,71,51,25,145)(11,139,123,107,167,45,95,219,178,203,72,60,26,154)(12,132,124,100,168,38,96,212,179,196,73,53,27,147)(13,141,125,109,169,47,81,221,180,205,74,62,28,156)(14,134,126,102,170,40,82,214,181,198,75,55,29,149)(15,143,127,111,171,33,83,223,182,207,76,64,30,158)(16,136,128,104,172,42,84,216,183,200,77,57,31,151), (1,160,9,152)(2,145,10,153)(3,146,11,154)(4,147,12,155)(5,148,13,156)(6,149,14,157)(7,150,15,158)(8,151,16,159)(17,130,25,138)(18,131,26,139)(19,132,27,140)(20,133,28,141)(21,134,29,142)(22,135,30,143)(23,136,31,144)(24,137,32,129)(33,190,41,182)(34,191,42,183)(35,192,43,184)(36,177,44,185)(37,178,45,186)(38,179,46,187)(39,180,47,188)(40,181,48,189)(49,120,57,128)(50,121,58,113)(51,122,59,114)(52,123,60,115)(53,124,61,116)(54,125,62,117)(55,126,63,118)(56,127,64,119)(65,100,73,108)(66,101,74,109)(67,102,75,110)(68,103,76,111)(69,104,77,112)(70,105,78,97)(71,106,79,98)(72,107,80,99)(81,221,89,213)(82,222,90,214)(83,223,91,215)(84,224,92,216)(85,209,93,217)(86,210,94,218)(87,211,95,219)(88,212,96,220)(161,197,169,205)(162,198,170,206)(163,199,171,207)(164,200,172,208)(165,201,173,193)(166,202,174,194)(167,203,175,195)(168,204,176,196)>;
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), (1,129,113,97,173,35,85,209,184,193,78,50,32,160)(2,138,114,106,174,44,86,218,185,202,79,59,17,153)(3,131,115,99,175,37,87,211,186,195,80,52,18,146)(4,140,116,108,176,46,88,220,187,204,65,61,19,155)(5,133,117,101,161,39,89,213,188,197,66,54,20,148)(6,142,118,110,162,48,90,222,189,206,67,63,21,157)(7,135,119,103,163,41,91,215,190,199,68,56,22,150)(8,144,120,112,164,34,92,224,191,208,69,49,23,159)(9,137,121,105,165,43,93,217,192,201,70,58,24,152)(10,130,122,98,166,36,94,210,177,194,71,51,25,145)(11,139,123,107,167,45,95,219,178,203,72,60,26,154)(12,132,124,100,168,38,96,212,179,196,73,53,27,147)(13,141,125,109,169,47,81,221,180,205,74,62,28,156)(14,134,126,102,170,40,82,214,181,198,75,55,29,149)(15,143,127,111,171,33,83,223,182,207,76,64,30,158)(16,136,128,104,172,42,84,216,183,200,77,57,31,151), (1,160,9,152)(2,145,10,153)(3,146,11,154)(4,147,12,155)(5,148,13,156)(6,149,14,157)(7,150,15,158)(8,151,16,159)(17,130,25,138)(18,131,26,139)(19,132,27,140)(20,133,28,141)(21,134,29,142)(22,135,30,143)(23,136,31,144)(24,137,32,129)(33,190,41,182)(34,191,42,183)(35,192,43,184)(36,177,44,185)(37,178,45,186)(38,179,46,187)(39,180,47,188)(40,181,48,189)(49,120,57,128)(50,121,58,113)(51,122,59,114)(52,123,60,115)(53,124,61,116)(54,125,62,117)(55,126,63,118)(56,127,64,119)(65,100,73,108)(66,101,74,109)(67,102,75,110)(68,103,76,111)(69,104,77,112)(70,105,78,97)(71,106,79,98)(72,107,80,99)(81,221,89,213)(82,222,90,214)(83,223,91,215)(84,224,92,216)(85,209,93,217)(86,210,94,218)(87,211,95,219)(88,212,96,220)(161,197,169,205)(162,198,170,206)(163,199,171,207)(164,200,172,208)(165,201,173,193)(166,202,174,194)(167,203,175,195)(168,204,176,196) );
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)], [(1,129,113,97,173,35,85,209,184,193,78,50,32,160),(2,138,114,106,174,44,86,218,185,202,79,59,17,153),(3,131,115,99,175,37,87,211,186,195,80,52,18,146),(4,140,116,108,176,46,88,220,187,204,65,61,19,155),(5,133,117,101,161,39,89,213,188,197,66,54,20,148),(6,142,118,110,162,48,90,222,189,206,67,63,21,157),(7,135,119,103,163,41,91,215,190,199,68,56,22,150),(8,144,120,112,164,34,92,224,191,208,69,49,23,159),(9,137,121,105,165,43,93,217,192,201,70,58,24,152),(10,130,122,98,166,36,94,210,177,194,71,51,25,145),(11,139,123,107,167,45,95,219,178,203,72,60,26,154),(12,132,124,100,168,38,96,212,179,196,73,53,27,147),(13,141,125,109,169,47,81,221,180,205,74,62,28,156),(14,134,126,102,170,40,82,214,181,198,75,55,29,149),(15,143,127,111,171,33,83,223,182,207,76,64,30,158),(16,136,128,104,172,42,84,216,183,200,77,57,31,151)], [(1,160,9,152),(2,145,10,153),(3,146,11,154),(4,147,12,155),(5,148,13,156),(6,149,14,157),(7,150,15,158),(8,151,16,159),(17,130,25,138),(18,131,26,139),(19,132,27,140),(20,133,28,141),(21,134,29,142),(22,135,30,143),(23,136,31,144),(24,137,32,129),(33,190,41,182),(34,191,42,183),(35,192,43,184),(36,177,44,185),(37,178,45,186),(38,179,46,187),(39,180,47,188),(40,181,48,189),(49,120,57,128),(50,121,58,113),(51,122,59,114),(52,123,60,115),(53,124,61,116),(54,125,62,117),(55,126,63,118),(56,127,64,119),(65,100,73,108),(66,101,74,109),(67,102,75,110),(68,103,76,111),(69,104,77,112),(70,105,78,97),(71,106,79,98),(72,107,80,99),(81,221,89,213),(82,222,90,214),(83,223,91,215),(84,224,92,216),(85,209,93,217),(86,210,94,218),(87,211,95,219),(88,212,96,220),(161,197,169,205),(162,198,170,206),(163,199,171,207),(164,200,172,208),(165,201,173,193),(166,202,174,194),(167,203,175,195),(168,204,176,196)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 14A | 14B | 14C | 14D | 14E | 14F | 16A | ··· | 16H | 16I | ··· | 16P | 16Q | 16R | 16S | 16T | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | ··· | 16 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 2 | 14 | 14 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D7 | D14 | D14 | C4×D7 | C4×D7 | D4○C16 | C8×D7 | C8×D7 | C16.12D14 |
| kernel | C16.12D14 | D7×C16 | C16⋊D7 | C2×C7⋊C16 | C7×M5(2) | D28.2C4 | C8⋊D7 | C4.Dic7 | C4○D28 | Dic14 | D28 | C7⋊D4 | M5(2) | C16 | C2×C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 3 | 6 | 3 | 6 | 6 | 8 | 12 | 12 | 12 |
Matrix representation of C16.12D14 ►in GL4(𝔽113) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 106 | 33 |
| 0 | 0 | 87 | 7 |
| 33 | 33 | 0 | 0 |
| 80 | 104 | 0 | 0 |
| 0 | 0 | 110 | 111 |
| 0 | 0 | 4 | 3 |
| 33 | 33 | 0 | 0 |
| 104 | 80 | 0 | 0 |
| 0 | 0 | 3 | 2 |
| 0 | 0 | 108 | 110 |
G:=sub<GL(4,GF(113))| [15,0,0,0,0,15,0,0,0,0,106,87,0,0,33,7],[33,80,0,0,33,104,0,0,0,0,110,4,0,0,111,3],[33,104,0,0,33,80,0,0,0,0,3,108,0,0,2,110] >;
C16.12D14 in GAP, Magma, Sage, TeX
C_{16}._{12}D_{14} % in TeX
G:=Group("C16.12D14"); // GroupNames label
G:=SmallGroup(448,441);
// by ID
G=gap.SmallGroup(448,441);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,58,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^16=b^14=1,c^2=a^8,b*a*b^-1=a^9,a*c=c*a,c*b*c^-1=a^8*b^-1>;
// generators/relations