metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.Dic13, Q8.Dic13, C52.42C23, C13⋊5(C8○D4), C52.36(C2×C4), C4○D4.3D13, (D4×C13).2C4, (C2×C4).58D26, (Q8×C13).2C4, C52.4C4⋊8C2, C4.5(C2×Dic13), (C2×C52).41C22, C26.40(C22×C4), C4.42(C22×D13), C13⋊2C8.13C22, C2.8(C22×Dic13), C22.1(C2×Dic13), (C2×C13⋊2C8)⋊7C2, (C2×C26).27(C2×C4), (C13×C4○D4).2C2, SmallGroup(416,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.Dic13
G = < a,b,c,d | a4=1, b2=c26=a2, d2=c13, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c25 >
Subgroups: 200 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, D4, Q8, C13, C2×C8, M4(2), C4○D4, C26, C26, C8○D4, C52, C52, C2×C26, C13⋊2C8, C13⋊2C8, C2×C52, D4×C13, Q8×C13, C2×C13⋊2C8, C52.4C4, C13×C4○D4, D4.Dic13
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D13, C8○D4, Dic13, D26, C2×Dic13, C22×D13, C22×Dic13, D4.Dic13
►On 208 points
Generators in S208
(1 91 27 65)(2 92 28 66)(3 93 29 67)(4 94 30 68)(5 95 31 69)(6 96 32 70)(7 97 33 71)(8 98 34 72)(9 99 35 73)(10 100 36 74)(11 101 37 75)(12 102 38 76)(13 103 39 77)(14 104 40 78)(15 53 41 79)(16 54 42 80)(17 55 43 81)(18 56 44 82)(19 57 45 83)(20 58 46 84)(21 59 47 85)(22 60 48 86)(23 61 49 87)(24 62 50 88)(25 63 51 89)(26 64 52 90)(105 170 131 196)(106 171 132 197)(107 172 133 198)(108 173 134 199)(109 174 135 200)(110 175 136 201)(111 176 137 202)(112 177 138 203)(113 178 139 204)(114 179 140 205)(115 180 141 206)(116 181 142 207)(117 182 143 208)(118 183 144 157)(119 184 145 158)(120 185 146 159)(121 186 147 160)(122 187 148 161)(123 188 149 162)(124 189 150 163)(125 190 151 164)(126 191 152 165)(127 192 153 166)(128 193 154 167)(129 194 155 168)(130 195 156 169)
(1 14 27 40)(2 15 28 41)(3 16 29 42)(4 17 30 43)(5 18 31 44)(6 19 32 45)(7 20 33 46)(8 21 34 47)(9 22 35 48)(10 23 36 49)(11 24 37 50)(12 25 38 51)(13 26 39 52)(53 92 79 66)(54 93 80 67)(55 94 81 68)(56 95 82 69)(57 96 83 70)(58 97 84 71)(59 98 85 72)(60 99 86 73)(61 100 87 74)(62 101 88 75)(63 102 89 76)(64 103 90 77)(65 104 91 78)(105 118 131 144)(106 119 132 145)(107 120 133 146)(108 121 134 147)(109 122 135 148)(110 123 136 149)(111 124 137 150)(112 125 138 151)(113 126 139 152)(114 127 140 153)(115 128 141 154)(116 129 142 155)(117 130 143 156)(157 196 183 170)(158 197 184 171)(159 198 185 172)(160 199 186 173)(161 200 187 174)(162 201 188 175)(163 202 189 176)(164 203 190 177)(165 204 191 178)(166 205 192 179)(167 206 193 180)(168 207 194 181)(169 208 195 182)
(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)
(1 105 14 118 27 131 40 144)(2 130 15 143 28 156 41 117)(3 155 16 116 29 129 42 142)(4 128 17 141 30 154 43 115)(5 153 18 114 31 127 44 140)(6 126 19 139 32 152 45 113)(7 151 20 112 33 125 46 138)(8 124 21 137 34 150 47 111)(9 149 22 110 35 123 48 136)(10 122 23 135 36 148 49 109)(11 147 24 108 37 121 50 134)(12 120 25 133 38 146 51 107)(13 145 26 106 39 119 52 132)(53 208 66 169 79 182 92 195)(54 181 67 194 80 207 93 168)(55 206 68 167 81 180 94 193)(56 179 69 192 82 205 95 166)(57 204 70 165 83 178 96 191)(58 177 71 190 84 203 97 164)(59 202 72 163 85 176 98 189)(60 175 73 188 86 201 99 162)(61 200 74 161 87 174 100 187)(62 173 75 186 88 199 101 160)(63 198 76 159 89 172 102 185)(64 171 77 184 90 197 103 158)(65 196 78 157 91 170 104 183)
G:=sub<Sym(208)| (1,91,27,65)(2,92,28,66)(3,93,29,67)(4,94,30,68)(5,95,31,69)(6,96,32,70)(7,97,33,71)(8,98,34,72)(9,99,35,73)(10,100,36,74)(11,101,37,75)(12,102,38,76)(13,103,39,77)(14,104,40,78)(15,53,41,79)(16,54,42,80)(17,55,43,81)(18,56,44,82)(19,57,45,83)(20,58,46,84)(21,59,47,85)(22,60,48,86)(23,61,49,87)(24,62,50,88)(25,63,51,89)(26,64,52,90)(105,170,131,196)(106,171,132,197)(107,172,133,198)(108,173,134,199)(109,174,135,200)(110,175,136,201)(111,176,137,202)(112,177,138,203)(113,178,139,204)(114,179,140,205)(115,180,141,206)(116,181,142,207)(117,182,143,208)(118,183,144,157)(119,184,145,158)(120,185,146,159)(121,186,147,160)(122,187,148,161)(123,188,149,162)(124,189,150,163)(125,190,151,164)(126,191,152,165)(127,192,153,166)(128,193,154,167)(129,194,155,168)(130,195,156,169), (1,14,27,40)(2,15,28,41)(3,16,29,42)(4,17,30,43)(5,18,31,44)(6,19,32,45)(7,20,33,46)(8,21,34,47)(9,22,35,48)(10,23,36,49)(11,24,37,50)(12,25,38,51)(13,26,39,52)(53,92,79,66)(54,93,80,67)(55,94,81,68)(56,95,82,69)(57,96,83,70)(58,97,84,71)(59,98,85,72)(60,99,86,73)(61,100,87,74)(62,101,88,75)(63,102,89,76)(64,103,90,77)(65,104,91,78)(105,118,131,144)(106,119,132,145)(107,120,133,146)(108,121,134,147)(109,122,135,148)(110,123,136,149)(111,124,137,150)(112,125,138,151)(113,126,139,152)(114,127,140,153)(115,128,141,154)(116,129,142,155)(117,130,143,156)(157,196,183,170)(158,197,184,171)(159,198,185,172)(160,199,186,173)(161,200,187,174)(162,201,188,175)(163,202,189,176)(164,203,190,177)(165,204,191,178)(166,205,192,179)(167,206,193,180)(168,207,194,181)(169,208,195,182), (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), (1,105,14,118,27,131,40,144)(2,130,15,143,28,156,41,117)(3,155,16,116,29,129,42,142)(4,128,17,141,30,154,43,115)(5,153,18,114,31,127,44,140)(6,126,19,139,32,152,45,113)(7,151,20,112,33,125,46,138)(8,124,21,137,34,150,47,111)(9,149,22,110,35,123,48,136)(10,122,23,135,36,148,49,109)(11,147,24,108,37,121,50,134)(12,120,25,133,38,146,51,107)(13,145,26,106,39,119,52,132)(53,208,66,169,79,182,92,195)(54,181,67,194,80,207,93,168)(55,206,68,167,81,180,94,193)(56,179,69,192,82,205,95,166)(57,204,70,165,83,178,96,191)(58,177,71,190,84,203,97,164)(59,202,72,163,85,176,98,189)(60,175,73,188,86,201,99,162)(61,200,74,161,87,174,100,187)(62,173,75,186,88,199,101,160)(63,198,76,159,89,172,102,185)(64,171,77,184,90,197,103,158)(65,196,78,157,91,170,104,183)>;
G:=Group( (1,91,27,65)(2,92,28,66)(3,93,29,67)(4,94,30,68)(5,95,31,69)(6,96,32,70)(7,97,33,71)(8,98,34,72)(9,99,35,73)(10,100,36,74)(11,101,37,75)(12,102,38,76)(13,103,39,77)(14,104,40,78)(15,53,41,79)(16,54,42,80)(17,55,43,81)(18,56,44,82)(19,57,45,83)(20,58,46,84)(21,59,47,85)(22,60,48,86)(23,61,49,87)(24,62,50,88)(25,63,51,89)(26,64,52,90)(105,170,131,196)(106,171,132,197)(107,172,133,198)(108,173,134,199)(109,174,135,200)(110,175,136,201)(111,176,137,202)(112,177,138,203)(113,178,139,204)(114,179,140,205)(115,180,141,206)(116,181,142,207)(117,182,143,208)(118,183,144,157)(119,184,145,158)(120,185,146,159)(121,186,147,160)(122,187,148,161)(123,188,149,162)(124,189,150,163)(125,190,151,164)(126,191,152,165)(127,192,153,166)(128,193,154,167)(129,194,155,168)(130,195,156,169), (1,14,27,40)(2,15,28,41)(3,16,29,42)(4,17,30,43)(5,18,31,44)(6,19,32,45)(7,20,33,46)(8,21,34,47)(9,22,35,48)(10,23,36,49)(11,24,37,50)(12,25,38,51)(13,26,39,52)(53,92,79,66)(54,93,80,67)(55,94,81,68)(56,95,82,69)(57,96,83,70)(58,97,84,71)(59,98,85,72)(60,99,86,73)(61,100,87,74)(62,101,88,75)(63,102,89,76)(64,103,90,77)(65,104,91,78)(105,118,131,144)(106,119,132,145)(107,120,133,146)(108,121,134,147)(109,122,135,148)(110,123,136,149)(111,124,137,150)(112,125,138,151)(113,126,139,152)(114,127,140,153)(115,128,141,154)(116,129,142,155)(117,130,143,156)(157,196,183,170)(158,197,184,171)(159,198,185,172)(160,199,186,173)(161,200,187,174)(162,201,188,175)(163,202,189,176)(164,203,190,177)(165,204,191,178)(166,205,192,179)(167,206,193,180)(168,207,194,181)(169,208,195,182), (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), (1,105,14,118,27,131,40,144)(2,130,15,143,28,156,41,117)(3,155,16,116,29,129,42,142)(4,128,17,141,30,154,43,115)(5,153,18,114,31,127,44,140)(6,126,19,139,32,152,45,113)(7,151,20,112,33,125,46,138)(8,124,21,137,34,150,47,111)(9,149,22,110,35,123,48,136)(10,122,23,135,36,148,49,109)(11,147,24,108,37,121,50,134)(12,120,25,133,38,146,51,107)(13,145,26,106,39,119,52,132)(53,208,66,169,79,182,92,195)(54,181,67,194,80,207,93,168)(55,206,68,167,81,180,94,193)(56,179,69,192,82,205,95,166)(57,204,70,165,83,178,96,191)(58,177,71,190,84,203,97,164)(59,202,72,163,85,176,98,189)(60,175,73,188,86,201,99,162)(61,200,74,161,87,174,100,187)(62,173,75,186,88,199,101,160)(63,198,76,159,89,172,102,185)(64,171,77,184,90,197,103,158)(65,196,78,157,91,170,104,183) );
G=PermutationGroup([[(1,91,27,65),(2,92,28,66),(3,93,29,67),(4,94,30,68),(5,95,31,69),(6,96,32,70),(7,97,33,71),(8,98,34,72),(9,99,35,73),(10,100,36,74),(11,101,37,75),(12,102,38,76),(13,103,39,77),(14,104,40,78),(15,53,41,79),(16,54,42,80),(17,55,43,81),(18,56,44,82),(19,57,45,83),(20,58,46,84),(21,59,47,85),(22,60,48,86),(23,61,49,87),(24,62,50,88),(25,63,51,89),(26,64,52,90),(105,170,131,196),(106,171,132,197),(107,172,133,198),(108,173,134,199),(109,174,135,200),(110,175,136,201),(111,176,137,202),(112,177,138,203),(113,178,139,204),(114,179,140,205),(115,180,141,206),(116,181,142,207),(117,182,143,208),(118,183,144,157),(119,184,145,158),(120,185,146,159),(121,186,147,160),(122,187,148,161),(123,188,149,162),(124,189,150,163),(125,190,151,164),(126,191,152,165),(127,192,153,166),(128,193,154,167),(129,194,155,168),(130,195,156,169)], [(1,14,27,40),(2,15,28,41),(3,16,29,42),(4,17,30,43),(5,18,31,44),(6,19,32,45),(7,20,33,46),(8,21,34,47),(9,22,35,48),(10,23,36,49),(11,24,37,50),(12,25,38,51),(13,26,39,52),(53,92,79,66),(54,93,80,67),(55,94,81,68),(56,95,82,69),(57,96,83,70),(58,97,84,71),(59,98,85,72),(60,99,86,73),(61,100,87,74),(62,101,88,75),(63,102,89,76),(64,103,90,77),(65,104,91,78),(105,118,131,144),(106,119,132,145),(107,120,133,146),(108,121,134,147),(109,122,135,148),(110,123,136,149),(111,124,137,150),(112,125,138,151),(113,126,139,152),(114,127,140,153),(115,128,141,154),(116,129,142,155),(117,130,143,156),(157,196,183,170),(158,197,184,171),(159,198,185,172),(160,199,186,173),(161,200,187,174),(162,201,188,175),(163,202,189,176),(164,203,190,177),(165,204,191,178),(166,205,192,179),(167,206,193,180),(168,207,194,181),(169,208,195,182)], [(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)], [(1,105,14,118,27,131,40,144),(2,130,15,143,28,156,41,117),(3,155,16,116,29,129,42,142),(4,128,17,141,30,154,43,115),(5,153,18,114,31,127,44,140),(6,126,19,139,32,152,45,113),(7,151,20,112,33,125,46,138),(8,124,21,137,34,150,47,111),(9,149,22,110,35,123,48,136),(10,122,23,135,36,148,49,109),(11,147,24,108,37,121,50,134),(12,120,25,133,38,146,51,107),(13,145,26,106,39,119,52,132),(53,208,66,169,79,182,92,195),(54,181,67,194,80,207,93,168),(55,206,68,167,81,180,94,193),(56,179,69,192,82,205,95,166),(57,204,70,165,83,178,96,191),(58,177,71,190,84,203,97,164),(59,202,72,163,85,176,98,189),(60,175,73,188,86,201,99,162),(61,200,74,161,87,174,100,187),(62,173,75,186,88,199,101,160),(63,198,76,159,89,172,102,185),(64,171,77,184,90,197,103,158),(65,196,78,157,91,170,104,183)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26X | 52A | ··· | 52L | 52M | ··· | 52AD |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D13 | C8○D4 | D26 | Dic13 | Dic13 | D4.Dic13 |
| kernel | D4.Dic13 | C2×C13⋊2C8 | C52.4C4 | C13×C4○D4 | D4×C13 | Q8×C13 | C4○D4 | C13 | C2×C4 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 6 | 4 | 18 | 18 | 6 | 12 |
Matrix representation of D4.Dic13 ►in GL4(𝔽313) generated by
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 288 | 197 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 288 | 0 |
| 0 | 0 | 286 | 25 |
| 0 | 1 | 0 | 0 |
| 312 | 24 | 0 | 0 |
| 0 | 0 | 288 | 0 |
| 0 | 0 | 0 | 288 |
| 104 | 252 | 0 | 0 |
| 244 | 209 | 0 | 0 |
| 0 | 0 | 188 | 0 |
| 0 | 0 | 0 | 188 |
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,288,0,0,0,197,25],[1,0,0,0,0,1,0,0,0,0,288,286,0,0,0,25],[0,312,0,0,1,24,0,0,0,0,288,0,0,0,0,288],[104,244,0,0,252,209,0,0,0,0,188,0,0,0,0,188] >;
D4.Dic13 in GAP, Magma, Sage, TeX
D_4.{\rm Dic}_{13} % in TeX
G:=Group("D4.Dic13"); // GroupNames label
G:=SmallGroup(416,169);
// by ID
G=gap.SmallGroup(416,169);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,188,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^26=a^2,d^2=c^13,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^25>;
// generators/relations