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G = C13×2- 1+4order 416 = 25·13

Direct product of C13 and 2- 1+4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C13×2- 1+4, C26.20C24, C52.52C23, C4○D44C26, (C2×Q8)⋊5C26, (Q8×C26)⋊12C2, D4.4(C2×C26), Q8.4(C2×C26), C2.5(C23×C26), (C2×C26).8C23, (C2×C52).71C22, C4.10(C22×C26), (D4×C13).14C22, C22.3(C22×C26), (Q8×C13).15C22, (C13×C4○D4)⋊9C2, (C2×C4).12(C2×C26), SmallGroup(416,232)

Series: Derived Chief Lower central Upper central

C1C2 — C13×2- 1+4
C1C2C26C2×C26D4×C13C13×C4○D4 — C13×2- 1+4
C1C2 — C13×2- 1+4
C1C26 — C13×2- 1+4

Generators and relations for C13×2- 1+4
 G = < a,b,c,d,e | a13=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2 [×5], C4 [×10], C22 [×5], C2×C4 [×15], D4 [×10], Q8 [×10], C13, C2×Q8 [×5], C4○D4 [×10], C26, C26 [×5], 2- 1+4, C52 [×10], C2×C26 [×5], C2×C52 [×15], D4×C13 [×10], Q8×C13 [×10], Q8×C26 [×5], C13×C4○D4 [×10], C13×2- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C13, C24, C26 [×15], 2- 1+4, C2×C26 [×35], C22×C26 [×15], C23×C26, C13×2- 1+4

Smallest permutation representation of C13×2- 1+4
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 73 202 64)(2 74 203 65)(3 75 204 53)(4 76 205 54)(5 77 206 55)(6 78 207 56)(7 66 208 57)(8 67 196 58)(9 68 197 59)(10 69 198 60)(11 70 199 61)(12 71 200 62)(13 72 201 63)(14 107 152 42)(15 108 153 43)(16 109 154 44)(17 110 155 45)(18 111 156 46)(19 112 144 47)(20 113 145 48)(21 114 146 49)(22 115 147 50)(23 116 148 51)(24 117 149 52)(25 105 150 40)(26 106 151 41)(27 95 134 176)(28 96 135 177)(29 97 136 178)(30 98 137 179)(31 99 138 180)(32 100 139 181)(33 101 140 182)(34 102 141 170)(35 103 142 171)(36 104 143 172)(37 92 131 173)(38 93 132 174)(39 94 133 175)(79 129 169 193)(80 130 157 194)(81 118 158 195)(82 119 159 183)(83 120 160 184)(84 121 161 185)(85 122 162 186)(86 123 163 187)(87 124 164 188)(88 125 165 189)(89 126 166 190)(90 127 167 191)(91 128 168 192)
(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 141)(35 142)(36 143)(37 131)(38 132)(39 133)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 75)(54 76)(55 77)(56 78)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)(118 195)(119 183)(120 184)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 193)(130 194)
(1 178 202 97)(2 179 203 98)(3 180 204 99)(4 181 205 100)(5 182 206 101)(6 170 207 102)(7 171 208 103)(8 172 196 104)(9 173 197 92)(10 174 198 93)(11 175 199 94)(12 176 200 95)(13 177 201 96)(14 168 152 91)(15 169 153 79)(16 157 154 80)(17 158 155 81)(18 159 156 82)(19 160 144 83)(20 161 145 84)(21 162 146 85)(22 163 147 86)(23 164 148 87)(24 165 149 88)(25 166 150 89)(26 167 151 90)(27 62 134 71)(28 63 135 72)(29 64 136 73)(30 65 137 74)(31 53 138 75)(32 54 139 76)(33 55 140 77)(34 56 141 78)(35 57 142 66)(36 58 143 67)(37 59 131 68)(38 60 132 69)(39 61 133 70)(40 126 105 190)(41 127 106 191)(42 128 107 192)(43 129 108 193)(44 130 109 194)(45 118 110 195)(46 119 111 183)(47 120 112 184)(48 121 113 185)(49 122 114 186)(50 123 115 187)(51 124 116 188)(52 125 117 189)
(1 83 202 160)(2 84 203 161)(3 85 204 162)(4 86 205 163)(5 87 206 164)(6 88 207 165)(7 89 208 166)(8 90 196 167)(9 91 197 168)(10 79 198 169)(11 80 199 157)(12 81 200 158)(13 82 201 159)(14 173 152 92)(15 174 153 93)(16 175 154 94)(17 176 155 95)(18 177 156 96)(19 178 144 97)(20 179 145 98)(21 180 146 99)(22 181 147 100)(23 182 148 101)(24 170 149 102)(25 171 150 103)(26 172 151 104)(27 45 134 110)(28 46 135 111)(29 47 136 112)(30 48 137 113)(31 49 138 114)(32 50 139 115)(33 51 140 116)(34 52 141 117)(35 40 142 105)(36 41 143 106)(37 42 131 107)(38 43 132 108)(39 44 133 109)(53 186 75 122)(54 187 76 123)(55 188 77 124)(56 189 78 125)(57 190 66 126)(58 191 67 127)(59 192 68 128)(60 193 69 129)(61 194 70 130)(62 195 71 118)(63 183 72 119)(64 184 73 120)(65 185 74 121)

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), (1,73,202,64)(2,74,203,65)(3,75,204,53)(4,76,205,54)(5,77,206,55)(6,78,207,56)(7,66,208,57)(8,67,196,58)(9,68,197,59)(10,69,198,60)(11,70,199,61)(12,71,200,62)(13,72,201,63)(14,107,152,42)(15,108,153,43)(16,109,154,44)(17,110,155,45)(18,111,156,46)(19,112,144,47)(20,113,145,48)(21,114,146,49)(22,115,147,50)(23,116,148,51)(24,117,149,52)(25,105,150,40)(26,106,151,41)(27,95,134,176)(28,96,135,177)(29,97,136,178)(30,98,137,179)(31,99,138,180)(32,100,139,181)(33,101,140,182)(34,102,141,170)(35,103,142,171)(36,104,143,172)(37,92,131,173)(38,93,132,174)(39,94,133,175)(79,129,169,193)(80,130,157,194)(81,118,158,195)(82,119,159,183)(83,120,160,184)(84,121,161,185)(85,122,162,186)(86,123,163,187)(87,124,164,188)(88,125,165,189)(89,126,166,190)(90,127,167,191)(91,128,168,192), (27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,141)(35,142)(36,143)(37,131)(38,132)(39,133)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(118,195)(119,183)(120,184)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194), (1,178,202,97)(2,179,203,98)(3,180,204,99)(4,181,205,100)(5,182,206,101)(6,170,207,102)(7,171,208,103)(8,172,196,104)(9,173,197,92)(10,174,198,93)(11,175,199,94)(12,176,200,95)(13,177,201,96)(14,168,152,91)(15,169,153,79)(16,157,154,80)(17,158,155,81)(18,159,156,82)(19,160,144,83)(20,161,145,84)(21,162,146,85)(22,163,147,86)(23,164,148,87)(24,165,149,88)(25,166,150,89)(26,167,151,90)(27,62,134,71)(28,63,135,72)(29,64,136,73)(30,65,137,74)(31,53,138,75)(32,54,139,76)(33,55,140,77)(34,56,141,78)(35,57,142,66)(36,58,143,67)(37,59,131,68)(38,60,132,69)(39,61,133,70)(40,126,105,190)(41,127,106,191)(42,128,107,192)(43,129,108,193)(44,130,109,194)(45,118,110,195)(46,119,111,183)(47,120,112,184)(48,121,113,185)(49,122,114,186)(50,123,115,187)(51,124,116,188)(52,125,117,189), (1,83,202,160)(2,84,203,161)(3,85,204,162)(4,86,205,163)(5,87,206,164)(6,88,207,165)(7,89,208,166)(8,90,196,167)(9,91,197,168)(10,79,198,169)(11,80,199,157)(12,81,200,158)(13,82,201,159)(14,173,152,92)(15,174,153,93)(16,175,154,94)(17,176,155,95)(18,177,156,96)(19,178,144,97)(20,179,145,98)(21,180,146,99)(22,181,147,100)(23,182,148,101)(24,170,149,102)(25,171,150,103)(26,172,151,104)(27,45,134,110)(28,46,135,111)(29,47,136,112)(30,48,137,113)(31,49,138,114)(32,50,139,115)(33,51,140,116)(34,52,141,117)(35,40,142,105)(36,41,143,106)(37,42,131,107)(38,43,132,108)(39,44,133,109)(53,186,75,122)(54,187,76,123)(55,188,77,124)(56,189,78,125)(57,190,66,126)(58,191,67,127)(59,192,68,128)(60,193,69,129)(61,194,70,130)(62,195,71,118)(63,183,72,119)(64,184,73,120)(65,185,74,121)>;

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), (1,73,202,64)(2,74,203,65)(3,75,204,53)(4,76,205,54)(5,77,206,55)(6,78,207,56)(7,66,208,57)(8,67,196,58)(9,68,197,59)(10,69,198,60)(11,70,199,61)(12,71,200,62)(13,72,201,63)(14,107,152,42)(15,108,153,43)(16,109,154,44)(17,110,155,45)(18,111,156,46)(19,112,144,47)(20,113,145,48)(21,114,146,49)(22,115,147,50)(23,116,148,51)(24,117,149,52)(25,105,150,40)(26,106,151,41)(27,95,134,176)(28,96,135,177)(29,97,136,178)(30,98,137,179)(31,99,138,180)(32,100,139,181)(33,101,140,182)(34,102,141,170)(35,103,142,171)(36,104,143,172)(37,92,131,173)(38,93,132,174)(39,94,133,175)(79,129,169,193)(80,130,157,194)(81,118,158,195)(82,119,159,183)(83,120,160,184)(84,121,161,185)(85,122,162,186)(86,123,163,187)(87,124,164,188)(88,125,165,189)(89,126,166,190)(90,127,167,191)(91,128,168,192), (27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,141)(35,142)(36,143)(37,131)(38,132)(39,133)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(118,195)(119,183)(120,184)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194), (1,178,202,97)(2,179,203,98)(3,180,204,99)(4,181,205,100)(5,182,206,101)(6,170,207,102)(7,171,208,103)(8,172,196,104)(9,173,197,92)(10,174,198,93)(11,175,199,94)(12,176,200,95)(13,177,201,96)(14,168,152,91)(15,169,153,79)(16,157,154,80)(17,158,155,81)(18,159,156,82)(19,160,144,83)(20,161,145,84)(21,162,146,85)(22,163,147,86)(23,164,148,87)(24,165,149,88)(25,166,150,89)(26,167,151,90)(27,62,134,71)(28,63,135,72)(29,64,136,73)(30,65,137,74)(31,53,138,75)(32,54,139,76)(33,55,140,77)(34,56,141,78)(35,57,142,66)(36,58,143,67)(37,59,131,68)(38,60,132,69)(39,61,133,70)(40,126,105,190)(41,127,106,191)(42,128,107,192)(43,129,108,193)(44,130,109,194)(45,118,110,195)(46,119,111,183)(47,120,112,184)(48,121,113,185)(49,122,114,186)(50,123,115,187)(51,124,116,188)(52,125,117,189), (1,83,202,160)(2,84,203,161)(3,85,204,162)(4,86,205,163)(5,87,206,164)(6,88,207,165)(7,89,208,166)(8,90,196,167)(9,91,197,168)(10,79,198,169)(11,80,199,157)(12,81,200,158)(13,82,201,159)(14,173,152,92)(15,174,153,93)(16,175,154,94)(17,176,155,95)(18,177,156,96)(19,178,144,97)(20,179,145,98)(21,180,146,99)(22,181,147,100)(23,182,148,101)(24,170,149,102)(25,171,150,103)(26,172,151,104)(27,45,134,110)(28,46,135,111)(29,47,136,112)(30,48,137,113)(31,49,138,114)(32,50,139,115)(33,51,140,116)(34,52,141,117)(35,40,142,105)(36,41,143,106)(37,42,131,107)(38,43,132,108)(39,44,133,109)(53,186,75,122)(54,187,76,123)(55,188,77,124)(56,189,78,125)(57,190,66,126)(58,191,67,127)(59,192,68,128)(60,193,69,129)(61,194,70,130)(62,195,71,118)(63,183,72,119)(64,184,73,120)(65,185,74,121) );

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)], [(1,73,202,64),(2,74,203,65),(3,75,204,53),(4,76,205,54),(5,77,206,55),(6,78,207,56),(7,66,208,57),(8,67,196,58),(9,68,197,59),(10,69,198,60),(11,70,199,61),(12,71,200,62),(13,72,201,63),(14,107,152,42),(15,108,153,43),(16,109,154,44),(17,110,155,45),(18,111,156,46),(19,112,144,47),(20,113,145,48),(21,114,146,49),(22,115,147,50),(23,116,148,51),(24,117,149,52),(25,105,150,40),(26,106,151,41),(27,95,134,176),(28,96,135,177),(29,97,136,178),(30,98,137,179),(31,99,138,180),(32,100,139,181),(33,101,140,182),(34,102,141,170),(35,103,142,171),(36,104,143,172),(37,92,131,173),(38,93,132,174),(39,94,133,175),(79,129,169,193),(80,130,157,194),(81,118,158,195),(82,119,159,183),(83,120,160,184),(84,121,161,185),(85,122,162,186),(86,123,163,187),(87,124,164,188),(88,125,165,189),(89,126,166,190),(90,127,167,191),(91,128,168,192)], [(27,134),(28,135),(29,136),(30,137),(31,138),(32,139),(33,140),(34,141),(35,142),(36,143),(37,131),(38,132),(39,133),(40,105),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,113),(49,114),(50,115),(51,116),(52,117),(53,75),(54,76),(55,77),(56,78),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,73),(65,74),(118,195),(119,183),(120,184),(121,185),(122,186),(123,187),(124,188),(125,189),(126,190),(127,191),(128,192),(129,193),(130,194)], [(1,178,202,97),(2,179,203,98),(3,180,204,99),(4,181,205,100),(5,182,206,101),(6,170,207,102),(7,171,208,103),(8,172,196,104),(9,173,197,92),(10,174,198,93),(11,175,199,94),(12,176,200,95),(13,177,201,96),(14,168,152,91),(15,169,153,79),(16,157,154,80),(17,158,155,81),(18,159,156,82),(19,160,144,83),(20,161,145,84),(21,162,146,85),(22,163,147,86),(23,164,148,87),(24,165,149,88),(25,166,150,89),(26,167,151,90),(27,62,134,71),(28,63,135,72),(29,64,136,73),(30,65,137,74),(31,53,138,75),(32,54,139,76),(33,55,140,77),(34,56,141,78),(35,57,142,66),(36,58,143,67),(37,59,131,68),(38,60,132,69),(39,61,133,70),(40,126,105,190),(41,127,106,191),(42,128,107,192),(43,129,108,193),(44,130,109,194),(45,118,110,195),(46,119,111,183),(47,120,112,184),(48,121,113,185),(49,122,114,186),(50,123,115,187),(51,124,116,188),(52,125,117,189)], [(1,83,202,160),(2,84,203,161),(3,85,204,162),(4,86,205,163),(5,87,206,164),(6,88,207,165),(7,89,208,166),(8,90,196,167),(9,91,197,168),(10,79,198,169),(11,80,199,157),(12,81,200,158),(13,82,201,159),(14,173,152,92),(15,174,153,93),(16,175,154,94),(17,176,155,95),(18,177,156,96),(19,178,144,97),(20,179,145,98),(21,180,146,99),(22,181,147,100),(23,182,148,101),(24,170,149,102),(25,171,150,103),(26,172,151,104),(27,45,134,110),(28,46,135,111),(29,47,136,112),(30,48,137,113),(31,49,138,114),(32,50,139,115),(33,51,140,116),(34,52,141,117),(35,40,142,105),(36,41,143,106),(37,42,131,107),(38,43,132,108),(39,44,133,109),(53,186,75,122),(54,187,76,123),(55,188,77,124),(56,189,78,125),(57,190,66,126),(58,191,67,127),(59,192,68,128),(60,193,69,129),(61,194,70,130),(62,195,71,118),(63,183,72,119),(64,184,73,120),(65,185,74,121)])

221 conjugacy classes

class 1 2A2B···2F4A···4J13A···13L26A···26L26M···26BT52A···52DP
order122···24···413···1326···2626···2652···52
size112···22···21···11···12···22···2

221 irreducible representations

dim11111144
type+++-
imageC1C2C2C13C26C262- 1+4C13×2- 1+4
kernelC13×2- 1+4Q8×C26C13×C4○D42- 1+4C2×Q8C4○D4C13C1
# reps15101260120112

Matrix representation of C13×2- 1+4 in GL4(𝔽53) generated by

16000
01600
00160
00016
,
45394836
1625365
10201052
22302926
,
1000
0100
438520
249052
,
232700
03000
9233026
013023
,
313100
102200
26482222
36364331
G:=sub<GL(4,GF(53))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[45,16,10,22,39,25,20,30,48,36,10,29,36,5,52,26],[1,0,43,2,0,1,8,49,0,0,52,0,0,0,0,52],[23,0,9,0,27,30,23,13,0,0,30,0,0,0,26,23],[31,10,26,36,31,22,48,36,0,0,22,43,0,0,22,31] >;

C13×2- 1+4 in GAP, Magma, Sage, TeX

C_{13}\times 2_-^{1+4}
% in TeX

G:=Group("C13xES-(2,2)");
// GroupNames label

G:=SmallGroup(416,232);
// by ID

G=gap.SmallGroup(416,232);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-13,-2,2521,1255,1916,950,5187]);
// Polycyclic

G:=Group<a,b,c,d,e|a^13=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations

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