direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×D13⋊C8, D26⋊3C8, Dic13.9C23, D13⋊(C2×C8), C26⋊1(C2×C8), C13⋊C8⋊4C22, C13⋊1(C22×C8), C52.19(C2×C4), (C2×C52).11C4, (C4×D13).7C4, D26.12(C2×C4), C26.1(C22×C4), (C22×D13).7C4, Dic13.14(C2×C4), (C4×D13).33C22, (C2×Dic13).54C22, (C2×C13⋊C8)⋊6C2, C4.20(C2×C13⋊C4), (C2×C4×D13).17C2, C2.1(C22×C13⋊C4), (C2×C4).11(C13⋊C4), (C2×C26).13(C2×C4), C22.15(C2×C13⋊C4), SmallGroup(416,199)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×D13⋊C8 |
Generators and relations for C2×D13⋊C8
G = < a,b,c,d | a2=b13=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >
Subgroups: 436 in 76 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C13, C2×C8, C22×C4, D13, C26, C26, C22×C8, Dic13, C52, D26, C2×C26, C13⋊C8, C4×D13, C2×Dic13, C2×C52, C22×D13, D13⋊C8, C2×C13⋊C8, C2×C4×D13, C2×D13⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C22×C8, C13⋊C4, C2×C13⋊C4, D13⋊C8, C22×C13⋊C4, C2×D13⋊C8
►On 208 points
Generators in S208
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 90)(28 91)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 169)(114 157)(115 158)(116 159)(117 160)(118 176)(119 177)(120 178)(121 179)(122 180)(123 181)(124 182)(125 170)(126 171)(127 172)(128 173)(129 174)(130 175)(131 188)(132 189)(133 190)(134 191)(135 192)(136 193)(137 194)(138 195)(139 183)(140 184)(141 185)(142 186)(143 187)(144 201)(145 202)(146 203)(147 204)(148 205)(149 206)(150 207)(151 208)(152 196)(153 197)(154 198)(155 199)(156 200)
(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 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 85)(28 84)(29 83)(30 82)(31 81)(32 80)(33 79)(34 91)(35 90)(36 89)(37 88)(38 87)(39 86)(40 92)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(105 165)(106 164)(107 163)(108 162)(109 161)(110 160)(111 159)(112 158)(113 157)(114 169)(115 168)(116 167)(117 166)(118 173)(119 172)(120 171)(121 170)(122 182)(123 181)(124 180)(125 179)(126 178)(127 177)(128 176)(129 175)(130 174)(131 191)(132 190)(133 189)(134 188)(135 187)(136 186)(137 185)(138 184)(139 183)(140 195)(141 194)(142 193)(143 192)(144 205)(145 204)(146 203)(147 202)(148 201)(149 200)(150 199)(151 198)(152 197)(153 196)(154 208)(155 207)(156 206)
(1 153 50 130 14 133 38 114)(2 148 49 122 15 141 37 106)(3 156 48 127 16 136 36 111)(4 151 47 119 17 131 35 116)(5 146 46 124 18 139 34 108)(6 154 45 129 19 134 33 113)(7 149 44 121 20 142 32 105)(8 144 43 126 21 137 31 110)(9 152 42 118 22 132 30 115)(10 147 41 123 23 140 29 107)(11 155 40 128 24 135 28 112)(12 150 52 120 25 143 27 117)(13 145 51 125 26 138 39 109)(53 197 96 175 66 190 88 157)(54 205 95 180 67 185 87 162)(55 200 94 172 68 193 86 167)(56 208 93 177 69 188 85 159)(57 203 92 182 70 183 84 164)(58 198 104 174 71 191 83 169)(59 206 103 179 72 186 82 161)(60 201 102 171 73 194 81 166)(61 196 101 176 74 189 80 158)(62 204 100 181 75 184 79 163)(63 199 99 173 76 192 91 168)(64 207 98 178 77 187 90 160)(65 202 97 170 78 195 89 165)
G:=sub<Sym(208)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,90)(28,91)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168)(113,169)(114,157)(115,158)(116,159)(117,160)(118,176)(119,177)(120,178)(121,179)(122,180)(123,181)(124,182)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,183)(140,184)(141,185)(142,186)(143,187)(144,201)(145,202)(146,203)(147,204)(148,205)(149,206)(150,207)(151,208)(152,196)(153,197)(154,198)(155,199)(156,200), (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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,85)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,92)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(105,165)(106,164)(107,163)(108,162)(109,161)(110,160)(111,159)(112,158)(113,157)(114,169)(115,168)(116,167)(117,166)(118,173)(119,172)(120,171)(121,170)(122,182)(123,181)(124,180)(125,179)(126,178)(127,177)(128,176)(129,175)(130,174)(131,191)(132,190)(133,189)(134,188)(135,187)(136,186)(137,185)(138,184)(139,183)(140,195)(141,194)(142,193)(143,192)(144,205)(145,204)(146,203)(147,202)(148,201)(149,200)(150,199)(151,198)(152,197)(153,196)(154,208)(155,207)(156,206), (1,153,50,130,14,133,38,114)(2,148,49,122,15,141,37,106)(3,156,48,127,16,136,36,111)(4,151,47,119,17,131,35,116)(5,146,46,124,18,139,34,108)(6,154,45,129,19,134,33,113)(7,149,44,121,20,142,32,105)(8,144,43,126,21,137,31,110)(9,152,42,118,22,132,30,115)(10,147,41,123,23,140,29,107)(11,155,40,128,24,135,28,112)(12,150,52,120,25,143,27,117)(13,145,51,125,26,138,39,109)(53,197,96,175,66,190,88,157)(54,205,95,180,67,185,87,162)(55,200,94,172,68,193,86,167)(56,208,93,177,69,188,85,159)(57,203,92,182,70,183,84,164)(58,198,104,174,71,191,83,169)(59,206,103,179,72,186,82,161)(60,201,102,171,73,194,81,166)(61,196,101,176,74,189,80,158)(62,204,100,181,75,184,79,163)(63,199,99,173,76,192,91,168)(64,207,98,178,77,187,90,160)(65,202,97,170,78,195,89,165)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,90)(28,91)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168)(113,169)(114,157)(115,158)(116,159)(117,160)(118,176)(119,177)(120,178)(121,179)(122,180)(123,181)(124,182)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,183)(140,184)(141,185)(142,186)(143,187)(144,201)(145,202)(146,203)(147,204)(148,205)(149,206)(150,207)(151,208)(152,196)(153,197)(154,198)(155,199)(156,200), (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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,85)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,92)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(105,165)(106,164)(107,163)(108,162)(109,161)(110,160)(111,159)(112,158)(113,157)(114,169)(115,168)(116,167)(117,166)(118,173)(119,172)(120,171)(121,170)(122,182)(123,181)(124,180)(125,179)(126,178)(127,177)(128,176)(129,175)(130,174)(131,191)(132,190)(133,189)(134,188)(135,187)(136,186)(137,185)(138,184)(139,183)(140,195)(141,194)(142,193)(143,192)(144,205)(145,204)(146,203)(147,202)(148,201)(149,200)(150,199)(151,198)(152,197)(153,196)(154,208)(155,207)(156,206), (1,153,50,130,14,133,38,114)(2,148,49,122,15,141,37,106)(3,156,48,127,16,136,36,111)(4,151,47,119,17,131,35,116)(5,146,46,124,18,139,34,108)(6,154,45,129,19,134,33,113)(7,149,44,121,20,142,32,105)(8,144,43,126,21,137,31,110)(9,152,42,118,22,132,30,115)(10,147,41,123,23,140,29,107)(11,155,40,128,24,135,28,112)(12,150,52,120,25,143,27,117)(13,145,51,125,26,138,39,109)(53,197,96,175,66,190,88,157)(54,205,95,180,67,185,87,162)(55,200,94,172,68,193,86,167)(56,208,93,177,69,188,85,159)(57,203,92,182,70,183,84,164)(58,198,104,174,71,191,83,169)(59,206,103,179,72,186,82,161)(60,201,102,171,73,194,81,166)(61,196,101,176,74,189,80,158)(62,204,100,181,75,184,79,163)(63,199,99,173,76,192,91,168)(64,207,98,178,77,187,90,160)(65,202,97,170,78,195,89,165) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,90),(28,91),(29,79),(30,80),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(105,161),(106,162),(107,163),(108,164),(109,165),(110,166),(111,167),(112,168),(113,169),(114,157),(115,158),(116,159),(117,160),(118,176),(119,177),(120,178),(121,179),(122,180),(123,181),(124,182),(125,170),(126,171),(127,172),(128,173),(129,174),(130,175),(131,188),(132,189),(133,190),(134,191),(135,192),(136,193),(137,194),(138,195),(139,183),(140,184),(141,185),(142,186),(143,187),(144,201),(145,202),(146,203),(147,204),(148,205),(149,206),(150,207),(151,208),(152,196),(153,197),(154,198),(155,199),(156,200)], [(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)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,85),(28,84),(29,83),(30,82),(31,81),(32,80),(33,79),(34,91),(35,90),(36,89),(37,88),(38,87),(39,86),(40,92),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(105,165),(106,164),(107,163),(108,162),(109,161),(110,160),(111,159),(112,158),(113,157),(114,169),(115,168),(116,167),(117,166),(118,173),(119,172),(120,171),(121,170),(122,182),(123,181),(124,180),(125,179),(126,178),(127,177),(128,176),(129,175),(130,174),(131,191),(132,190),(133,189),(134,188),(135,187),(136,186),(137,185),(138,184),(139,183),(140,195),(141,194),(142,193),(143,192),(144,205),(145,204),(146,203),(147,202),(148,201),(149,200),(150,199),(151,198),(152,197),(153,196),(154,208),(155,207),(156,206)], [(1,153,50,130,14,133,38,114),(2,148,49,122,15,141,37,106),(3,156,48,127,16,136,36,111),(4,151,47,119,17,131,35,116),(5,146,46,124,18,139,34,108),(6,154,45,129,19,134,33,113),(7,149,44,121,20,142,32,105),(8,144,43,126,21,137,31,110),(9,152,42,118,22,132,30,115),(10,147,41,123,23,140,29,107),(11,155,40,128,24,135,28,112),(12,150,52,120,25,143,27,117),(13,145,51,125,26,138,39,109),(53,197,96,175,66,190,88,157),(54,205,95,180,67,185,87,162),(55,200,94,172,68,193,86,167),(56,208,93,177,69,188,85,159),(57,203,92,182,70,183,84,164),(58,198,104,174,71,191,83,169),(59,206,103,179,72,186,82,161),(60,201,102,171,73,194,81,166),(61,196,101,176,74,189,80,158),(62,204,100,181,75,184,79,163),(63,199,99,173,76,192,91,168),(64,207,98,178,77,187,90,160),(65,202,97,170,78,195,89,165)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8P | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | ··· | 13 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C13⋊C4 | C2×C13⋊C4 | C2×C13⋊C4 | D13⋊C8 |
| kernel | C2×D13⋊C8 | D13⋊C8 | C2×C13⋊C8 | C2×C4×D13 | C4×D13 | C2×C52 | C22×D13 | D26 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 16 | 3 | 6 | 3 | 12 |
Matrix representation of C2×D13⋊C8 ►in GL5(𝔽313)
| 312 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 312 | 312 | 312 | 70 |
| 0 | 1 | 0 | 0 | 31 |
| 0 | 0 | 1 | 0 | 282 |
| 0 | 0 | 0 | 1 | 242 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 30 | 62 | 61 |
| 0 | 30 | 312 | 39 | 39 |
| 0 | 210 | 0 | 272 | 273 |
| 0 | 241 | 1 | 211 | 283 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 284 | 225 | 293 | 19 |
| 0 | 70 | 88 | 271 | 271 |
| 0 | 124 | 143 | 44 | 5 |
| 0 | 172 | 0 | 100 | 210 |
G:=sub<GL(5,GF(313))| [312,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,312,1,0,0,0,312,0,1,0,0,312,0,0,1,0,70,31,282,242],[1,0,0,0,0,0,72,30,210,241,0,30,312,0,1,0,62,39,272,211,0,61,39,273,283],[1,0,0,0,0,0,284,70,124,172,0,225,88,143,0,0,293,271,44,100,0,19,271,5,210] >;
C2×D13⋊C8 in GAP, Magma, Sage, TeX
C_2\times D_{13}\rtimes C_8 % in TeX
G:=Group("C2xD13:C8"); // GroupNames label
G:=SmallGroup(416,199);
// by ID
G=gap.SmallGroup(416,199);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,86,69,9221,1751]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^13=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations