metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.10D26, C26.10C24, C52.24C23, D26.5C23, C13⋊12- 1+4, D52.13C22, Dic13.6C23, Dic26.13C22, (C2×Q8)⋊5D13, (Q8×C26)⋊7C2, (Q8×D13)⋊4C2, (C2×C4).24D26, D52⋊C2⋊4C2, D52⋊5C2⋊6C2, (C2×C52).48C22, (C2×C26).68C23, (C4×D13).5C22, C4.24(C22×D13), C2.11(C23×D13), C13⋊D4.2C22, (Q8×C13).10C22, C22.7(C22×D13), SmallGroup(416,221)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.10D26
G = < a,b,c,d | a4=1, b2=c26=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c25 >
Subgroups: 864 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, Q8, C13, C2×Q8, C2×Q8, C4○D4, D13, C26, C26, 2- 1+4, Dic13, C52, D26, C2×C26, Dic26, C4×D13, D52, C13⋊D4, C2×C52, Q8×C13, D52⋊5C2, Q8×D13, D52⋊C2, Q8×C26, Q8.10D26
Quotients: C1, C2, C22, C23, C24, D13, 2- 1+4, D26, C22×D13, C23×D13, Q8.10D26
►On 208 points
Generators in S208
(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 66 79 92)(54 67 80 93)(55 68 81 94)(56 69 82 95)(57 70 83 96)(58 71 84 97)(59 72 85 98)(60 73 86 99)(61 74 87 100)(62 75 88 101)(63 76 89 102)(64 77 90 103)(65 78 91 104)(105 144 131 118)(106 145 132 119)(107 146 133 120)(108 147 134 121)(109 148 135 122)(110 149 136 123)(111 150 137 124)(112 151 138 125)(113 152 139 126)(114 153 140 127)(115 154 141 128)(116 155 142 129)(117 156 143 130)(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)
(1 103 27 77)(2 78 28 104)(3 53 29 79)(4 80 30 54)(5 55 31 81)(6 82 32 56)(7 57 33 83)(8 84 34 58)(9 59 35 85)(10 86 36 60)(11 61 37 87)(12 88 38 62)(13 63 39 89)(14 90 40 64)(15 65 41 91)(16 92 42 66)(17 67 43 93)(18 94 44 68)(19 69 45 95)(20 96 46 70)(21 71 47 97)(22 98 48 72)(23 73 49 99)(24 100 50 74)(25 75 51 101)(26 102 52 76)(105 191 131 165)(106 166 132 192)(107 193 133 167)(108 168 134 194)(109 195 135 169)(110 170 136 196)(111 197 137 171)(112 172 138 198)(113 199 139 173)(114 174 140 200)(115 201 141 175)(116 176 142 202)(117 203 143 177)(118 178 144 204)(119 205 145 179)(120 180 146 206)(121 207 147 181)(122 182 148 208)(123 157 149 183)(124 184 150 158)(125 159 151 185)(126 186 152 160)(127 161 153 187)(128 188 154 162)(129 163 155 189)(130 190 156 164)
(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)
(1 109 27 135)(2 134 28 108)(3 107 29 133)(4 132 30 106)(5 105 31 131)(6 130 32 156)(7 155 33 129)(8 128 34 154)(9 153 35 127)(10 126 36 152)(11 151 37 125)(12 124 38 150)(13 149 39 123)(14 122 40 148)(15 147 41 121)(16 120 42 146)(17 145 43 119)(18 118 44 144)(19 143 45 117)(20 116 46 142)(21 141 47 115)(22 114 48 140)(23 139 49 113)(24 112 50 138)(25 137 51 111)(26 110 52 136)(53 167 79 193)(54 192 80 166)(55 165 81 191)(56 190 82 164)(57 163 83 189)(58 188 84 162)(59 161 85 187)(60 186 86 160)(61 159 87 185)(62 184 88 158)(63 157 89 183)(64 182 90 208)(65 207 91 181)(66 180 92 206)(67 205 93 179)(68 178 94 204)(69 203 95 177)(70 176 96 202)(71 201 97 175)(72 174 98 200)(73 199 99 173)(74 172 100 198)(75 197 101 171)(76 170 102 196)(77 195 103 169)(78 168 104 194)
G:=sub<Sym(208)| (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)(53,66,79,92)(54,67,80,93)(55,68,81,94)(56,69,82,95)(57,70,83,96)(58,71,84,97)(59,72,85,98)(60,73,86,99)(61,74,87,100)(62,75,88,101)(63,76,89,102)(64,77,90,103)(65,78,91,104)(105,144,131,118)(106,145,132,119)(107,146,133,120)(108,147,134,121)(109,148,135,122)(110,149,136,123)(111,150,137,124)(112,151,138,125)(113,152,139,126)(114,153,140,127)(115,154,141,128)(116,155,142,129)(117,156,143,130)(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), (1,103,27,77)(2,78,28,104)(3,53,29,79)(4,80,30,54)(5,55,31,81)(6,82,32,56)(7,57,33,83)(8,84,34,58)(9,59,35,85)(10,86,36,60)(11,61,37,87)(12,88,38,62)(13,63,39,89)(14,90,40,64)(15,65,41,91)(16,92,42,66)(17,67,43,93)(18,94,44,68)(19,69,45,95)(20,96,46,70)(21,71,47,97)(22,98,48,72)(23,73,49,99)(24,100,50,74)(25,75,51,101)(26,102,52,76)(105,191,131,165)(106,166,132,192)(107,193,133,167)(108,168,134,194)(109,195,135,169)(110,170,136,196)(111,197,137,171)(112,172,138,198)(113,199,139,173)(114,174,140,200)(115,201,141,175)(116,176,142,202)(117,203,143,177)(118,178,144,204)(119,205,145,179)(120,180,146,206)(121,207,147,181)(122,182,148,208)(123,157,149,183)(124,184,150,158)(125,159,151,185)(126,186,152,160)(127,161,153,187)(128,188,154,162)(129,163,155,189)(130,190,156,164), (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), (1,109,27,135)(2,134,28,108)(3,107,29,133)(4,132,30,106)(5,105,31,131)(6,130,32,156)(7,155,33,129)(8,128,34,154)(9,153,35,127)(10,126,36,152)(11,151,37,125)(12,124,38,150)(13,149,39,123)(14,122,40,148)(15,147,41,121)(16,120,42,146)(17,145,43,119)(18,118,44,144)(19,143,45,117)(20,116,46,142)(21,141,47,115)(22,114,48,140)(23,139,49,113)(24,112,50,138)(25,137,51,111)(26,110,52,136)(53,167,79,193)(54,192,80,166)(55,165,81,191)(56,190,82,164)(57,163,83,189)(58,188,84,162)(59,161,85,187)(60,186,86,160)(61,159,87,185)(62,184,88,158)(63,157,89,183)(64,182,90,208)(65,207,91,181)(66,180,92,206)(67,205,93,179)(68,178,94,204)(69,203,95,177)(70,176,96,202)(71,201,97,175)(72,174,98,200)(73,199,99,173)(74,172,100,198)(75,197,101,171)(76,170,102,196)(77,195,103,169)(78,168,104,194)>;
G:=Group( (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)(53,66,79,92)(54,67,80,93)(55,68,81,94)(56,69,82,95)(57,70,83,96)(58,71,84,97)(59,72,85,98)(60,73,86,99)(61,74,87,100)(62,75,88,101)(63,76,89,102)(64,77,90,103)(65,78,91,104)(105,144,131,118)(106,145,132,119)(107,146,133,120)(108,147,134,121)(109,148,135,122)(110,149,136,123)(111,150,137,124)(112,151,138,125)(113,152,139,126)(114,153,140,127)(115,154,141,128)(116,155,142,129)(117,156,143,130)(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), (1,103,27,77)(2,78,28,104)(3,53,29,79)(4,80,30,54)(5,55,31,81)(6,82,32,56)(7,57,33,83)(8,84,34,58)(9,59,35,85)(10,86,36,60)(11,61,37,87)(12,88,38,62)(13,63,39,89)(14,90,40,64)(15,65,41,91)(16,92,42,66)(17,67,43,93)(18,94,44,68)(19,69,45,95)(20,96,46,70)(21,71,47,97)(22,98,48,72)(23,73,49,99)(24,100,50,74)(25,75,51,101)(26,102,52,76)(105,191,131,165)(106,166,132,192)(107,193,133,167)(108,168,134,194)(109,195,135,169)(110,170,136,196)(111,197,137,171)(112,172,138,198)(113,199,139,173)(114,174,140,200)(115,201,141,175)(116,176,142,202)(117,203,143,177)(118,178,144,204)(119,205,145,179)(120,180,146,206)(121,207,147,181)(122,182,148,208)(123,157,149,183)(124,184,150,158)(125,159,151,185)(126,186,152,160)(127,161,153,187)(128,188,154,162)(129,163,155,189)(130,190,156,164), (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), (1,109,27,135)(2,134,28,108)(3,107,29,133)(4,132,30,106)(5,105,31,131)(6,130,32,156)(7,155,33,129)(8,128,34,154)(9,153,35,127)(10,126,36,152)(11,151,37,125)(12,124,38,150)(13,149,39,123)(14,122,40,148)(15,147,41,121)(16,120,42,146)(17,145,43,119)(18,118,44,144)(19,143,45,117)(20,116,46,142)(21,141,47,115)(22,114,48,140)(23,139,49,113)(24,112,50,138)(25,137,51,111)(26,110,52,136)(53,167,79,193)(54,192,80,166)(55,165,81,191)(56,190,82,164)(57,163,83,189)(58,188,84,162)(59,161,85,187)(60,186,86,160)(61,159,87,185)(62,184,88,158)(63,157,89,183)(64,182,90,208)(65,207,91,181)(66,180,92,206)(67,205,93,179)(68,178,94,204)(69,203,95,177)(70,176,96,202)(71,201,97,175)(72,174,98,200)(73,199,99,173)(74,172,100,198)(75,197,101,171)(76,170,102,196)(77,195,103,169)(78,168,104,194) );
G=PermutationGroup([[(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),(53,66,79,92),(54,67,80,93),(55,68,81,94),(56,69,82,95),(57,70,83,96),(58,71,84,97),(59,72,85,98),(60,73,86,99),(61,74,87,100),(62,75,88,101),(63,76,89,102),(64,77,90,103),(65,78,91,104),(105,144,131,118),(106,145,132,119),(107,146,133,120),(108,147,134,121),(109,148,135,122),(110,149,136,123),(111,150,137,124),(112,151,138,125),(113,152,139,126),(114,153,140,127),(115,154,141,128),(116,155,142,129),(117,156,143,130),(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)], [(1,103,27,77),(2,78,28,104),(3,53,29,79),(4,80,30,54),(5,55,31,81),(6,82,32,56),(7,57,33,83),(8,84,34,58),(9,59,35,85),(10,86,36,60),(11,61,37,87),(12,88,38,62),(13,63,39,89),(14,90,40,64),(15,65,41,91),(16,92,42,66),(17,67,43,93),(18,94,44,68),(19,69,45,95),(20,96,46,70),(21,71,47,97),(22,98,48,72),(23,73,49,99),(24,100,50,74),(25,75,51,101),(26,102,52,76),(105,191,131,165),(106,166,132,192),(107,193,133,167),(108,168,134,194),(109,195,135,169),(110,170,136,196),(111,197,137,171),(112,172,138,198),(113,199,139,173),(114,174,140,200),(115,201,141,175),(116,176,142,202),(117,203,143,177),(118,178,144,204),(119,205,145,179),(120,180,146,206),(121,207,147,181),(122,182,148,208),(123,157,149,183),(124,184,150,158),(125,159,151,185),(126,186,152,160),(127,161,153,187),(128,188,154,162),(129,163,155,189),(130,190,156,164)], [(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)], [(1,109,27,135),(2,134,28,108),(3,107,29,133),(4,132,30,106),(5,105,31,131),(6,130,32,156),(7,155,33,129),(8,128,34,154),(9,153,35,127),(10,126,36,152),(11,151,37,125),(12,124,38,150),(13,149,39,123),(14,122,40,148),(15,147,41,121),(16,120,42,146),(17,145,43,119),(18,118,44,144),(19,143,45,117),(20,116,46,142),(21,141,47,115),(22,114,48,140),(23,139,49,113),(24,112,50,138),(25,137,51,111),(26,110,52,136),(53,167,79,193),(54,192,80,166),(55,165,81,191),(56,190,82,164),(57,163,83,189),(58,188,84,162),(59,161,85,187),(60,186,86,160),(61,159,87,185),(62,184,88,158),(63,157,89,183),(64,182,90,208),(65,207,91,181),(66,180,92,206),(67,205,93,179),(68,178,94,204),(69,203,95,177),(70,176,96,202),(71,201,97,175),(72,174,98,200),(73,199,99,173),(74,172,100,198),(75,197,101,171),(76,170,102,196),(77,195,103,169),(78,168,104,194)]])
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | D13 | D26 | D26 | 2- 1+4 | Q8.10D26 |
| kernel | Q8.10D26 | D52⋊5C2 | Q8×D13 | D52⋊C2 | Q8×C26 | C2×Q8 | C2×C4 | Q8 | C13 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 6 | 18 | 24 | 1 | 12 |
Matrix representation of Q8.10D26 ►in GL4(𝔽53) generated by
| 1 | 12 | 44 | 0 |
| 44 | 52 | 14 | 39 |
| 0 | 0 | 0 | 52 |
| 0 | 0 | 1 | 0 |
| 8 | 45 | 0 | 47 |
| 28 | 45 | 33 | 20 |
| 0 | 0 | 17 | 9 |
| 0 | 0 | 9 | 36 |
| 42 | 27 | 25 | 32 |
| 46 | 11 | 5 | 48 |
| 0 | 0 | 0 | 24 |
| 0 | 0 | 29 | 0 |
| 5 | 7 | 42 | 34 |
| 0 | 0 | 29 | 0 |
| 0 | 42 | 0 | 0 |
| 46 | 11 | 5 | 48 |
G:=sub<GL(4,GF(53))| [1,44,0,0,12,52,0,0,44,14,0,1,0,39,52,0],[8,28,0,0,45,45,0,0,0,33,17,9,47,20,9,36],[42,46,0,0,27,11,0,0,25,5,0,29,32,48,24,0],[5,0,0,46,7,0,42,11,42,29,0,5,34,0,0,48] >;
Q8.10D26 in GAP, Magma, Sage, TeX
Q_8._{10}D_{26} % in TeX
G:=Group("Q8.10D26"); // GroupNames label
G:=SmallGroup(416,221);
// by ID
G=gap.SmallGroup(416,221);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,188,86,579,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^26=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^25>;
// generators/relations