direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×M5(2), C4.C104, C16⋊3C26, C208⋊7C2, C52.7C8, C8.2C52, C22.C104, C104.11C4, C104.29C22, (C2×C8).8C26, (C2×C26).3C8, (C2×C4).5C52, C8.8(C2×C26), C26.22(C2×C8), C4.12(C2×C52), (C2×C52).25C4, C52.70(C2×C4), C2.3(C2×C104), (C2×C104).18C2, SmallGroup(416,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×M5(2)
G = < a,b,c | a13=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C13×M5(2)
►On 208 points
Generators in S208
(1 200 110 62 171 120 76 42 135 85 188 149 17)(2 201 111 63 172 121 77 43 136 86 189 150 18)(3 202 112 64 173 122 78 44 137 87 190 151 19)(4 203 97 49 174 123 79 45 138 88 191 152 20)(5 204 98 50 175 124 80 46 139 89 192 153 21)(6 205 99 51 176 125 65 47 140 90 177 154 22)(7 206 100 52 161 126 66 48 141 91 178 155 23)(8 207 101 53 162 127 67 33 142 92 179 156 24)(9 208 102 54 163 128 68 34 143 93 180 157 25)(10 193 103 55 164 113 69 35 144 94 181 158 26)(11 194 104 56 165 114 70 36 129 95 182 159 27)(12 195 105 57 166 115 71 37 130 96 183 160 28)(13 196 106 58 167 116 72 38 131 81 184 145 29)(14 197 107 59 168 117 73 39 132 82 185 146 30)(15 198 108 60 169 118 74 40 133 83 186 147 31)(16 199 109 61 170 119 75 41 134 84 187 148 32)
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)(113 121)(115 123)(117 125)(119 127)(130 138)(132 140)(134 142)(136 144)(146 154)(148 156)(150 158)(152 160)(162 170)(164 172)(166 174)(168 176)(177 185)(179 187)(181 189)(183 191)(193 201)(195 203)(197 205)(199 207)
G:=sub<Sym(208)| (1,200,110,62,171,120,76,42,135,85,188,149,17)(2,201,111,63,172,121,77,43,136,86,189,150,18)(3,202,112,64,173,122,78,44,137,87,190,151,19)(4,203,97,49,174,123,79,45,138,88,191,152,20)(5,204,98,50,175,124,80,46,139,89,192,153,21)(6,205,99,51,176,125,65,47,140,90,177,154,22)(7,206,100,52,161,126,66,48,141,91,178,155,23)(8,207,101,53,162,127,67,33,142,92,179,156,24)(9,208,102,54,163,128,68,34,143,93,180,157,25)(10,193,103,55,164,113,69,35,144,94,181,158,26)(11,194,104,56,165,114,70,36,129,95,182,159,27)(12,195,105,57,166,115,71,37,130,96,183,160,28)(13,196,106,58,167,116,72,38,131,81,184,145,29)(14,197,107,59,168,117,73,39,132,82,185,146,30)(15,198,108,60,169,118,74,40,133,83,186,147,31)(16,199,109,61,170,119,75,41,134,84,187,148,32), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111)(113,121)(115,123)(117,125)(119,127)(130,138)(132,140)(134,142)(136,144)(146,154)(148,156)(150,158)(152,160)(162,170)(164,172)(166,174)(168,176)(177,185)(179,187)(181,189)(183,191)(193,201)(195,203)(197,205)(199,207)>;
G:=Group( (1,200,110,62,171,120,76,42,135,85,188,149,17)(2,201,111,63,172,121,77,43,136,86,189,150,18)(3,202,112,64,173,122,78,44,137,87,190,151,19)(4,203,97,49,174,123,79,45,138,88,191,152,20)(5,204,98,50,175,124,80,46,139,89,192,153,21)(6,205,99,51,176,125,65,47,140,90,177,154,22)(7,206,100,52,161,126,66,48,141,91,178,155,23)(8,207,101,53,162,127,67,33,142,92,179,156,24)(9,208,102,54,163,128,68,34,143,93,180,157,25)(10,193,103,55,164,113,69,35,144,94,181,158,26)(11,194,104,56,165,114,70,36,129,95,182,159,27)(12,195,105,57,166,115,71,37,130,96,183,160,28)(13,196,106,58,167,116,72,38,131,81,184,145,29)(14,197,107,59,168,117,73,39,132,82,185,146,30)(15,198,108,60,169,118,74,40,133,83,186,147,31)(16,199,109,61,170,119,75,41,134,84,187,148,32), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111)(113,121)(115,123)(117,125)(119,127)(130,138)(132,140)(134,142)(136,144)(146,154)(148,156)(150,158)(152,160)(162,170)(164,172)(166,174)(168,176)(177,185)(179,187)(181,189)(183,191)(193,201)(195,203)(197,205)(199,207) );
G=PermutationGroup([[(1,200,110,62,171,120,76,42,135,85,188,149,17),(2,201,111,63,172,121,77,43,136,86,189,150,18),(3,202,112,64,173,122,78,44,137,87,190,151,19),(4,203,97,49,174,123,79,45,138,88,191,152,20),(5,204,98,50,175,124,80,46,139,89,192,153,21),(6,205,99,51,176,125,65,47,140,90,177,154,22),(7,206,100,52,161,126,66,48,141,91,178,155,23),(8,207,101,53,162,127,67,33,142,92,179,156,24),(9,208,102,54,163,128,68,34,143,93,180,157,25),(10,193,103,55,164,113,69,35,144,94,181,158,26),(11,194,104,56,165,114,70,36,129,95,182,159,27),(12,195,105,57,166,115,71,37,130,96,183,160,28),(13,196,106,58,167,116,72,38,131,81,184,145,29),(14,197,107,59,168,117,73,39,132,82,185,146,30),(15,198,108,60,169,118,74,40,133,83,186,147,31),(16,199,109,61,170,119,75,41,134,84,187,148,32)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47),(49,57),(51,59),(53,61),(55,63),(65,73),(67,75),(69,77),(71,79),(82,90),(84,92),(86,94),(88,96),(97,105),(99,107),(101,109),(103,111),(113,121),(115,123),(117,125),(119,127),(130,138),(132,140),(134,142),(136,144),(146,154),(148,156),(150,158),(152,160),(162,170),(164,172),(166,174),(168,176),(177,185),(179,187),(181,189),(183,191),(193,201),(195,203),(197,205),(199,207)]])
260 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 13A | ··· | 13L | 16A | ··· | 16H | 26A | ··· | 26L | 26M | ··· | 26X | 52A | ··· | 52X | 52Y | ··· | 52AJ | 104A | ··· | 104AV | 104AW | ··· | 104BT | 208A | ··· | 208CR |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 16 | ··· | 16 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
260 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C13 | C26 | C26 | C52 | C52 | C104 | C104 | M5(2) | C13×M5(2) |
| kernel | C13×M5(2) | C208 | C2×C104 | C104 | C2×C52 | C52 | C2×C26 | M5(2) | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 12 | 24 | 12 | 24 | 24 | 48 | 48 | 4 | 48 |
Matrix representation of C13×M5(2) ►in GL2(𝔽1249) generated by
| 933 | 0 |
| 0 | 933 |
| 1022 | 1247 |
| 354 | 227 |
| 1 | 0 |
| 1022 | 1248 |
G:=sub<GL(2,GF(1249))| [933,0,0,933],[1022,354,1247,227],[1,1022,0,1248] >;
C13×M5(2) in GAP, Magma, Sage, TeX
C_{13}\times M_5(2) % in TeX
G:=Group("C13xM5(2)"); // GroupNames label
G:=SmallGroup(416,60);
// by ID
G=gap.SmallGroup(416,60);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,312,2521,69,88]);
// Polycyclic
G:=Group<a,b,c|a^13=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations
Export