metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.Dic13, D4.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), C26.40(C22×C4), (C2×C52).41C22, 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 Q8.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 [×3], C4, C4 [×3], C22 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C13, C2×C8 [×3], M4(2) [×3], C4○D4, C26, C26 [×3], C8○D4, C52, C52 [×3], C2×C26 [×3], C13⋊2C8, C13⋊2C8 [×3], C2×C52 [×3], D4×C13 [×3], Q8×C13, C2×C13⋊2C8 [×3], C52.4C4 [×3], C13×C4○D4, Q8.Dic13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, D13, C8○D4, Dic13 [×4], D26 [×3], C2×Dic13 [×6], C22×D13, C22×Dic13, Q8.Dic13
►On 208 points
Generators in S208
(1 61 27 87)(2 62 28 88)(3 63 29 89)(4 64 30 90)(5 65 31 91)(6 66 32 92)(7 67 33 93)(8 68 34 94)(9 69 35 95)(10 70 36 96)(11 71 37 97)(12 72 38 98)(13 73 39 99)(14 74 40 100)(15 75 41 101)(16 76 42 102)(17 77 43 103)(18 78 44 104)(19 79 45 53)(20 80 46 54)(21 81 47 55)(22 82 48 56)(23 83 49 57)(24 84 50 58)(25 85 51 59)(26 86 52 60)(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)
(1 144 14 105 27 118 40 131)(2 117 15 130 28 143 41 156)(3 142 16 155 29 116 42 129)(4 115 17 128 30 141 43 154)(5 140 18 153 31 114 44 127)(6 113 19 126 32 139 45 152)(7 138 20 151 33 112 46 125)(8 111 21 124 34 137 47 150)(9 136 22 149 35 110 48 123)(10 109 23 122 36 135 49 148)(11 134 24 147 37 108 50 121)(12 107 25 120 38 133 51 146)(13 132 26 145 39 106 52 119)(53 165 66 178 79 191 92 204)(54 190 67 203 80 164 93 177)(55 163 68 176 81 189 94 202)(56 188 69 201 82 162 95 175)(57 161 70 174 83 187 96 200)(58 186 71 199 84 160 97 173)(59 159 72 172 85 185 98 198)(60 184 73 197 86 158 99 171)(61 157 74 170 87 183 100 196)(62 182 75 195 88 208 101 169)(63 207 76 168 89 181 102 194)(64 180 77 193 90 206 103 167)(65 205 78 166 91 179 104 192)
G:=sub<Sym(208)| (1,61,27,87)(2,62,28,88)(3,63,29,89)(4,64,30,90)(5,65,31,91)(6,66,32,92)(7,67,33,93)(8,68,34,94)(9,69,35,95)(10,70,36,96)(11,71,37,97)(12,72,38,98)(13,73,39,99)(14,74,40,100)(15,75,41,101)(16,76,42,102)(17,77,43,103)(18,78,44,104)(19,79,45,53)(20,80,46,54)(21,81,47,55)(22,82,48,56)(23,83,49,57)(24,84,50,58)(25,85,51,59)(26,86,52,60)(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), (1,144,14,105,27,118,40,131)(2,117,15,130,28,143,41,156)(3,142,16,155,29,116,42,129)(4,115,17,128,30,141,43,154)(5,140,18,153,31,114,44,127)(6,113,19,126,32,139,45,152)(7,138,20,151,33,112,46,125)(8,111,21,124,34,137,47,150)(9,136,22,149,35,110,48,123)(10,109,23,122,36,135,49,148)(11,134,24,147,37,108,50,121)(12,107,25,120,38,133,51,146)(13,132,26,145,39,106,52,119)(53,165,66,178,79,191,92,204)(54,190,67,203,80,164,93,177)(55,163,68,176,81,189,94,202)(56,188,69,201,82,162,95,175)(57,161,70,174,83,187,96,200)(58,186,71,199,84,160,97,173)(59,159,72,172,85,185,98,198)(60,184,73,197,86,158,99,171)(61,157,74,170,87,183,100,196)(62,182,75,195,88,208,101,169)(63,207,76,168,89,181,102,194)(64,180,77,193,90,206,103,167)(65,205,78,166,91,179,104,192)>;
G:=Group( (1,61,27,87)(2,62,28,88)(3,63,29,89)(4,64,30,90)(5,65,31,91)(6,66,32,92)(7,67,33,93)(8,68,34,94)(9,69,35,95)(10,70,36,96)(11,71,37,97)(12,72,38,98)(13,73,39,99)(14,74,40,100)(15,75,41,101)(16,76,42,102)(17,77,43,103)(18,78,44,104)(19,79,45,53)(20,80,46,54)(21,81,47,55)(22,82,48,56)(23,83,49,57)(24,84,50,58)(25,85,51,59)(26,86,52,60)(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), (1,144,14,105,27,118,40,131)(2,117,15,130,28,143,41,156)(3,142,16,155,29,116,42,129)(4,115,17,128,30,141,43,154)(5,140,18,153,31,114,44,127)(6,113,19,126,32,139,45,152)(7,138,20,151,33,112,46,125)(8,111,21,124,34,137,47,150)(9,136,22,149,35,110,48,123)(10,109,23,122,36,135,49,148)(11,134,24,147,37,108,50,121)(12,107,25,120,38,133,51,146)(13,132,26,145,39,106,52,119)(53,165,66,178,79,191,92,204)(54,190,67,203,80,164,93,177)(55,163,68,176,81,189,94,202)(56,188,69,201,82,162,95,175)(57,161,70,174,83,187,96,200)(58,186,71,199,84,160,97,173)(59,159,72,172,85,185,98,198)(60,184,73,197,86,158,99,171)(61,157,74,170,87,183,100,196)(62,182,75,195,88,208,101,169)(63,207,76,168,89,181,102,194)(64,180,77,193,90,206,103,167)(65,205,78,166,91,179,104,192) );
G=PermutationGroup([(1,61,27,87),(2,62,28,88),(3,63,29,89),(4,64,30,90),(5,65,31,91),(6,66,32,92),(7,67,33,93),(8,68,34,94),(9,69,35,95),(10,70,36,96),(11,71,37,97),(12,72,38,98),(13,73,39,99),(14,74,40,100),(15,75,41,101),(16,76,42,102),(17,77,43,103),(18,78,44,104),(19,79,45,53),(20,80,46,54),(21,81,47,55),(22,82,48,56),(23,83,49,57),(24,84,50,58),(25,85,51,59),(26,86,52,60),(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)], [(1,144,14,105,27,118,40,131),(2,117,15,130,28,143,41,156),(3,142,16,155,29,116,42,129),(4,115,17,128,30,141,43,154),(5,140,18,153,31,114,44,127),(6,113,19,126,32,139,45,152),(7,138,20,151,33,112,46,125),(8,111,21,124,34,137,47,150),(9,136,22,149,35,110,48,123),(10,109,23,122,36,135,49,148),(11,134,24,147,37,108,50,121),(12,107,25,120,38,133,51,146),(13,132,26,145,39,106,52,119),(53,165,66,178,79,191,92,204),(54,190,67,203,80,164,93,177),(55,163,68,176,81,189,94,202),(56,188,69,201,82,162,95,175),(57,161,70,174,83,187,96,200),(58,186,71,199,84,160,97,173),(59,159,72,172,85,185,98,198),(60,184,73,197,86,158,99,171),(61,157,74,170,87,183,100,196),(62,182,75,195,88,208,101,169),(63,207,76,168,89,181,102,194),(64,180,77,193,90,206,103,167),(65,205,78,166,91,179,104,192)])
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 | Q8.Dic13 |
| kernel | Q8.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 Q8.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] >;
Q8.Dic13 in GAP, Magma, Sage, TeX
Q_8.{\rm Dic}_{13} % in TeX
G:=Group("Q8.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