direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C8.C4, C8.1C52, C52.68D4, C104.10C4, M4(2).2C26, C4.8(C2×C52), (C2×C8).5C26, (C2×C26).2Q8, C52.66(C2×C4), C22.(Q8×C13), C4.19(D4×C13), C26.21(C4⋊C4), (C2×C104).15C2, (C2×C52).119C22, (C13×M4(2)).4C2, C2.5(C13×C4⋊C4), (C2×C4).22(C2×C26), SmallGroup(416,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C8.C4
G = < a,b,c | a13=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13×C8.C4
►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 41 71 64 97 195 15 141)(2 42 72 65 98 183 16 142)(3 43 73 53 99 184 17 143)(4 44 74 54 100 185 18 131)(5 45 75 55 101 186 19 132)(6 46 76 56 102 187 20 133)(7 47 77 57 103 188 21 134)(8 48 78 58 104 189 22 135)(9 49 66 59 92 190 23 136)(10 50 67 60 93 191 24 137)(11 51 68 61 94 192 25 138)(12 52 69 62 95 193 26 139)(13 40 70 63 96 194 14 140)(27 145 178 205 88 123 110 164)(28 146 179 206 89 124 111 165)(29 147 180 207 90 125 112 166)(30 148 181 208 91 126 113 167)(31 149 182 196 79 127 114 168)(32 150 170 197 80 128 115 169)(33 151 171 198 81 129 116 157)(34 152 172 199 82 130 117 158)(35 153 173 200 83 118 105 159)(36 154 174 201 84 119 106 160)(37 155 175 202 85 120 107 161)(38 156 176 203 86 121 108 162)(39 144 177 204 87 122 109 163)
(1 149 71 168 97 127 15 196)(2 150 72 169 98 128 16 197)(3 151 73 157 99 129 17 198)(4 152 74 158 100 130 18 199)(5 153 75 159 101 118 19 200)(6 154 76 160 102 119 20 201)(7 155 77 161 103 120 21 202)(8 156 78 162 104 121 22 203)(9 144 66 163 92 122 23 204)(10 145 67 164 93 123 24 205)(11 146 68 165 94 124 25 206)(12 147 69 166 95 125 26 207)(13 148 70 167 96 126 14 208)(27 60 110 191 88 137 178 50)(28 61 111 192 89 138 179 51)(29 62 112 193 90 139 180 52)(30 63 113 194 91 140 181 40)(31 64 114 195 79 141 182 41)(32 65 115 183 80 142 170 42)(33 53 116 184 81 143 171 43)(34 54 117 185 82 131 172 44)(35 55 105 186 83 132 173 45)(36 56 106 187 84 133 174 46)(37 57 107 188 85 134 175 47)(38 58 108 189 86 135 176 48)(39 59 109 190 87 136 177 49)
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), (1,41,71,64,97,195,15,141)(2,42,72,65,98,183,16,142)(3,43,73,53,99,184,17,143)(4,44,74,54,100,185,18,131)(5,45,75,55,101,186,19,132)(6,46,76,56,102,187,20,133)(7,47,77,57,103,188,21,134)(8,48,78,58,104,189,22,135)(9,49,66,59,92,190,23,136)(10,50,67,60,93,191,24,137)(11,51,68,61,94,192,25,138)(12,52,69,62,95,193,26,139)(13,40,70,63,96,194,14,140)(27,145,178,205,88,123,110,164)(28,146,179,206,89,124,111,165)(29,147,180,207,90,125,112,166)(30,148,181,208,91,126,113,167)(31,149,182,196,79,127,114,168)(32,150,170,197,80,128,115,169)(33,151,171,198,81,129,116,157)(34,152,172,199,82,130,117,158)(35,153,173,200,83,118,105,159)(36,154,174,201,84,119,106,160)(37,155,175,202,85,120,107,161)(38,156,176,203,86,121,108,162)(39,144,177,204,87,122,109,163), (1,149,71,168,97,127,15,196)(2,150,72,169,98,128,16,197)(3,151,73,157,99,129,17,198)(4,152,74,158,100,130,18,199)(5,153,75,159,101,118,19,200)(6,154,76,160,102,119,20,201)(7,155,77,161,103,120,21,202)(8,156,78,162,104,121,22,203)(9,144,66,163,92,122,23,204)(10,145,67,164,93,123,24,205)(11,146,68,165,94,124,25,206)(12,147,69,166,95,125,26,207)(13,148,70,167,96,126,14,208)(27,60,110,191,88,137,178,50)(28,61,111,192,89,138,179,51)(29,62,112,193,90,139,180,52)(30,63,113,194,91,140,181,40)(31,64,114,195,79,141,182,41)(32,65,115,183,80,142,170,42)(33,53,116,184,81,143,171,43)(34,54,117,185,82,131,172,44)(35,55,105,186,83,132,173,45)(36,56,106,187,84,133,174,46)(37,57,107,188,85,134,175,47)(38,58,108,189,86,135,176,48)(39,59,109,190,87,136,177,49)>;
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), (1,41,71,64,97,195,15,141)(2,42,72,65,98,183,16,142)(3,43,73,53,99,184,17,143)(4,44,74,54,100,185,18,131)(5,45,75,55,101,186,19,132)(6,46,76,56,102,187,20,133)(7,47,77,57,103,188,21,134)(8,48,78,58,104,189,22,135)(9,49,66,59,92,190,23,136)(10,50,67,60,93,191,24,137)(11,51,68,61,94,192,25,138)(12,52,69,62,95,193,26,139)(13,40,70,63,96,194,14,140)(27,145,178,205,88,123,110,164)(28,146,179,206,89,124,111,165)(29,147,180,207,90,125,112,166)(30,148,181,208,91,126,113,167)(31,149,182,196,79,127,114,168)(32,150,170,197,80,128,115,169)(33,151,171,198,81,129,116,157)(34,152,172,199,82,130,117,158)(35,153,173,200,83,118,105,159)(36,154,174,201,84,119,106,160)(37,155,175,202,85,120,107,161)(38,156,176,203,86,121,108,162)(39,144,177,204,87,122,109,163), (1,149,71,168,97,127,15,196)(2,150,72,169,98,128,16,197)(3,151,73,157,99,129,17,198)(4,152,74,158,100,130,18,199)(5,153,75,159,101,118,19,200)(6,154,76,160,102,119,20,201)(7,155,77,161,103,120,21,202)(8,156,78,162,104,121,22,203)(9,144,66,163,92,122,23,204)(10,145,67,164,93,123,24,205)(11,146,68,165,94,124,25,206)(12,147,69,166,95,125,26,207)(13,148,70,167,96,126,14,208)(27,60,110,191,88,137,178,50)(28,61,111,192,89,138,179,51)(29,62,112,193,90,139,180,52)(30,63,113,194,91,140,181,40)(31,64,114,195,79,141,182,41)(32,65,115,183,80,142,170,42)(33,53,116,184,81,143,171,43)(34,54,117,185,82,131,172,44)(35,55,105,186,83,132,173,45)(36,56,106,187,84,133,174,46)(37,57,107,188,85,134,175,47)(38,58,108,189,86,135,176,48)(39,59,109,190,87,136,177,49) );
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)], [(1,41,71,64,97,195,15,141),(2,42,72,65,98,183,16,142),(3,43,73,53,99,184,17,143),(4,44,74,54,100,185,18,131),(5,45,75,55,101,186,19,132),(6,46,76,56,102,187,20,133),(7,47,77,57,103,188,21,134),(8,48,78,58,104,189,22,135),(9,49,66,59,92,190,23,136),(10,50,67,60,93,191,24,137),(11,51,68,61,94,192,25,138),(12,52,69,62,95,193,26,139),(13,40,70,63,96,194,14,140),(27,145,178,205,88,123,110,164),(28,146,179,206,89,124,111,165),(29,147,180,207,90,125,112,166),(30,148,181,208,91,126,113,167),(31,149,182,196,79,127,114,168),(32,150,170,197,80,128,115,169),(33,151,171,198,81,129,116,157),(34,152,172,199,82,130,117,158),(35,153,173,200,83,118,105,159),(36,154,174,201,84,119,106,160),(37,155,175,202,85,120,107,161),(38,156,176,203,86,121,108,162),(39,144,177,204,87,122,109,163)], [(1,149,71,168,97,127,15,196),(2,150,72,169,98,128,16,197),(3,151,73,157,99,129,17,198),(4,152,74,158,100,130,18,199),(5,153,75,159,101,118,19,200),(6,154,76,160,102,119,20,201),(7,155,77,161,103,120,21,202),(8,156,78,162,104,121,22,203),(9,144,66,163,92,122,23,204),(10,145,67,164,93,123,24,205),(11,146,68,165,94,124,25,206),(12,147,69,166,95,125,26,207),(13,148,70,167,96,126,14,208),(27,60,110,191,88,137,178,50),(28,61,111,192,89,138,179,51),(29,62,112,193,90,139,180,52),(30,63,113,194,91,140,181,40),(31,64,114,195,79,141,182,41),(32,65,115,183,80,142,170,42),(33,53,116,184,81,143,171,43),(34,54,117,185,82,131,172,44),(35,55,105,186,83,132,173,45),(36,56,106,187,84,133,174,46),(37,57,107,188,85,134,175,47),(38,58,108,189,86,135,176,48),(39,59,109,190,87,136,177,49)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 52A | ··· | 52X | 52Y | ··· | 52AJ | 104A | ··· | 104AV | 104AW | ··· | 104CR |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C13 | C26 | C26 | C52 | D4 | Q8 | C8.C4 | D4×C13 | Q8×C13 | C13×C8.C4 |
| kernel | C13×C8.C4 | C2×C104 | C13×M4(2) | C104 | C8.C4 | C2×C8 | M4(2) | C8 | C52 | C2×C26 | C13 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 4 | 12 | 12 | 24 | 48 | 1 | 1 | 4 | 12 | 12 | 48 |
Matrix representation of C13×C8.C4 ►in GL2(𝔽313) generated by
| 234 | 0 |
| 0 | 234 |
| 308 | 0 |
| 134 | 125 |
| 205 | 311 |
| 29 | 108 |
G:=sub<GL(2,GF(313))| [234,0,0,234],[308,134,0,125],[205,29,311,108] >;
C13×C8.C4 in GAP, Magma, Sage, TeX
C_{13}\times C_8.C_4 % in TeX
G:=Group("C13xC8.C4"); // GroupNames label
G:=SmallGroup(416,58);
// by ID
G=gap.SmallGroup(416,58);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,319,6243,117,88]);
// Polycyclic
G:=Group<a,b,c|a^13=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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