metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic25, C100.4C4, C4.15D50, C25⋊4M4(2), C20.54D10, C20.4Dic5, C22.Dic25, C100.15C22, C25⋊2C8⋊5C2, (C2×C50).5C4, (C2×C4).2D25, (C2×C100).4C2, C50.14(C2×C4), (C2×C20).10D5, C5.(C4.Dic5), (C2×C10).5Dic5, C2.3(C2×Dic25), C10.14(C2×Dic5), SmallGroup(400,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic25
G = < a,b,c | a4=1, b50=a2, c2=b25, ab=ba, cac-1=a-1, cbc-1=b49 >
Smallest permutation representation of C4.Dic25
►On 200 points
Generators in S200
(1 26 51 76)(2 27 52 77)(3 28 53 78)(4 29 54 79)(5 30 55 80)(6 31 56 81)(7 32 57 82)(8 33 58 83)(9 34 59 84)(10 35 60 85)(11 36 61 86)(12 37 62 87)(13 38 63 88)(14 39 64 89)(15 40 65 90)(16 41 66 91)(17 42 67 92)(18 43 68 93)(19 44 69 94)(20 45 70 95)(21 46 71 96)(22 47 72 97)(23 48 73 98)(24 49 74 99)(25 50 75 100)(101 176 151 126)(102 177 152 127)(103 178 153 128)(104 179 154 129)(105 180 155 130)(106 181 156 131)(107 182 157 132)(108 183 158 133)(109 184 159 134)(110 185 160 135)(111 186 161 136)(112 187 162 137)(113 188 163 138)(114 189 164 139)(115 190 165 140)(116 191 166 141)(117 192 167 142)(118 193 168 143)(119 194 169 144)(120 195 170 145)(121 196 171 146)(122 197 172 147)(123 198 173 148)(124 199 174 149)(125 200 175 150)
(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)
(1 196 26 121 51 146 76 171)(2 145 27 170 52 195 77 120)(3 194 28 119 53 144 78 169)(4 143 29 168 54 193 79 118)(5 192 30 117 55 142 80 167)(6 141 31 166 56 191 81 116)(7 190 32 115 57 140 82 165)(8 139 33 164 58 189 83 114)(9 188 34 113 59 138 84 163)(10 137 35 162 60 187 85 112)(11 186 36 111 61 136 86 161)(12 135 37 160 62 185 87 110)(13 184 38 109 63 134 88 159)(14 133 39 158 64 183 89 108)(15 182 40 107 65 132 90 157)(16 131 41 156 66 181 91 106)(17 180 42 105 67 130 92 155)(18 129 43 154 68 179 93 104)(19 178 44 103 69 128 94 153)(20 127 45 152 70 177 95 102)(21 176 46 101 71 126 96 151)(22 125 47 150 72 175 97 200)(23 174 48 199 73 124 98 149)(24 123 49 148 74 173 99 198)(25 172 50 197 75 122 100 147)
G:=sub<Sym(200)| (1,26,51,76)(2,27,52,77)(3,28,53,78)(4,29,54,79)(5,30,55,80)(6,31,56,81)(7,32,57,82)(8,33,58,83)(9,34,59,84)(10,35,60,85)(11,36,61,86)(12,37,62,87)(13,38,63,88)(14,39,64,89)(15,40,65,90)(16,41,66,91)(17,42,67,92)(18,43,68,93)(19,44,69,94)(20,45,70,95)(21,46,71,96)(22,47,72,97)(23,48,73,98)(24,49,74,99)(25,50,75,100)(101,176,151,126)(102,177,152,127)(103,178,153,128)(104,179,154,129)(105,180,155,130)(106,181,156,131)(107,182,157,132)(108,183,158,133)(109,184,159,134)(110,185,160,135)(111,186,161,136)(112,187,162,137)(113,188,163,138)(114,189,164,139)(115,190,165,140)(116,191,166,141)(117,192,167,142)(118,193,168,143)(119,194,169,144)(120,195,170,145)(121,196,171,146)(122,197,172,147)(123,198,173,148)(124,199,174,149)(125,200,175,150), (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), (1,196,26,121,51,146,76,171)(2,145,27,170,52,195,77,120)(3,194,28,119,53,144,78,169)(4,143,29,168,54,193,79,118)(5,192,30,117,55,142,80,167)(6,141,31,166,56,191,81,116)(7,190,32,115,57,140,82,165)(8,139,33,164,58,189,83,114)(9,188,34,113,59,138,84,163)(10,137,35,162,60,187,85,112)(11,186,36,111,61,136,86,161)(12,135,37,160,62,185,87,110)(13,184,38,109,63,134,88,159)(14,133,39,158,64,183,89,108)(15,182,40,107,65,132,90,157)(16,131,41,156,66,181,91,106)(17,180,42,105,67,130,92,155)(18,129,43,154,68,179,93,104)(19,178,44,103,69,128,94,153)(20,127,45,152,70,177,95,102)(21,176,46,101,71,126,96,151)(22,125,47,150,72,175,97,200)(23,174,48,199,73,124,98,149)(24,123,49,148,74,173,99,198)(25,172,50,197,75,122,100,147)>;
G:=Group( (1,26,51,76)(2,27,52,77)(3,28,53,78)(4,29,54,79)(5,30,55,80)(6,31,56,81)(7,32,57,82)(8,33,58,83)(9,34,59,84)(10,35,60,85)(11,36,61,86)(12,37,62,87)(13,38,63,88)(14,39,64,89)(15,40,65,90)(16,41,66,91)(17,42,67,92)(18,43,68,93)(19,44,69,94)(20,45,70,95)(21,46,71,96)(22,47,72,97)(23,48,73,98)(24,49,74,99)(25,50,75,100)(101,176,151,126)(102,177,152,127)(103,178,153,128)(104,179,154,129)(105,180,155,130)(106,181,156,131)(107,182,157,132)(108,183,158,133)(109,184,159,134)(110,185,160,135)(111,186,161,136)(112,187,162,137)(113,188,163,138)(114,189,164,139)(115,190,165,140)(116,191,166,141)(117,192,167,142)(118,193,168,143)(119,194,169,144)(120,195,170,145)(121,196,171,146)(122,197,172,147)(123,198,173,148)(124,199,174,149)(125,200,175,150), (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), (1,196,26,121,51,146,76,171)(2,145,27,170,52,195,77,120)(3,194,28,119,53,144,78,169)(4,143,29,168,54,193,79,118)(5,192,30,117,55,142,80,167)(6,141,31,166,56,191,81,116)(7,190,32,115,57,140,82,165)(8,139,33,164,58,189,83,114)(9,188,34,113,59,138,84,163)(10,137,35,162,60,187,85,112)(11,186,36,111,61,136,86,161)(12,135,37,160,62,185,87,110)(13,184,38,109,63,134,88,159)(14,133,39,158,64,183,89,108)(15,182,40,107,65,132,90,157)(16,131,41,156,66,181,91,106)(17,180,42,105,67,130,92,155)(18,129,43,154,68,179,93,104)(19,178,44,103,69,128,94,153)(20,127,45,152,70,177,95,102)(21,176,46,101,71,126,96,151)(22,125,47,150,72,175,97,200)(23,174,48,199,73,124,98,149)(24,123,49,148,74,173,99,198)(25,172,50,197,75,122,100,147) );
G=PermutationGroup([[(1,26,51,76),(2,27,52,77),(3,28,53,78),(4,29,54,79),(5,30,55,80),(6,31,56,81),(7,32,57,82),(8,33,58,83),(9,34,59,84),(10,35,60,85),(11,36,61,86),(12,37,62,87),(13,38,63,88),(14,39,64,89),(15,40,65,90),(16,41,66,91),(17,42,67,92),(18,43,68,93),(19,44,69,94),(20,45,70,95),(21,46,71,96),(22,47,72,97),(23,48,73,98),(24,49,74,99),(25,50,75,100),(101,176,151,126),(102,177,152,127),(103,178,153,128),(104,179,154,129),(105,180,155,130),(106,181,156,131),(107,182,157,132),(108,183,158,133),(109,184,159,134),(110,185,160,135),(111,186,161,136),(112,187,162,137),(113,188,163,138),(114,189,164,139),(115,190,165,140),(116,191,166,141),(117,192,167,142),(118,193,168,143),(119,194,169,144),(120,195,170,145),(121,196,171,146),(122,197,172,147),(123,198,173,148),(124,199,174,149),(125,200,175,150)], [(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)], [(1,196,26,121,51,146,76,171),(2,145,27,170,52,195,77,120),(3,194,28,119,53,144,78,169),(4,143,29,168,54,193,79,118),(5,192,30,117,55,142,80,167),(6,141,31,166,56,191,81,116),(7,190,32,115,57,140,82,165),(8,139,33,164,58,189,83,114),(9,188,34,113,59,138,84,163),(10,137,35,162,60,187,85,112),(11,186,36,111,61,136,86,161),(12,135,37,160,62,185,87,110),(13,184,38,109,63,134,88,159),(14,133,39,158,64,183,89,108),(15,182,40,107,65,132,90,157),(16,131,41,156,66,181,91,106),(17,180,42,105,67,130,92,155),(18,129,43,154,68,179,93,104),(19,178,44,103,69,128,94,153),(20,127,45,152,70,177,95,102),(21,176,46,101,71,126,96,151),(22,125,47,150,72,175,97,200),(23,174,48,199,73,124,98,149),(24,123,49,148,74,173,99,198),(25,172,50,197,75,122,100,147)]])
106 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 25A | ··· | 25J | 50A | ··· | 50AD | 100A | ··· | 100AN |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 50 | 50 | 50 | 50 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C4 | C4 | D5 | M4(2) | Dic5 | D10 | Dic5 | D25 | C4.Dic5 | Dic25 | D50 | Dic25 | C4.Dic25 |
| kernel | C4.Dic25 | C25⋊2C8 | C2×C100 | C100 | C2×C50 | C2×C20 | C25 | C20 | C20 | C2×C10 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 8 | 10 | 10 | 10 | 40 |
Matrix representation of C4.Dic25 ►in GL2(𝔽401) generated by
| 20 | 0 |
| 120 | 381 |
| 285 | 0 |
| 91 | 121 |
| 18 | 395 |
| 318 | 383 |
G:=sub<GL(2,GF(401))| [20,120,0,381],[285,91,0,121],[18,318,395,383] >;
C4.Dic25 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{25} % in TeX
G:=Group("C4.Dic25"); // GroupNames label
G:=SmallGroup(400,10);
// by ID
G=gap.SmallGroup(400,10);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^50=a^2,c^2=b^25,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^49>;
// generators/relations
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