direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C4.10D4, C52.59D4, M4(2).1C26, (C2×C4).C52, (C2×C52).13C4, C4.10(D4×C13), (C2×Q8).1C26, (Q8×C26).6C2, C22.4(C2×C52), (C2×C52).60C22, C26.34(C22⋊C4), (C13×M4(2)).3C2, (C2×C4).2(C2×C26), (C2×C26).41(C2×C4), C2.5(C13×C22⋊C4), SmallGroup(416,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4.10D4
G = < a,b,c,d | a13=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C13×C4.10D4
►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)
(1 184 204 116)(2 185 205 117)(3 186 206 105)(4 187 207 106)(5 188 208 107)(6 189 196 108)(7 190 197 109)(8 191 198 110)(9 192 199 111)(10 193 200 112)(11 194 201 113)(12 195 202 114)(13 183 203 115)(14 168 30 69)(15 169 31 70)(16 157 32 71)(17 158 33 72)(18 159 34 73)(19 160 35 74)(20 161 36 75)(21 162 37 76)(22 163 38 77)(23 164 39 78)(24 165 27 66)(25 166 28 67)(26 167 29 68)(40 59 177 90)(41 60 178 91)(42 61 179 79)(43 62 180 80)(44 63 181 81)(45 64 182 82)(46 65 170 83)(47 53 171 84)(48 54 172 85)(49 55 173 86)(50 56 174 87)(51 57 175 88)(52 58 176 89)(92 140 147 127)(93 141 148 128)(94 142 149 129)(95 143 150 130)(96 131 151 118)(97 132 152 119)(98 133 153 120)(99 134 154 121)(100 135 155 122)(101 136 156 123)(102 137 144 124)(103 138 145 125)(104 139 146 126)
(1 63 184 44 204 81 116 181)(2 64 185 45 205 82 117 182)(3 65 186 46 206 83 105 170)(4 53 187 47 207 84 106 171)(5 54 188 48 208 85 107 172)(6 55 189 49 196 86 108 173)(7 56 190 50 197 87 109 174)(8 57 191 51 198 88 110 175)(9 58 192 52 199 89 111 176)(10 59 193 40 200 90 112 177)(11 60 194 41 201 91 113 178)(12 61 195 42 202 79 114 179)(13 62 183 43 203 80 115 180)(14 141 69 148 30 128 168 93)(15 142 70 149 31 129 169 94)(16 143 71 150 32 130 157 95)(17 131 72 151 33 118 158 96)(18 132 73 152 34 119 159 97)(19 133 74 153 35 120 160 98)(20 134 75 154 36 121 161 99)(21 135 76 155 37 122 162 100)(22 136 77 156 38 123 163 101)(23 137 78 144 39 124 164 102)(24 138 66 145 27 125 165 103)(25 139 67 146 28 126 166 104)(26 140 68 147 29 127 167 92)
(1 75 116 36 204 161 184 20)(2 76 117 37 205 162 185 21)(3 77 105 38 206 163 186 22)(4 78 106 39 207 164 187 23)(5 66 107 27 208 165 188 24)(6 67 108 28 196 166 189 25)(7 68 109 29 197 167 190 26)(8 69 110 30 198 168 191 14)(9 70 111 31 199 169 192 15)(10 71 112 32 200 157 193 16)(11 72 113 33 201 158 194 17)(12 73 114 34 202 159 195 18)(13 74 115 35 203 160 183 19)(40 143 90 95 177 130 59 150)(41 131 91 96 178 118 60 151)(42 132 79 97 179 119 61 152)(43 133 80 98 180 120 62 153)(44 134 81 99 181 121 63 154)(45 135 82 100 182 122 64 155)(46 136 83 101 170 123 65 156)(47 137 84 102 171 124 53 144)(48 138 85 103 172 125 54 145)(49 139 86 104 173 126 55 146)(50 140 87 92 174 127 56 147)(51 141 88 93 175 128 57 148)(52 142 89 94 176 129 58 149)
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), (1,184,204,116)(2,185,205,117)(3,186,206,105)(4,187,207,106)(5,188,208,107)(6,189,196,108)(7,190,197,109)(8,191,198,110)(9,192,199,111)(10,193,200,112)(11,194,201,113)(12,195,202,114)(13,183,203,115)(14,168,30,69)(15,169,31,70)(16,157,32,71)(17,158,33,72)(18,159,34,73)(19,160,35,74)(20,161,36,75)(21,162,37,76)(22,163,38,77)(23,164,39,78)(24,165,27,66)(25,166,28,67)(26,167,29,68)(40,59,177,90)(41,60,178,91)(42,61,179,79)(43,62,180,80)(44,63,181,81)(45,64,182,82)(46,65,170,83)(47,53,171,84)(48,54,172,85)(49,55,173,86)(50,56,174,87)(51,57,175,88)(52,58,176,89)(92,140,147,127)(93,141,148,128)(94,142,149,129)(95,143,150,130)(96,131,151,118)(97,132,152,119)(98,133,153,120)(99,134,154,121)(100,135,155,122)(101,136,156,123)(102,137,144,124)(103,138,145,125)(104,139,146,126), (1,63,184,44,204,81,116,181)(2,64,185,45,205,82,117,182)(3,65,186,46,206,83,105,170)(4,53,187,47,207,84,106,171)(5,54,188,48,208,85,107,172)(6,55,189,49,196,86,108,173)(7,56,190,50,197,87,109,174)(8,57,191,51,198,88,110,175)(9,58,192,52,199,89,111,176)(10,59,193,40,200,90,112,177)(11,60,194,41,201,91,113,178)(12,61,195,42,202,79,114,179)(13,62,183,43,203,80,115,180)(14,141,69,148,30,128,168,93)(15,142,70,149,31,129,169,94)(16,143,71,150,32,130,157,95)(17,131,72,151,33,118,158,96)(18,132,73,152,34,119,159,97)(19,133,74,153,35,120,160,98)(20,134,75,154,36,121,161,99)(21,135,76,155,37,122,162,100)(22,136,77,156,38,123,163,101)(23,137,78,144,39,124,164,102)(24,138,66,145,27,125,165,103)(25,139,67,146,28,126,166,104)(26,140,68,147,29,127,167,92), (1,75,116,36,204,161,184,20)(2,76,117,37,205,162,185,21)(3,77,105,38,206,163,186,22)(4,78,106,39,207,164,187,23)(5,66,107,27,208,165,188,24)(6,67,108,28,196,166,189,25)(7,68,109,29,197,167,190,26)(8,69,110,30,198,168,191,14)(9,70,111,31,199,169,192,15)(10,71,112,32,200,157,193,16)(11,72,113,33,201,158,194,17)(12,73,114,34,202,159,195,18)(13,74,115,35,203,160,183,19)(40,143,90,95,177,130,59,150)(41,131,91,96,178,118,60,151)(42,132,79,97,179,119,61,152)(43,133,80,98,180,120,62,153)(44,134,81,99,181,121,63,154)(45,135,82,100,182,122,64,155)(46,136,83,101,170,123,65,156)(47,137,84,102,171,124,53,144)(48,138,85,103,172,125,54,145)(49,139,86,104,173,126,55,146)(50,140,87,92,174,127,56,147)(51,141,88,93,175,128,57,148)(52,142,89,94,176,129,58,149)>;
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), (1,184,204,116)(2,185,205,117)(3,186,206,105)(4,187,207,106)(5,188,208,107)(6,189,196,108)(7,190,197,109)(8,191,198,110)(9,192,199,111)(10,193,200,112)(11,194,201,113)(12,195,202,114)(13,183,203,115)(14,168,30,69)(15,169,31,70)(16,157,32,71)(17,158,33,72)(18,159,34,73)(19,160,35,74)(20,161,36,75)(21,162,37,76)(22,163,38,77)(23,164,39,78)(24,165,27,66)(25,166,28,67)(26,167,29,68)(40,59,177,90)(41,60,178,91)(42,61,179,79)(43,62,180,80)(44,63,181,81)(45,64,182,82)(46,65,170,83)(47,53,171,84)(48,54,172,85)(49,55,173,86)(50,56,174,87)(51,57,175,88)(52,58,176,89)(92,140,147,127)(93,141,148,128)(94,142,149,129)(95,143,150,130)(96,131,151,118)(97,132,152,119)(98,133,153,120)(99,134,154,121)(100,135,155,122)(101,136,156,123)(102,137,144,124)(103,138,145,125)(104,139,146,126), (1,63,184,44,204,81,116,181)(2,64,185,45,205,82,117,182)(3,65,186,46,206,83,105,170)(4,53,187,47,207,84,106,171)(5,54,188,48,208,85,107,172)(6,55,189,49,196,86,108,173)(7,56,190,50,197,87,109,174)(8,57,191,51,198,88,110,175)(9,58,192,52,199,89,111,176)(10,59,193,40,200,90,112,177)(11,60,194,41,201,91,113,178)(12,61,195,42,202,79,114,179)(13,62,183,43,203,80,115,180)(14,141,69,148,30,128,168,93)(15,142,70,149,31,129,169,94)(16,143,71,150,32,130,157,95)(17,131,72,151,33,118,158,96)(18,132,73,152,34,119,159,97)(19,133,74,153,35,120,160,98)(20,134,75,154,36,121,161,99)(21,135,76,155,37,122,162,100)(22,136,77,156,38,123,163,101)(23,137,78,144,39,124,164,102)(24,138,66,145,27,125,165,103)(25,139,67,146,28,126,166,104)(26,140,68,147,29,127,167,92), (1,75,116,36,204,161,184,20)(2,76,117,37,205,162,185,21)(3,77,105,38,206,163,186,22)(4,78,106,39,207,164,187,23)(5,66,107,27,208,165,188,24)(6,67,108,28,196,166,189,25)(7,68,109,29,197,167,190,26)(8,69,110,30,198,168,191,14)(9,70,111,31,199,169,192,15)(10,71,112,32,200,157,193,16)(11,72,113,33,201,158,194,17)(12,73,114,34,202,159,195,18)(13,74,115,35,203,160,183,19)(40,143,90,95,177,130,59,150)(41,131,91,96,178,118,60,151)(42,132,79,97,179,119,61,152)(43,133,80,98,180,120,62,153)(44,134,81,99,181,121,63,154)(45,135,82,100,182,122,64,155)(46,136,83,101,170,123,65,156)(47,137,84,102,171,124,53,144)(48,138,85,103,172,125,54,145)(49,139,86,104,173,126,55,146)(50,140,87,92,174,127,56,147)(51,141,88,93,175,128,57,148)(52,142,89,94,176,129,58,149) );
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)], [(1,184,204,116),(2,185,205,117),(3,186,206,105),(4,187,207,106),(5,188,208,107),(6,189,196,108),(7,190,197,109),(8,191,198,110),(9,192,199,111),(10,193,200,112),(11,194,201,113),(12,195,202,114),(13,183,203,115),(14,168,30,69),(15,169,31,70),(16,157,32,71),(17,158,33,72),(18,159,34,73),(19,160,35,74),(20,161,36,75),(21,162,37,76),(22,163,38,77),(23,164,39,78),(24,165,27,66),(25,166,28,67),(26,167,29,68),(40,59,177,90),(41,60,178,91),(42,61,179,79),(43,62,180,80),(44,63,181,81),(45,64,182,82),(46,65,170,83),(47,53,171,84),(48,54,172,85),(49,55,173,86),(50,56,174,87),(51,57,175,88),(52,58,176,89),(92,140,147,127),(93,141,148,128),(94,142,149,129),(95,143,150,130),(96,131,151,118),(97,132,152,119),(98,133,153,120),(99,134,154,121),(100,135,155,122),(101,136,156,123),(102,137,144,124),(103,138,145,125),(104,139,146,126)], [(1,63,184,44,204,81,116,181),(2,64,185,45,205,82,117,182),(3,65,186,46,206,83,105,170),(4,53,187,47,207,84,106,171),(5,54,188,48,208,85,107,172),(6,55,189,49,196,86,108,173),(7,56,190,50,197,87,109,174),(8,57,191,51,198,88,110,175),(9,58,192,52,199,89,111,176),(10,59,193,40,200,90,112,177),(11,60,194,41,201,91,113,178),(12,61,195,42,202,79,114,179),(13,62,183,43,203,80,115,180),(14,141,69,148,30,128,168,93),(15,142,70,149,31,129,169,94),(16,143,71,150,32,130,157,95),(17,131,72,151,33,118,158,96),(18,132,73,152,34,119,159,97),(19,133,74,153,35,120,160,98),(20,134,75,154,36,121,161,99),(21,135,76,155,37,122,162,100),(22,136,77,156,38,123,163,101),(23,137,78,144,39,124,164,102),(24,138,66,145,27,125,165,103),(25,139,67,146,28,126,166,104),(26,140,68,147,29,127,167,92)], [(1,75,116,36,204,161,184,20),(2,76,117,37,205,162,185,21),(3,77,105,38,206,163,186,22),(4,78,106,39,207,164,187,23),(5,66,107,27,208,165,188,24),(6,67,108,28,196,166,189,25),(7,68,109,29,197,167,190,26),(8,69,110,30,198,168,191,14),(9,70,111,31,199,169,192,15),(10,71,112,32,200,157,193,16),(11,72,113,33,201,158,194,17),(12,73,114,34,202,159,195,18),(13,74,115,35,203,160,183,19),(40,143,90,95,177,130,59,150),(41,131,91,96,178,118,60,151),(42,132,79,97,179,119,61,152),(43,133,80,98,180,120,62,153),(44,134,81,99,181,121,63,154),(45,135,82,100,182,122,64,155),(46,136,83,101,170,123,65,156),(47,137,84,102,171,124,53,144),(48,138,85,103,172,125,54,145),(49,139,86,104,173,126,55,146),(50,140,87,92,174,127,56,147),(51,141,88,93,175,128,57,148),(52,142,89,94,176,129,58,149)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 52A | ··· | 52X | 52Y | ··· | 52AV | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C13 | C26 | C26 | C52 | D4 | D4×C13 | C4.10D4 | C13×C4.10D4 |
| kernel | C13×C4.10D4 | C13×M4(2) | Q8×C26 | C2×C52 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | C52 | C4 | C13 | C1 |
| # reps | 1 | 2 | 1 | 4 | 12 | 24 | 12 | 48 | 2 | 24 | 1 | 12 |
Matrix representation of C13×C4.10D4 ►in GL4(𝔽313) generated by
| 44 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 0 | 44 |
| 1 | 311 | 0 | 0 |
| 1 | 312 | 0 | 0 |
| 175 | 138 | 0 | 312 |
| 0 | 138 | 1 | 0 |
| 175 | 0 | 311 | 0 |
| 0 | 0 | 312 | 1 |
| 288 | 1 | 138 | 0 |
| 289 | 0 | 138 | 0 |
| 245 | 0 | 292 | 206 |
| 116 | 0 | 43 | 249 |
| 175 | 167 | 197 | 184 |
| 278 | 210 | 197 | 184 |
G:=sub<GL(4,GF(313))| [44,0,0,0,0,44,0,0,0,0,44,0,0,0,0,44],[1,1,175,0,311,312,138,138,0,0,0,1,0,0,312,0],[175,0,288,289,0,0,1,0,311,312,138,138,0,1,0,0],[245,116,175,278,0,0,167,210,292,43,197,197,206,249,184,184] >;
C13×C4.10D4 in GAP, Magma, Sage, TeX
C_{13}\times C_4._{10}D_4 % in TeX
G:=Group("C13xC4.10D4"); // GroupNames label
G:=SmallGroup(416,51);
// by ID
G=gap.SmallGroup(416,51);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,1255,6243,4690,88]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
Export