metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.12D52, C52.47D4, M4(2).2D13, (C2×C4).2D26, C22.5(C4×D13), C52.4C4.3C2, C4.22(C13⋊D4), C13⋊2(C4.10D4), (C2×C52).14C22, (C2×Dic13).1C4, (C2×Dic26).7C2, C26.20(C22⋊C4), (C13×M4(2)).2C2, C2.10(D26⋊C4), (C2×C26).23(C2×C4), SmallGroup(416,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.12D52
G = < a,b,c | a4=1, b52=a2, c2=bab-1=a-1, ac=ca, cbc-1=a-1b51 >
Smallest permutation representation of C4.12D52
►On 208 points
Generators in S208
(1 27 53 79)(2 80 54 28)(3 29 55 81)(4 82 56 30)(5 31 57 83)(6 84 58 32)(7 33 59 85)(8 86 60 34)(9 35 61 87)(10 88 62 36)(11 37 63 89)(12 90 64 38)(13 39 65 91)(14 92 66 40)(15 41 67 93)(16 94 68 42)(17 43 69 95)(18 96 70 44)(19 45 71 97)(20 98 72 46)(21 47 73 99)(22 100 74 48)(23 49 75 101)(24 102 76 50)(25 51 77 103)(26 104 78 52)(105 183 157 131)(106 132 158 184)(107 185 159 133)(108 134 160 186)(109 187 161 135)(110 136 162 188)(111 189 163 137)(112 138 164 190)(113 191 165 139)(114 140 166 192)(115 193 167 141)(116 142 168 194)(117 195 169 143)(118 144 170 196)(119 197 171 145)(120 146 172 198)(121 199 173 147)(122 148 174 200)(123 201 175 149)(124 150 176 202)(125 203 177 151)(126 152 178 204)(127 205 179 153)(128 154 180 206)(129 207 181 155)(130 156 182 208)
(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)
(1 201 79 123 53 149 27 175)(2 122 28 200 54 174 80 148)(3 199 81 121 55 147 29 173)(4 120 30 198 56 172 82 146)(5 197 83 119 57 145 31 171)(6 118 32 196 58 170 84 144)(7 195 85 117 59 143 33 169)(8 116 34 194 60 168 86 142)(9 193 87 115 61 141 35 167)(10 114 36 192 62 166 88 140)(11 191 89 113 63 139 37 165)(12 112 38 190 64 164 90 138)(13 189 91 111 65 137 39 163)(14 110 40 188 66 162 92 136)(15 187 93 109 67 135 41 161)(16 108 42 186 68 160 94 134)(17 185 95 107 69 133 43 159)(18 106 44 184 70 158 96 132)(19 183 97 105 71 131 45 157)(20 208 46 182 72 156 98 130)(21 181 99 207 73 129 47 155)(22 206 48 180 74 154 100 128)(23 179 101 205 75 127 49 153)(24 204 50 178 76 152 102 126)(25 177 103 203 77 125 51 151)(26 202 52 176 78 150 104 124)
G:=sub<Sym(208)| (1,27,53,79)(2,80,54,28)(3,29,55,81)(4,82,56,30)(5,31,57,83)(6,84,58,32)(7,33,59,85)(8,86,60,34)(9,35,61,87)(10,88,62,36)(11,37,63,89)(12,90,64,38)(13,39,65,91)(14,92,66,40)(15,41,67,93)(16,94,68,42)(17,43,69,95)(18,96,70,44)(19,45,71,97)(20,98,72,46)(21,47,73,99)(22,100,74,48)(23,49,75,101)(24,102,76,50)(25,51,77,103)(26,104,78,52)(105,183,157,131)(106,132,158,184)(107,185,159,133)(108,134,160,186)(109,187,161,135)(110,136,162,188)(111,189,163,137)(112,138,164,190)(113,191,165,139)(114,140,166,192)(115,193,167,141)(116,142,168,194)(117,195,169,143)(118,144,170,196)(119,197,171,145)(120,146,172,198)(121,199,173,147)(122,148,174,200)(123,201,175,149)(124,150,176,202)(125,203,177,151)(126,152,178,204)(127,205,179,153)(128,154,180,206)(129,207,181,155)(130,156,182,208), (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), (1,201,79,123,53,149,27,175)(2,122,28,200,54,174,80,148)(3,199,81,121,55,147,29,173)(4,120,30,198,56,172,82,146)(5,197,83,119,57,145,31,171)(6,118,32,196,58,170,84,144)(7,195,85,117,59,143,33,169)(8,116,34,194,60,168,86,142)(9,193,87,115,61,141,35,167)(10,114,36,192,62,166,88,140)(11,191,89,113,63,139,37,165)(12,112,38,190,64,164,90,138)(13,189,91,111,65,137,39,163)(14,110,40,188,66,162,92,136)(15,187,93,109,67,135,41,161)(16,108,42,186,68,160,94,134)(17,185,95,107,69,133,43,159)(18,106,44,184,70,158,96,132)(19,183,97,105,71,131,45,157)(20,208,46,182,72,156,98,130)(21,181,99,207,73,129,47,155)(22,206,48,180,74,154,100,128)(23,179,101,205,75,127,49,153)(24,204,50,178,76,152,102,126)(25,177,103,203,77,125,51,151)(26,202,52,176,78,150,104,124)>;
G:=Group( (1,27,53,79)(2,80,54,28)(3,29,55,81)(4,82,56,30)(5,31,57,83)(6,84,58,32)(7,33,59,85)(8,86,60,34)(9,35,61,87)(10,88,62,36)(11,37,63,89)(12,90,64,38)(13,39,65,91)(14,92,66,40)(15,41,67,93)(16,94,68,42)(17,43,69,95)(18,96,70,44)(19,45,71,97)(20,98,72,46)(21,47,73,99)(22,100,74,48)(23,49,75,101)(24,102,76,50)(25,51,77,103)(26,104,78,52)(105,183,157,131)(106,132,158,184)(107,185,159,133)(108,134,160,186)(109,187,161,135)(110,136,162,188)(111,189,163,137)(112,138,164,190)(113,191,165,139)(114,140,166,192)(115,193,167,141)(116,142,168,194)(117,195,169,143)(118,144,170,196)(119,197,171,145)(120,146,172,198)(121,199,173,147)(122,148,174,200)(123,201,175,149)(124,150,176,202)(125,203,177,151)(126,152,178,204)(127,205,179,153)(128,154,180,206)(129,207,181,155)(130,156,182,208), (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), (1,201,79,123,53,149,27,175)(2,122,28,200,54,174,80,148)(3,199,81,121,55,147,29,173)(4,120,30,198,56,172,82,146)(5,197,83,119,57,145,31,171)(6,118,32,196,58,170,84,144)(7,195,85,117,59,143,33,169)(8,116,34,194,60,168,86,142)(9,193,87,115,61,141,35,167)(10,114,36,192,62,166,88,140)(11,191,89,113,63,139,37,165)(12,112,38,190,64,164,90,138)(13,189,91,111,65,137,39,163)(14,110,40,188,66,162,92,136)(15,187,93,109,67,135,41,161)(16,108,42,186,68,160,94,134)(17,185,95,107,69,133,43,159)(18,106,44,184,70,158,96,132)(19,183,97,105,71,131,45,157)(20,208,46,182,72,156,98,130)(21,181,99,207,73,129,47,155)(22,206,48,180,74,154,100,128)(23,179,101,205,75,127,49,153)(24,204,50,178,76,152,102,126)(25,177,103,203,77,125,51,151)(26,202,52,176,78,150,104,124) );
G=PermutationGroup([[(1,27,53,79),(2,80,54,28),(3,29,55,81),(4,82,56,30),(5,31,57,83),(6,84,58,32),(7,33,59,85),(8,86,60,34),(9,35,61,87),(10,88,62,36),(11,37,63,89),(12,90,64,38),(13,39,65,91),(14,92,66,40),(15,41,67,93),(16,94,68,42),(17,43,69,95),(18,96,70,44),(19,45,71,97),(20,98,72,46),(21,47,73,99),(22,100,74,48),(23,49,75,101),(24,102,76,50),(25,51,77,103),(26,104,78,52),(105,183,157,131),(106,132,158,184),(107,185,159,133),(108,134,160,186),(109,187,161,135),(110,136,162,188),(111,189,163,137),(112,138,164,190),(113,191,165,139),(114,140,166,192),(115,193,167,141),(116,142,168,194),(117,195,169,143),(118,144,170,196),(119,197,171,145),(120,146,172,198),(121,199,173,147),(122,148,174,200),(123,201,175,149),(124,150,176,202),(125,203,177,151),(126,152,178,204),(127,205,179,153),(128,154,180,206),(129,207,181,155),(130,156,182,208)], [(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)], [(1,201,79,123,53,149,27,175),(2,122,28,200,54,174,80,148),(3,199,81,121,55,147,29,173),(4,120,30,198,56,172,82,146),(5,197,83,119,57,145,31,171),(6,118,32,196,58,170,84,144),(7,195,85,117,59,143,33,169),(8,116,34,194,60,168,86,142),(9,193,87,115,61,141,35,167),(10,114,36,192,62,166,88,140),(11,191,89,113,63,139,37,165),(12,112,38,190,64,164,90,138),(13,189,91,111,65,137,39,163),(14,110,40,188,66,162,92,136),(15,187,93,109,67,135,41,161),(16,108,42,186,68,160,94,134),(17,185,95,107,69,133,43,159),(18,106,44,184,70,158,96,132),(19,183,97,105,71,131,45,157),(20,208,46,182,72,156,98,130),(21,181,99,207,73,129,47,155),(22,206,48,180,74,154,100,128),(23,179,101,205,75,127,49,153),(24,204,50,178,76,152,102,126),(25,177,103,203,77,125,51,151),(26,202,52,176,78,150,104,124)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| 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 | 52 | 52 | 4 | 4 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D13 | D26 | D52 | C13⋊D4 | C4×D13 | C4.10D4 | C4.12D52 |
| kernel | C4.12D52 | C52.4C4 | C13×M4(2) | C2×Dic26 | C2×Dic13 | C52 | M4(2) | C2×C4 | C4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 6 | 6 | 12 | 12 | 12 | 1 | 12 |
Matrix representation of C4.12D52 ►in GL4(𝔽313) generated by
| 18 | 271 | 0 | 0 |
| 45 | 295 | 0 | 0 |
| 215 | 175 | 93 | 42 |
| 228 | 59 | 271 | 220 |
| 84 | 288 | 232 | 143 |
| 228 | 152 | 23 | 312 |
| 179 | 184 | 228 | 25 |
| 157 | 74 | 288 | 162 |
| 86 | 247 | 169 | 0 |
| 3 | 227 | 123 | 144 |
| 224 | 91 | 288 | 247 |
| 167 | 5 | 0 | 25 |
G:=sub<GL(4,GF(313))| [18,45,215,228,271,295,175,59,0,0,93,271,0,0,42,220],[84,228,179,157,288,152,184,74,232,23,228,288,143,312,25,162],[86,3,224,167,247,227,91,5,169,123,288,0,0,144,247,25] >;
C4.12D52 in GAP, Magma, Sage, TeX
C_4._{12}D_{52} % in TeX
G:=Group("C4.12D52"); // GroupNames label
G:=SmallGroup(416,31);
// by ID
G=gap.SmallGroup(416,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,121,31,362,86,297,13829]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^52=a^2,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^51>;
// generators/relations
Export