metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56.3C4, C28.56D8, C8.23D28, C56.39D4, Dic28.3C4, C8.14(C4×D7), C56.12(C2×C4), C8.C4⋊2D7, (C2×C28).97D4, C4.6(D14⋊C4), C4.29(D4⋊D7), (C2×C8).249D14, C7⋊1(D8.C4), (C2×C14).6SD16, D56⋊7C2.3C2, C28.6(C22⋊C4), (C2×C56).37C22, C22.1(Q8⋊D7), C14.9(D4⋊C4), C2.11(C14.D8), (C2×C7⋊C16)⋊1C2, (C7×C8.C4)⋊1C2, (C2×C4).118(C7⋊D4), SmallGroup(448,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic28.C4
G = < a,b,c | a56=1, b2=c4=a28, bab-1=a-1, cac-1=a15, cbc-1=a49b >
2C2
56C2
28C22
28C4
2C14
8D7
4C8
14Q8
14D4
28D4
28C2×C4
4Dic7
4D14
7Q16
7D8
14C16
14SD16
14C4○D4
2D28
4C56
Smallest permutation representation of Dic28.C4
►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)
(1 148 29 120)(2 147 30 119)(3 146 31 118)(4 145 32 117)(5 144 33 116)(6 143 34 115)(7 142 35 114)(8 141 36 113)(9 140 37 168)(10 139 38 167)(11 138 39 166)(12 137 40 165)(13 136 41 164)(14 135 42 163)(15 134 43 162)(16 133 44 161)(17 132 45 160)(18 131 46 159)(19 130 47 158)(20 129 48 157)(21 128 49 156)(22 127 50 155)(23 126 51 154)(24 125 52 153)(25 124 53 152)(26 123 54 151)(27 122 55 150)(28 121 56 149)(57 211 85 183)(58 210 86 182)(59 209 87 181)(60 208 88 180)(61 207 89 179)(62 206 90 178)(63 205 91 177)(64 204 92 176)(65 203 93 175)(66 202 94 174)(67 201 95 173)(68 200 96 172)(69 199 97 171)(70 198 98 170)(71 197 99 169)(72 196 100 224)(73 195 101 223)(74 194 102 222)(75 193 103 221)(76 192 104 220)(77 191 105 219)(78 190 106 218)(79 189 107 217)(80 188 108 216)(81 187 109 215)(82 186 110 214)(83 185 111 213)(84 184 112 212)
(1 71 43 85 29 99 15 57)(2 86 44 100 30 58 16 72)(3 101 45 59 31 73 17 87)(4 60 46 74 32 88 18 102)(5 75 47 89 33 103 19 61)(6 90 48 104 34 62 20 76)(7 105 49 63 35 77 21 91)(8 64 50 78 36 92 22 106)(9 79 51 93 37 107 23 65)(10 94 52 108 38 66 24 80)(11 109 53 67 39 81 25 95)(12 68 54 82 40 96 26 110)(13 83 55 97 41 111 27 69)(14 98 56 112 42 70 28 84)(113 183 127 169 141 211 155 197)(114 198 128 184 142 170 156 212)(115 213 129 199 143 185 157 171)(116 172 130 214 144 200 158 186)(117 187 131 173 145 215 159 201)(118 202 132 188 146 174 160 216)(119 217 133 203 147 189 161 175)(120 176 134 218 148 204 162 190)(121 191 135 177 149 219 163 205)(122 206 136 192 150 178 164 220)(123 221 137 207 151 193 165 179)(124 180 138 222 152 208 166 194)(125 195 139 181 153 223 167 209)(126 210 140 196 154 182 168 224)
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), (1,148,29,120)(2,147,30,119)(3,146,31,118)(4,145,32,117)(5,144,33,116)(6,143,34,115)(7,142,35,114)(8,141,36,113)(9,140,37,168)(10,139,38,167)(11,138,39,166)(12,137,40,165)(13,136,41,164)(14,135,42,163)(15,134,43,162)(16,133,44,161)(17,132,45,160)(18,131,46,159)(19,130,47,158)(20,129,48,157)(21,128,49,156)(22,127,50,155)(23,126,51,154)(24,125,52,153)(25,124,53,152)(26,123,54,151)(27,122,55,150)(28,121,56,149)(57,211,85,183)(58,210,86,182)(59,209,87,181)(60,208,88,180)(61,207,89,179)(62,206,90,178)(63,205,91,177)(64,204,92,176)(65,203,93,175)(66,202,94,174)(67,201,95,173)(68,200,96,172)(69,199,97,171)(70,198,98,170)(71,197,99,169)(72,196,100,224)(73,195,101,223)(74,194,102,222)(75,193,103,221)(76,192,104,220)(77,191,105,219)(78,190,106,218)(79,189,107,217)(80,188,108,216)(81,187,109,215)(82,186,110,214)(83,185,111,213)(84,184,112,212), (1,71,43,85,29,99,15,57)(2,86,44,100,30,58,16,72)(3,101,45,59,31,73,17,87)(4,60,46,74,32,88,18,102)(5,75,47,89,33,103,19,61)(6,90,48,104,34,62,20,76)(7,105,49,63,35,77,21,91)(8,64,50,78,36,92,22,106)(9,79,51,93,37,107,23,65)(10,94,52,108,38,66,24,80)(11,109,53,67,39,81,25,95)(12,68,54,82,40,96,26,110)(13,83,55,97,41,111,27,69)(14,98,56,112,42,70,28,84)(113,183,127,169,141,211,155,197)(114,198,128,184,142,170,156,212)(115,213,129,199,143,185,157,171)(116,172,130,214,144,200,158,186)(117,187,131,173,145,215,159,201)(118,202,132,188,146,174,160,216)(119,217,133,203,147,189,161,175)(120,176,134,218,148,204,162,190)(121,191,135,177,149,219,163,205)(122,206,136,192,150,178,164,220)(123,221,137,207,151,193,165,179)(124,180,138,222,152,208,166,194)(125,195,139,181,153,223,167,209)(126,210,140,196,154,182,168,224)>;
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), (1,148,29,120)(2,147,30,119)(3,146,31,118)(4,145,32,117)(5,144,33,116)(6,143,34,115)(7,142,35,114)(8,141,36,113)(9,140,37,168)(10,139,38,167)(11,138,39,166)(12,137,40,165)(13,136,41,164)(14,135,42,163)(15,134,43,162)(16,133,44,161)(17,132,45,160)(18,131,46,159)(19,130,47,158)(20,129,48,157)(21,128,49,156)(22,127,50,155)(23,126,51,154)(24,125,52,153)(25,124,53,152)(26,123,54,151)(27,122,55,150)(28,121,56,149)(57,211,85,183)(58,210,86,182)(59,209,87,181)(60,208,88,180)(61,207,89,179)(62,206,90,178)(63,205,91,177)(64,204,92,176)(65,203,93,175)(66,202,94,174)(67,201,95,173)(68,200,96,172)(69,199,97,171)(70,198,98,170)(71,197,99,169)(72,196,100,224)(73,195,101,223)(74,194,102,222)(75,193,103,221)(76,192,104,220)(77,191,105,219)(78,190,106,218)(79,189,107,217)(80,188,108,216)(81,187,109,215)(82,186,110,214)(83,185,111,213)(84,184,112,212), (1,71,43,85,29,99,15,57)(2,86,44,100,30,58,16,72)(3,101,45,59,31,73,17,87)(4,60,46,74,32,88,18,102)(5,75,47,89,33,103,19,61)(6,90,48,104,34,62,20,76)(7,105,49,63,35,77,21,91)(8,64,50,78,36,92,22,106)(9,79,51,93,37,107,23,65)(10,94,52,108,38,66,24,80)(11,109,53,67,39,81,25,95)(12,68,54,82,40,96,26,110)(13,83,55,97,41,111,27,69)(14,98,56,112,42,70,28,84)(113,183,127,169,141,211,155,197)(114,198,128,184,142,170,156,212)(115,213,129,199,143,185,157,171)(116,172,130,214,144,200,158,186)(117,187,131,173,145,215,159,201)(118,202,132,188,146,174,160,216)(119,217,133,203,147,189,161,175)(120,176,134,218,148,204,162,190)(121,191,135,177,149,219,163,205)(122,206,136,192,150,178,164,220)(123,221,137,207,151,193,165,179)(124,180,138,222,152,208,166,194)(125,195,139,181,153,223,167,209)(126,210,140,196,154,182,168,224) );
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)], [(1,148,29,120),(2,147,30,119),(3,146,31,118),(4,145,32,117),(5,144,33,116),(6,143,34,115),(7,142,35,114),(8,141,36,113),(9,140,37,168),(10,139,38,167),(11,138,39,166),(12,137,40,165),(13,136,41,164),(14,135,42,163),(15,134,43,162),(16,133,44,161),(17,132,45,160),(18,131,46,159),(19,130,47,158),(20,129,48,157),(21,128,49,156),(22,127,50,155),(23,126,51,154),(24,125,52,153),(25,124,53,152),(26,123,54,151),(27,122,55,150),(28,121,56,149),(57,211,85,183),(58,210,86,182),(59,209,87,181),(60,208,88,180),(61,207,89,179),(62,206,90,178),(63,205,91,177),(64,204,92,176),(65,203,93,175),(66,202,94,174),(67,201,95,173),(68,200,96,172),(69,199,97,171),(70,198,98,170),(71,197,99,169),(72,196,100,224),(73,195,101,223),(74,194,102,222),(75,193,103,221),(76,192,104,220),(77,191,105,219),(78,190,106,218),(79,189,107,217),(80,188,108,216),(81,187,109,215),(82,186,110,214),(83,185,111,213),(84,184,112,212)], [(1,71,43,85,29,99,15,57),(2,86,44,100,30,58,16,72),(3,101,45,59,31,73,17,87),(4,60,46,74,32,88,18,102),(5,75,47,89,33,103,19,61),(6,90,48,104,34,62,20,76),(7,105,49,63,35,77,21,91),(8,64,50,78,36,92,22,106),(9,79,51,93,37,107,23,65),(10,94,52,108,38,66,24,80),(11,109,53,67,39,81,25,95),(12,68,54,82,40,96,26,110),(13,83,55,97,41,111,27,69),(14,98,56,112,42,70,28,84),(113,183,127,169,141,211,155,197),(114,198,128,184,142,170,156,212),(115,213,129,199,143,185,157,171),(116,172,130,214,144,200,158,186),(117,187,131,173,145,215,159,201),(118,202,132,188,146,174,160,216),(119,217,133,203,147,189,161,175),(120,176,134,218,148,204,162,190),(121,191,135,177,149,219,163,205),(122,206,136,192,150,178,164,220),(123,221,137,207,151,193,165,179),(124,180,138,222,152,208,166,194),(125,195,139,181,153,223,167,209),(126,210,140,196,154,182,168,224)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | 14B | 14C | 14D | 14E | 14F | 16A | ··· | 16H | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 56 | 1 | 1 | 2 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4×D7 | D28 | C7⋊D4 | D8.C4 | D4⋊D7 | Q8⋊D7 | Dic28.C4 |
| kernel | Dic28.C4 | C2×C7⋊C16 | C7×C8.C4 | D56⋊7C2 | D56 | Dic28 | C56 | C2×C28 | C8.C4 | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 8 | 3 | 3 | 12 |
Matrix representation of Dic28.C4 ►in GL4(𝔽113) generated by
| 44 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 24 | 13 |
| 0 | 0 | 1 | 10 |
| 0 | 71 | 0 | 0 |
| 35 | 0 | 0 | 0 |
| 0 | 0 | 63 | 3 |
| 0 | 0 | 71 | 50 |
| 0 | 18 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 0 | 84 | 43 |
| 0 | 0 | 12 | 29 |
G:=sub<GL(4,GF(113))| [44,0,0,0,0,18,0,0,0,0,24,1,0,0,13,10],[0,35,0,0,71,0,0,0,0,0,63,71,0,0,3,50],[0,18,0,0,18,0,0,0,0,0,84,12,0,0,43,29] >;
Dic28.C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{28}.C_4 % in TeX
G:=Group("Dic28.C4"); // GroupNames label
G:=SmallGroup(448,54);
// by ID
G=gap.SmallGroup(448,54);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,184,675,346,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^2=c^4=a^28,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^49*b>;
// generators/relations
Export