direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C42⋊C2, C42⋊1C26, C4⋊C4⋊6C26, (C4×C52)⋊2C2, (C2×C4)⋊4C52, (C2×C52)⋊14C4, C4.9(C2×C52), C52.67(C2×C4), C22⋊C4.3C26, C2.3(C22×C52), C23.6(C2×C26), (C22×C4).4C26, C22.5(C2×C52), C26.38(C4○D4), (C22×C52).14C2, (C2×C52).79C22, (C2×C26).72C23, C26.44(C22×C4), C22.6(C22×C26), (C22×C26).25C22, (C13×C4⋊C4)⋊15C2, C2.1(C13×C4○D4), (C2×C4).14(C2×C26), (C2×C26).42(C2×C4), (C13×C22⋊C4).6C2, SmallGroup(416,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C42⋊C2
G = < a,b,c,d | a13=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, C26, C26, C26, C42⋊C2, C52, C52, C2×C26, C2×C26, C2×C26, C2×C52, C2×C52, C22×C26, C4×C52, C13×C22⋊C4, C13×C4⋊C4, C22×C52, C13×C42⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C13, C22×C4, C4○D4, C26, C42⋊C2, C52, C2×C26, C2×C52, C22×C26, C22×C52, C13×C4○D4, C13×C42⋊C2
►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)
(1 173 121 86)(2 174 122 87)(3 175 123 88)(4 176 124 89)(5 177 125 90)(6 178 126 91)(7 179 127 79)(8 180 128 80)(9 181 129 81)(10 182 130 82)(11 170 118 83)(12 171 119 84)(13 172 120 85)(14 165 192 60)(15 166 193 61)(16 167 194 62)(17 168 195 63)(18 169 183 64)(19 157 184 65)(20 158 185 53)(21 159 186 54)(22 160 187 55)(23 161 188 56)(24 162 189 57)(25 163 190 58)(26 164 191 59)(27 156 71 102)(28 144 72 103)(29 145 73 104)(30 146 74 92)(31 147 75 93)(32 148 76 94)(33 149 77 95)(34 150 78 96)(35 151 66 97)(36 152 67 98)(37 153 68 99)(38 154 69 100)(39 155 70 101)(40 113 131 196)(41 114 132 197)(42 115 133 198)(43 116 134 199)(44 117 135 200)(45 105 136 201)(46 106 137 202)(47 107 138 203)(48 108 139 204)(49 109 140 205)(50 110 141 206)(51 111 142 207)(52 112 143 208)
(1 33 162 136)(2 34 163 137)(3 35 164 138)(4 36 165 139)(5 37 166 140)(6 38 167 141)(7 39 168 142)(8 27 169 143)(9 28 157 131)(10 29 158 132)(11 30 159 133)(12 31 160 134)(13 32 161 135)(14 108 89 98)(15 109 90 99)(16 110 91 100)(17 111 79 101)(18 112 80 102)(19 113 81 103)(20 114 82 104)(21 115 83 92)(22 116 84 93)(23 117 85 94)(24 105 86 95)(25 106 87 96)(26 107 88 97)(40 129 72 65)(41 130 73 53)(42 118 74 54)(43 119 75 55)(44 120 76 56)(45 121 77 57)(46 122 78 58)(47 123 66 59)(48 124 67 60)(49 125 68 61)(50 126 69 62)(51 127 70 63)(52 128 71 64)(144 184 196 181)(145 185 197 182)(146 186 198 170)(147 187 199 171)(148 188 200 172)(149 189 201 173)(150 190 202 174)(151 191 203 175)(152 192 204 176)(153 193 205 177)(154 194 206 178)(155 195 207 179)(156 183 208 180)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 118)(12 119)(13 120)(14 176)(15 177)(16 178)(17 179)(18 180)(19 181)(20 182)(21 170)(22 171)(23 172)(24 173)(25 174)(26 175)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 66)(36 67)(37 68)(38 69)(39 70)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 158)(54 159)(55 160)(56 161)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 169)(65 157)(79 195)(80 183)(81 184)(82 185)(83 186)(84 187)(85 188)(86 189)(87 190)(88 191)(89 192)(90 193)(91 194)(92 198)(93 199)(94 200)(95 201)(96 202)(97 203)(98 204)(99 205)(100 206)(101 207)(102 208)(103 196)(104 197)(105 149)(106 150)(107 151)(108 152)(109 153)(110 154)(111 155)(112 156)(113 144)(114 145)(115 146)(116 147)(117 148)
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), (1,173,121,86)(2,174,122,87)(3,175,123,88)(4,176,124,89)(5,177,125,90)(6,178,126,91)(7,179,127,79)(8,180,128,80)(9,181,129,81)(10,182,130,82)(11,170,118,83)(12,171,119,84)(13,172,120,85)(14,165,192,60)(15,166,193,61)(16,167,194,62)(17,168,195,63)(18,169,183,64)(19,157,184,65)(20,158,185,53)(21,159,186,54)(22,160,187,55)(23,161,188,56)(24,162,189,57)(25,163,190,58)(26,164,191,59)(27,156,71,102)(28,144,72,103)(29,145,73,104)(30,146,74,92)(31,147,75,93)(32,148,76,94)(33,149,77,95)(34,150,78,96)(35,151,66,97)(36,152,67,98)(37,153,68,99)(38,154,69,100)(39,155,70,101)(40,113,131,196)(41,114,132,197)(42,115,133,198)(43,116,134,199)(44,117,135,200)(45,105,136,201)(46,106,137,202)(47,107,138,203)(48,108,139,204)(49,109,140,205)(50,110,141,206)(51,111,142,207)(52,112,143,208), (1,33,162,136)(2,34,163,137)(3,35,164,138)(4,36,165,139)(5,37,166,140)(6,38,167,141)(7,39,168,142)(8,27,169,143)(9,28,157,131)(10,29,158,132)(11,30,159,133)(12,31,160,134)(13,32,161,135)(14,108,89,98)(15,109,90,99)(16,110,91,100)(17,111,79,101)(18,112,80,102)(19,113,81,103)(20,114,82,104)(21,115,83,92)(22,116,84,93)(23,117,85,94)(24,105,86,95)(25,106,87,96)(26,107,88,97)(40,129,72,65)(41,130,73,53)(42,118,74,54)(43,119,75,55)(44,120,76,56)(45,121,77,57)(46,122,78,58)(47,123,66,59)(48,124,67,60)(49,125,68,61)(50,126,69,62)(51,127,70,63)(52,128,71,64)(144,184,196,181)(145,185,197,182)(146,186,198,170)(147,187,199,171)(148,188,200,172)(149,189,201,173)(150,190,202,174)(151,191,203,175)(152,192,204,176)(153,193,205,177)(154,194,206,178)(155,195,207,179)(156,183,208,180), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,118)(12,119)(13,120)(14,176)(15,177)(16,178)(17,179)(18,180)(19,181)(20,182)(21,170)(22,171)(23,172)(24,173)(25,174)(26,175)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,66)(36,67)(37,68)(38,69)(39,70)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,158)(54,159)(55,160)(56,161)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(65,157)(79,195)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,194)(92,198)(93,199)(94,200)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,196)(104,197)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,144)(114,145)(115,146)(116,147)(117,148)>;
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), (1,173,121,86)(2,174,122,87)(3,175,123,88)(4,176,124,89)(5,177,125,90)(6,178,126,91)(7,179,127,79)(8,180,128,80)(9,181,129,81)(10,182,130,82)(11,170,118,83)(12,171,119,84)(13,172,120,85)(14,165,192,60)(15,166,193,61)(16,167,194,62)(17,168,195,63)(18,169,183,64)(19,157,184,65)(20,158,185,53)(21,159,186,54)(22,160,187,55)(23,161,188,56)(24,162,189,57)(25,163,190,58)(26,164,191,59)(27,156,71,102)(28,144,72,103)(29,145,73,104)(30,146,74,92)(31,147,75,93)(32,148,76,94)(33,149,77,95)(34,150,78,96)(35,151,66,97)(36,152,67,98)(37,153,68,99)(38,154,69,100)(39,155,70,101)(40,113,131,196)(41,114,132,197)(42,115,133,198)(43,116,134,199)(44,117,135,200)(45,105,136,201)(46,106,137,202)(47,107,138,203)(48,108,139,204)(49,109,140,205)(50,110,141,206)(51,111,142,207)(52,112,143,208), (1,33,162,136)(2,34,163,137)(3,35,164,138)(4,36,165,139)(5,37,166,140)(6,38,167,141)(7,39,168,142)(8,27,169,143)(9,28,157,131)(10,29,158,132)(11,30,159,133)(12,31,160,134)(13,32,161,135)(14,108,89,98)(15,109,90,99)(16,110,91,100)(17,111,79,101)(18,112,80,102)(19,113,81,103)(20,114,82,104)(21,115,83,92)(22,116,84,93)(23,117,85,94)(24,105,86,95)(25,106,87,96)(26,107,88,97)(40,129,72,65)(41,130,73,53)(42,118,74,54)(43,119,75,55)(44,120,76,56)(45,121,77,57)(46,122,78,58)(47,123,66,59)(48,124,67,60)(49,125,68,61)(50,126,69,62)(51,127,70,63)(52,128,71,64)(144,184,196,181)(145,185,197,182)(146,186,198,170)(147,187,199,171)(148,188,200,172)(149,189,201,173)(150,190,202,174)(151,191,203,175)(152,192,204,176)(153,193,205,177)(154,194,206,178)(155,195,207,179)(156,183,208,180), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,118)(12,119)(13,120)(14,176)(15,177)(16,178)(17,179)(18,180)(19,181)(20,182)(21,170)(22,171)(23,172)(24,173)(25,174)(26,175)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,66)(36,67)(37,68)(38,69)(39,70)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,158)(54,159)(55,160)(56,161)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(65,157)(79,195)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,194)(92,198)(93,199)(94,200)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,196)(104,197)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,144)(114,145)(115,146)(116,147)(117,148) );
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)], [(1,173,121,86),(2,174,122,87),(3,175,123,88),(4,176,124,89),(5,177,125,90),(6,178,126,91),(7,179,127,79),(8,180,128,80),(9,181,129,81),(10,182,130,82),(11,170,118,83),(12,171,119,84),(13,172,120,85),(14,165,192,60),(15,166,193,61),(16,167,194,62),(17,168,195,63),(18,169,183,64),(19,157,184,65),(20,158,185,53),(21,159,186,54),(22,160,187,55),(23,161,188,56),(24,162,189,57),(25,163,190,58),(26,164,191,59),(27,156,71,102),(28,144,72,103),(29,145,73,104),(30,146,74,92),(31,147,75,93),(32,148,76,94),(33,149,77,95),(34,150,78,96),(35,151,66,97),(36,152,67,98),(37,153,68,99),(38,154,69,100),(39,155,70,101),(40,113,131,196),(41,114,132,197),(42,115,133,198),(43,116,134,199),(44,117,135,200),(45,105,136,201),(46,106,137,202),(47,107,138,203),(48,108,139,204),(49,109,140,205),(50,110,141,206),(51,111,142,207),(52,112,143,208)], [(1,33,162,136),(2,34,163,137),(3,35,164,138),(4,36,165,139),(5,37,166,140),(6,38,167,141),(7,39,168,142),(8,27,169,143),(9,28,157,131),(10,29,158,132),(11,30,159,133),(12,31,160,134),(13,32,161,135),(14,108,89,98),(15,109,90,99),(16,110,91,100),(17,111,79,101),(18,112,80,102),(19,113,81,103),(20,114,82,104),(21,115,83,92),(22,116,84,93),(23,117,85,94),(24,105,86,95),(25,106,87,96),(26,107,88,97),(40,129,72,65),(41,130,73,53),(42,118,74,54),(43,119,75,55),(44,120,76,56),(45,121,77,57),(46,122,78,58),(47,123,66,59),(48,124,67,60),(49,125,68,61),(50,126,69,62),(51,127,70,63),(52,128,71,64),(144,184,196,181),(145,185,197,182),(146,186,198,170),(147,187,199,171),(148,188,200,172),(149,189,201,173),(150,190,202,174),(151,191,203,175),(152,192,204,176),(153,193,205,177),(154,194,206,178),(155,195,207,179),(156,183,208,180)], [(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,118),(12,119),(13,120),(14,176),(15,177),(16,178),(17,179),(18,180),(19,181),(20,182),(21,170),(22,171),(23,172),(24,173),(25,174),(26,175),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,66),(36,67),(37,68),(38,69),(39,70),(40,131),(41,132),(42,133),(43,134),(44,135),(45,136),(46,137),(47,138),(48,139),(49,140),(50,141),(51,142),(52,143),(53,158),(54,159),(55,160),(56,161),(57,162),(58,163),(59,164),(60,165),(61,166),(62,167),(63,168),(64,169),(65,157),(79,195),(80,183),(81,184),(82,185),(83,186),(84,187),(85,188),(86,189),(87,190),(88,191),(89,192),(90,193),(91,194),(92,198),(93,199),(94,200),(95,201),(96,202),(97,203),(98,204),(99,205),(100,206),(101,207),(102,208),(103,196),(104,197),(105,149),(106,150),(107,151),(108,152),(109,153),(110,154),(111,155),(112,156),(113,144),(114,145),(115,146),(116,147),(117,148)]])
260 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 52A | ··· | 52AV | 52AW | ··· | 52FL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
260 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C13 | C26 | C26 | C26 | C26 | C52 | C4○D4 | C13×C4○D4 |
| kernel | C13×C42⋊C2 | C4×C52 | C13×C22⋊C4 | C13×C4⋊C4 | C22×C52 | C2×C52 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C26 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 12 | 24 | 24 | 24 | 12 | 96 | 4 | 48 |
Matrix representation of C13×C42⋊C2 ►in GL3(𝔽53) generated by
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 23 | 0 | 0 |
| 0 | 52 | 51 |
| 0 | 1 | 1 |
| 52 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 30 |
| 1 | 0 | 0 |
| 0 | 52 | 0 |
| 0 | 1 | 1 |
G:=sub<GL(3,GF(53))| [1,0,0,0,13,0,0,0,13],[23,0,0,0,52,1,0,51,1],[52,0,0,0,30,0,0,0,30],[1,0,0,0,52,1,0,0,1] >;
C13×C42⋊C2 in GAP, Magma, Sage, TeX
C_{13}\times C_4^2\rtimes C_2 % in TeX
G:=Group("C13xC4^2:C2"); // GroupNames label
G:=SmallGroup(416,178);
// by ID
G=gap.SmallGroup(416,178);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,482]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations