metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.16D8, C28.15SD16, C42.217D14, C4.5(D4⋊D7), C14.56(C2×D8), C4⋊1D4.3D7, C7⋊3(C4.4D8), C28⋊2Q8⋊19C2, (C2×D4).53D14, (C2×C28).146D4, C4.3(D4.D7), C28.74(C4○D4), D4⋊Dic7⋊20C2, C14.57(C2×SD16), C4.22(D4⋊2D7), (C4×C28).118C22, (C2×C28).388C23, (D4×C14).69C22, C14.43(C4.4D4), C4⋊Dic7.154C22, C2.10(C28.17D4), (C4×C7⋊C8)⋊14C2, C2.11(C2×D4⋊D7), (C7×C4⋊1D4).2C2, C2.11(C2×D4.D7), (C2×C14).519(C2×D4), (C2×C7⋊C8).256C22, (C2×C4).130(C7⋊D4), (C2×C4).486(C22×D7), C22.192(C2×C7⋊D4), SmallGroup(448,604)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.16D8
G = < a,b,c | a28=b8=1, c2=a14, bab-1=a13, cac-1=a-1, cbc-1=a14b-1 >
Subgroups: 492 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C7×D4, C22×C14, C4.4D8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×Dic14, D4×C14, D4×C14, C4×C7⋊C8, D4⋊Dic7, C28⋊2Q8, C7×C4⋊1D4, C28.16D8
Quotients: C1, C2, C22, D4, C23, D7, D8, SD16, C2×D4, C4○D4, D14, C4.4D4, C2×D8, C2×SD16, C7⋊D4, C22×D7, C4.4D8, D4⋊D7, D4.D7, D4⋊2D7, C2×C7⋊D4, C2×D4⋊D7, C2×D4.D7, C28.17D4, C28.16D8
►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 83 149 87 39 216 178 137)(2 68 150 100 40 201 179 122)(3 81 151 85 41 214 180 135)(4 66 152 98 42 199 181 120)(5 79 153 111 43 212 182 133)(6 64 154 96 44 197 183 118)(7 77 155 109 45 210 184 131)(8 62 156 94 46 223 185 116)(9 75 157 107 47 208 186 129)(10 60 158 92 48 221 187 114)(11 73 159 105 49 206 188 127)(12 58 160 90 50 219 189 140)(13 71 161 103 51 204 190 125)(14 84 162 88 52 217 191 138)(15 69 163 101 53 202 192 123)(16 82 164 86 54 215 193 136)(17 67 165 99 55 200 194 121)(18 80 166 112 56 213 195 134)(19 65 167 97 29 198 196 119)(20 78 168 110 30 211 169 132)(21 63 141 95 31 224 170 117)(22 76 142 108 32 209 171 130)(23 61 143 93 33 222 172 115)(24 74 144 106 34 207 173 128)(25 59 145 91 35 220 174 113)(26 72 146 104 36 205 175 126)(27 57 147 89 37 218 176 139)(28 70 148 102 38 203 177 124)
(1 116 15 130)(2 115 16 129)(3 114 17 128)(4 113 18 127)(5 140 19 126)(6 139 20 125)(7 138 21 124)(8 137 22 123)(9 136 23 122)(10 135 24 121)(11 134 25 120)(12 133 26 119)(13 132 27 118)(14 131 28 117)(29 104 43 90)(30 103 44 89)(31 102 45 88)(32 101 46 87)(33 100 47 86)(34 99 48 85)(35 98 49 112)(36 97 50 111)(37 96 51 110)(38 95 52 109)(39 94 53 108)(40 93 54 107)(41 92 55 106)(42 91 56 105)(57 169 71 183)(58 196 72 182)(59 195 73 181)(60 194 74 180)(61 193 75 179)(62 192 76 178)(63 191 77 177)(64 190 78 176)(65 189 79 175)(66 188 80 174)(67 187 81 173)(68 186 82 172)(69 185 83 171)(70 184 84 170)(141 203 155 217)(142 202 156 216)(143 201 157 215)(144 200 158 214)(145 199 159 213)(146 198 160 212)(147 197 161 211)(148 224 162 210)(149 223 163 209)(150 222 164 208)(151 221 165 207)(152 220 166 206)(153 219 167 205)(154 218 168 204)
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), (1,83,149,87,39,216,178,137)(2,68,150,100,40,201,179,122)(3,81,151,85,41,214,180,135)(4,66,152,98,42,199,181,120)(5,79,153,111,43,212,182,133)(6,64,154,96,44,197,183,118)(7,77,155,109,45,210,184,131)(8,62,156,94,46,223,185,116)(9,75,157,107,47,208,186,129)(10,60,158,92,48,221,187,114)(11,73,159,105,49,206,188,127)(12,58,160,90,50,219,189,140)(13,71,161,103,51,204,190,125)(14,84,162,88,52,217,191,138)(15,69,163,101,53,202,192,123)(16,82,164,86,54,215,193,136)(17,67,165,99,55,200,194,121)(18,80,166,112,56,213,195,134)(19,65,167,97,29,198,196,119)(20,78,168,110,30,211,169,132)(21,63,141,95,31,224,170,117)(22,76,142,108,32,209,171,130)(23,61,143,93,33,222,172,115)(24,74,144,106,34,207,173,128)(25,59,145,91,35,220,174,113)(26,72,146,104,36,205,175,126)(27,57,147,89,37,218,176,139)(28,70,148,102,38,203,177,124), (1,116,15,130)(2,115,16,129)(3,114,17,128)(4,113,18,127)(5,140,19,126)(6,139,20,125)(7,138,21,124)(8,137,22,123)(9,136,23,122)(10,135,24,121)(11,134,25,120)(12,133,26,119)(13,132,27,118)(14,131,28,117)(29,104,43,90)(30,103,44,89)(31,102,45,88)(32,101,46,87)(33,100,47,86)(34,99,48,85)(35,98,49,112)(36,97,50,111)(37,96,51,110)(38,95,52,109)(39,94,53,108)(40,93,54,107)(41,92,55,106)(42,91,56,105)(57,169,71,183)(58,196,72,182)(59,195,73,181)(60,194,74,180)(61,193,75,179)(62,192,76,178)(63,191,77,177)(64,190,78,176)(65,189,79,175)(66,188,80,174)(67,187,81,173)(68,186,82,172)(69,185,83,171)(70,184,84,170)(141,203,155,217)(142,202,156,216)(143,201,157,215)(144,200,158,214)(145,199,159,213)(146,198,160,212)(147,197,161,211)(148,224,162,210)(149,223,163,209)(150,222,164,208)(151,221,165,207)(152,220,166,206)(153,219,167,205)(154,218,168,204)>;
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), (1,83,149,87,39,216,178,137)(2,68,150,100,40,201,179,122)(3,81,151,85,41,214,180,135)(4,66,152,98,42,199,181,120)(5,79,153,111,43,212,182,133)(6,64,154,96,44,197,183,118)(7,77,155,109,45,210,184,131)(8,62,156,94,46,223,185,116)(9,75,157,107,47,208,186,129)(10,60,158,92,48,221,187,114)(11,73,159,105,49,206,188,127)(12,58,160,90,50,219,189,140)(13,71,161,103,51,204,190,125)(14,84,162,88,52,217,191,138)(15,69,163,101,53,202,192,123)(16,82,164,86,54,215,193,136)(17,67,165,99,55,200,194,121)(18,80,166,112,56,213,195,134)(19,65,167,97,29,198,196,119)(20,78,168,110,30,211,169,132)(21,63,141,95,31,224,170,117)(22,76,142,108,32,209,171,130)(23,61,143,93,33,222,172,115)(24,74,144,106,34,207,173,128)(25,59,145,91,35,220,174,113)(26,72,146,104,36,205,175,126)(27,57,147,89,37,218,176,139)(28,70,148,102,38,203,177,124), (1,116,15,130)(2,115,16,129)(3,114,17,128)(4,113,18,127)(5,140,19,126)(6,139,20,125)(7,138,21,124)(8,137,22,123)(9,136,23,122)(10,135,24,121)(11,134,25,120)(12,133,26,119)(13,132,27,118)(14,131,28,117)(29,104,43,90)(30,103,44,89)(31,102,45,88)(32,101,46,87)(33,100,47,86)(34,99,48,85)(35,98,49,112)(36,97,50,111)(37,96,51,110)(38,95,52,109)(39,94,53,108)(40,93,54,107)(41,92,55,106)(42,91,56,105)(57,169,71,183)(58,196,72,182)(59,195,73,181)(60,194,74,180)(61,193,75,179)(62,192,76,178)(63,191,77,177)(64,190,78,176)(65,189,79,175)(66,188,80,174)(67,187,81,173)(68,186,82,172)(69,185,83,171)(70,184,84,170)(141,203,155,217)(142,202,156,216)(143,201,157,215)(144,200,158,214)(145,199,159,213)(146,198,160,212)(147,197,161,211)(148,224,162,210)(149,223,163,209)(150,222,164,208)(151,221,165,207)(152,220,166,206)(153,219,167,205)(154,218,168,204) );
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)], [(1,83,149,87,39,216,178,137),(2,68,150,100,40,201,179,122),(3,81,151,85,41,214,180,135),(4,66,152,98,42,199,181,120),(5,79,153,111,43,212,182,133),(6,64,154,96,44,197,183,118),(7,77,155,109,45,210,184,131),(8,62,156,94,46,223,185,116),(9,75,157,107,47,208,186,129),(10,60,158,92,48,221,187,114),(11,73,159,105,49,206,188,127),(12,58,160,90,50,219,189,140),(13,71,161,103,51,204,190,125),(14,84,162,88,52,217,191,138),(15,69,163,101,53,202,192,123),(16,82,164,86,54,215,193,136),(17,67,165,99,55,200,194,121),(18,80,166,112,56,213,195,134),(19,65,167,97,29,198,196,119),(20,78,168,110,30,211,169,132),(21,63,141,95,31,224,170,117),(22,76,142,108,32,209,171,130),(23,61,143,93,33,222,172,115),(24,74,144,106,34,207,173,128),(25,59,145,91,35,220,174,113),(26,72,146,104,36,205,175,126),(27,57,147,89,37,218,176,139),(28,70,148,102,38,203,177,124)], [(1,116,15,130),(2,115,16,129),(3,114,17,128),(4,113,18,127),(5,140,19,126),(6,139,20,125),(7,138,21,124),(8,137,22,123),(9,136,23,122),(10,135,24,121),(11,134,25,120),(12,133,26,119),(13,132,27,118),(14,131,28,117),(29,104,43,90),(30,103,44,89),(31,102,45,88),(32,101,46,87),(33,100,47,86),(34,99,48,85),(35,98,49,112),(36,97,50,111),(37,96,51,110),(38,95,52,109),(39,94,53,108),(40,93,54,107),(41,92,55,106),(42,91,56,105),(57,169,71,183),(58,196,72,182),(59,195,73,181),(60,194,74,180),(61,193,75,179),(62,192,76,178),(63,191,77,177),(64,190,78,176),(65,189,79,175),(66,188,80,174),(67,187,81,173),(68,186,82,172),(69,185,83,171),(70,184,84,170),(141,203,155,217),(142,202,156,216),(143,201,157,215),(144,200,158,214),(145,199,159,213),(146,198,160,212),(147,197,161,211),(148,224,162,210),(149,223,163,209),(150,222,164,208),(151,221,165,207),(152,220,166,206),(153,219,167,205),(154,218,168,204)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 56 | 56 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | D8 | SD16 | C4○D4 | D14 | D14 | C7⋊D4 | D4⋊D7 | D4.D7 | D4⋊2D7 |
| kernel | C28.16D8 | C4×C7⋊C8 | D4⋊Dic7 | C28⋊2Q8 | C7×C4⋊1D4 | C2×C28 | C4⋊1D4 | C28 | C28 | C28 | C42 | C2×D4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 4 | 4 | 3 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of C28.16D8 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 111 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 83 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 51 | 91 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 26 | 0 | 0 |
| 0 | 0 | 100 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 83 |
| 0 | 0 | 0 | 0 | 49 | 0 |
| 62 | 22 | 0 | 0 | 0 | 0 |
| 77 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 30 |
| 0 | 0 | 0 | 0 | 49 | 0 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,1,0,0,0,0,111,1,0,0,0,0,0,0,83,0,0,0,0,0,0,64],[51,36,0,0,0,0,91,0,0,0,0,0,0,0,26,100,0,0,0,0,26,0,0,0,0,0,0,0,0,49,0,0,0,0,83,0],[62,77,0,0,0,0,22,51,0,0,0,0,0,0,0,13,0,0,0,0,26,0,0,0,0,0,0,0,0,49,0,0,0,0,30,0] >;
C28.16D8 in GAP, Magma, Sage, TeX
C_{28}._{16}D_8 % in TeX
G:=Group("C28.16D8"); // GroupNames label
G:=SmallGroup(448,604);
// by ID
G=gap.SmallGroup(448,604);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,64,590,135,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=a^14*b^-1>;
// generators/relations