direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C26, C26.12C23, C52.20C22, (C2×C4).3C26, C4.4(C2×C26), (C2×C52).9C2, C22.4(C2×C26), C2.2(C22×C26), (C2×C26).15C22, SmallGroup(208,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C26
G = < a,b,c | a26=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C26
►Regular action on 208 points
Generators in S208
(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 33 53)(2 106 34 54)(3 107 35 55)(4 108 36 56)(5 109 37 57)(6 110 38 58)(7 111 39 59)(8 112 40 60)(9 113 41 61)(10 114 42 62)(11 115 43 63)(12 116 44 64)(13 117 45 65)(14 118 46 66)(15 119 47 67)(16 120 48 68)(17 121 49 69)(18 122 50 70)(19 123 51 71)(20 124 52 72)(21 125 27 73)(22 126 28 74)(23 127 29 75)(24 128 30 76)(25 129 31 77)(26 130 32 78)(79 153 173 197)(80 154 174 198)(81 155 175 199)(82 156 176 200)(83 131 177 201)(84 132 178 202)(85 133 179 203)(86 134 180 204)(87 135 181 205)(88 136 182 206)(89 137 157 207)(90 138 158 208)(91 139 159 183)(92 140 160 184)(93 141 161 185)(94 142 162 186)(95 143 163 187)(96 144 164 188)(97 145 165 189)(98 146 166 190)(99 147 167 191)(100 148 168 192)(101 149 169 193)(102 150 170 194)(103 151 171 195)(104 152 172 196)
(1 93 33 161)(2 94 34 162)(3 95 35 163)(4 96 36 164)(5 97 37 165)(6 98 38 166)(7 99 39 167)(8 100 40 168)(9 101 41 169)(10 102 42 170)(11 103 43 171)(12 104 44 172)(13 79 45 173)(14 80 46 174)(15 81 47 175)(16 82 48 176)(17 83 49 177)(18 84 50 178)(19 85 51 179)(20 86 52 180)(21 87 27 181)(22 88 28 182)(23 89 29 157)(24 90 30 158)(25 91 31 159)(26 92 32 160)(53 141 105 185)(54 142 106 186)(55 143 107 187)(56 144 108 188)(57 145 109 189)(58 146 110 190)(59 147 111 191)(60 148 112 192)(61 149 113 193)(62 150 114 194)(63 151 115 195)(64 152 116 196)(65 153 117 197)(66 154 118 198)(67 155 119 199)(68 156 120 200)(69 131 121 201)(70 132 122 202)(71 133 123 203)(72 134 124 204)(73 135 125 205)(74 136 126 206)(75 137 127 207)(76 138 128 208)(77 139 129 183)(78 140 130 184)
G:=sub<Sym(208)| (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), (1,105,33,53)(2,106,34,54)(3,107,35,55)(4,108,36,56)(5,109,37,57)(6,110,38,58)(7,111,39,59)(8,112,40,60)(9,113,41,61)(10,114,42,62)(11,115,43,63)(12,116,44,64)(13,117,45,65)(14,118,46,66)(15,119,47,67)(16,120,48,68)(17,121,49,69)(18,122,50,70)(19,123,51,71)(20,124,52,72)(21,125,27,73)(22,126,28,74)(23,127,29,75)(24,128,30,76)(25,129,31,77)(26,130,32,78)(79,153,173,197)(80,154,174,198)(81,155,175,199)(82,156,176,200)(83,131,177,201)(84,132,178,202)(85,133,179,203)(86,134,180,204)(87,135,181,205)(88,136,182,206)(89,137,157,207)(90,138,158,208)(91,139,159,183)(92,140,160,184)(93,141,161,185)(94,142,162,186)(95,143,163,187)(96,144,164,188)(97,145,165,189)(98,146,166,190)(99,147,167,191)(100,148,168,192)(101,149,169,193)(102,150,170,194)(103,151,171,195)(104,152,172,196), (1,93,33,161)(2,94,34,162)(3,95,35,163)(4,96,36,164)(5,97,37,165)(6,98,38,166)(7,99,39,167)(8,100,40,168)(9,101,41,169)(10,102,42,170)(11,103,43,171)(12,104,44,172)(13,79,45,173)(14,80,46,174)(15,81,47,175)(16,82,48,176)(17,83,49,177)(18,84,50,178)(19,85,51,179)(20,86,52,180)(21,87,27,181)(22,88,28,182)(23,89,29,157)(24,90,30,158)(25,91,31,159)(26,92,32,160)(53,141,105,185)(54,142,106,186)(55,143,107,187)(56,144,108,188)(57,145,109,189)(58,146,110,190)(59,147,111,191)(60,148,112,192)(61,149,113,193)(62,150,114,194)(63,151,115,195)(64,152,116,196)(65,153,117,197)(66,154,118,198)(67,155,119,199)(68,156,120,200)(69,131,121,201)(70,132,122,202)(71,133,123,203)(72,134,124,204)(73,135,125,205)(74,136,126,206)(75,137,127,207)(76,138,128,208)(77,139,129,183)(78,140,130,184)>;
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), (1,105,33,53)(2,106,34,54)(3,107,35,55)(4,108,36,56)(5,109,37,57)(6,110,38,58)(7,111,39,59)(8,112,40,60)(9,113,41,61)(10,114,42,62)(11,115,43,63)(12,116,44,64)(13,117,45,65)(14,118,46,66)(15,119,47,67)(16,120,48,68)(17,121,49,69)(18,122,50,70)(19,123,51,71)(20,124,52,72)(21,125,27,73)(22,126,28,74)(23,127,29,75)(24,128,30,76)(25,129,31,77)(26,130,32,78)(79,153,173,197)(80,154,174,198)(81,155,175,199)(82,156,176,200)(83,131,177,201)(84,132,178,202)(85,133,179,203)(86,134,180,204)(87,135,181,205)(88,136,182,206)(89,137,157,207)(90,138,158,208)(91,139,159,183)(92,140,160,184)(93,141,161,185)(94,142,162,186)(95,143,163,187)(96,144,164,188)(97,145,165,189)(98,146,166,190)(99,147,167,191)(100,148,168,192)(101,149,169,193)(102,150,170,194)(103,151,171,195)(104,152,172,196), (1,93,33,161)(2,94,34,162)(3,95,35,163)(4,96,36,164)(5,97,37,165)(6,98,38,166)(7,99,39,167)(8,100,40,168)(9,101,41,169)(10,102,42,170)(11,103,43,171)(12,104,44,172)(13,79,45,173)(14,80,46,174)(15,81,47,175)(16,82,48,176)(17,83,49,177)(18,84,50,178)(19,85,51,179)(20,86,52,180)(21,87,27,181)(22,88,28,182)(23,89,29,157)(24,90,30,158)(25,91,31,159)(26,92,32,160)(53,141,105,185)(54,142,106,186)(55,143,107,187)(56,144,108,188)(57,145,109,189)(58,146,110,190)(59,147,111,191)(60,148,112,192)(61,149,113,193)(62,150,114,194)(63,151,115,195)(64,152,116,196)(65,153,117,197)(66,154,118,198)(67,155,119,199)(68,156,120,200)(69,131,121,201)(70,132,122,202)(71,133,123,203)(72,134,124,204)(73,135,125,205)(74,136,126,206)(75,137,127,207)(76,138,128,208)(77,139,129,183)(78,140,130,184) );
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)], [(1,105,33,53),(2,106,34,54),(3,107,35,55),(4,108,36,56),(5,109,37,57),(6,110,38,58),(7,111,39,59),(8,112,40,60),(9,113,41,61),(10,114,42,62),(11,115,43,63),(12,116,44,64),(13,117,45,65),(14,118,46,66),(15,119,47,67),(16,120,48,68),(17,121,49,69),(18,122,50,70),(19,123,51,71),(20,124,52,72),(21,125,27,73),(22,126,28,74),(23,127,29,75),(24,128,30,76),(25,129,31,77),(26,130,32,78),(79,153,173,197),(80,154,174,198),(81,155,175,199),(82,156,176,200),(83,131,177,201),(84,132,178,202),(85,133,179,203),(86,134,180,204),(87,135,181,205),(88,136,182,206),(89,137,157,207),(90,138,158,208),(91,139,159,183),(92,140,160,184),(93,141,161,185),(94,142,162,186),(95,143,163,187),(96,144,164,188),(97,145,165,189),(98,146,166,190),(99,147,167,191),(100,148,168,192),(101,149,169,193),(102,150,170,194),(103,151,171,195),(104,152,172,196)], [(1,93,33,161),(2,94,34,162),(3,95,35,163),(4,96,36,164),(5,97,37,165),(6,98,38,166),(7,99,39,167),(8,100,40,168),(9,101,41,169),(10,102,42,170),(11,103,43,171),(12,104,44,172),(13,79,45,173),(14,80,46,174),(15,81,47,175),(16,82,48,176),(17,83,49,177),(18,84,50,178),(19,85,51,179),(20,86,52,180),(21,87,27,181),(22,88,28,182),(23,89,29,157),(24,90,30,158),(25,91,31,159),(26,92,32,160),(53,141,105,185),(54,142,106,186),(55,143,107,187),(56,144,108,188),(57,145,109,189),(58,146,110,190),(59,147,111,191),(60,148,112,192),(61,149,113,193),(62,150,114,194),(63,151,115,195),(64,152,116,196),(65,153,117,197),(66,154,118,198),(67,155,119,199),(68,156,120,200),(69,131,121,201),(70,132,122,202),(71,133,123,203),(72,134,124,204),(73,135,125,205),(74,136,126,206),(75,137,127,207),(76,138,128,208),(77,139,129,183),(78,140,130,184)]])
Q8×C26 is a maximal subgroup of
Q8⋊Dic13 C52.10D4 Q8.D26 Dic13⋊Q8 D26⋊3Q8 C52.23D4 Q8.10D26
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 13A | ··· | 13L | 26A | ··· | 26AJ | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | Q8 | Q8×C13 |
| kernel | Q8×C26 | C2×C52 | Q8×C13 | C2×Q8 | C2×C4 | Q8 | C26 | C2 |
| # reps | 1 | 3 | 4 | 12 | 36 | 48 | 2 | 24 |
Matrix representation of Q8×C26 ►in GL3(𝔽53) generated by
| 52 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 52 | 0 |
| 52 | 0 | 0 |
| 0 | 50 | 19 |
| 0 | 19 | 3 |
G:=sub<GL(3,GF(53))| [52,0,0,0,4,0,0,0,4],[1,0,0,0,0,52,0,1,0],[52,0,0,0,50,19,0,19,3] >;
Q8×C26 in GAP, Magma, Sage, TeX
Q_8\times C_{26} % in TeX
G:=Group("Q8xC26"); // GroupNames label
G:=SmallGroup(208,47);
// by ID
G=gap.SmallGroup(208,47);
# by ID
G:=PCGroup([5,-2,-2,-2,-13,-2,520,1061,526]);
// Polycyclic
G:=Group<a,b,c|a^26=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export