metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D25, C200⋊4C2, C40.7D5, D50.1C4, C4.13D50, C25⋊3M4(2), C20.52D10, Dic25.1C4, C100.13C22, C25⋊2C8⋊4C2, C50.9(C2×C4), C5.(C8⋊D5), C2.3(C4×D25), (C4×D25).2C2, C10.13(C4×D5), SmallGroup(400,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D25
G = < a,b,c | a8=b25=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D25
►On 200 points
Generators in S200
(1 182 97 146 26 159 75 107)(2 183 98 147 27 160 51 108)(3 184 99 148 28 161 52 109)(4 185 100 149 29 162 53 110)(5 186 76 150 30 163 54 111)(6 187 77 126 31 164 55 112)(7 188 78 127 32 165 56 113)(8 189 79 128 33 166 57 114)(9 190 80 129 34 167 58 115)(10 191 81 130 35 168 59 116)(11 192 82 131 36 169 60 117)(12 193 83 132 37 170 61 118)(13 194 84 133 38 171 62 119)(14 195 85 134 39 172 63 120)(15 196 86 135 40 173 64 121)(16 197 87 136 41 174 65 122)(17 198 88 137 42 175 66 123)(18 199 89 138 43 151 67 124)(19 200 90 139 44 152 68 125)(20 176 91 140 45 153 69 101)(21 177 92 141 46 154 70 102)(22 178 93 142 47 155 71 103)(23 179 94 143 48 156 72 104)(24 180 95 144 49 157 73 105)(25 181 96 145 50 158 74 106)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(74 75)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(93 100)(94 99)(95 98)(96 97)(101 126)(102 150)(103 149)(104 148)(105 147)(106 146)(107 145)(108 144)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 135)(118 134)(119 133)(120 132)(121 131)(122 130)(123 129)(124 128)(125 127)(151 189)(152 188)(153 187)(154 186)(155 185)(156 184)(157 183)(158 182)(159 181)(160 180)(161 179)(162 178)(163 177)(164 176)(165 200)(166 199)(167 198)(168 197)(169 196)(170 195)(171 194)(172 193)(173 192)(174 191)(175 190)
G:=sub<Sym(200)| (1,182,97,146,26,159,75,107)(2,183,98,147,27,160,51,108)(3,184,99,148,28,161,52,109)(4,185,100,149,29,162,53,110)(5,186,76,150,30,163,54,111)(6,187,77,126,31,164,55,112)(7,188,78,127,32,165,56,113)(8,189,79,128,33,166,57,114)(9,190,80,129,34,167,58,115)(10,191,81,130,35,168,59,116)(11,192,82,131,36,169,60,117)(12,193,83,132,37,170,61,118)(13,194,84,133,38,171,62,119)(14,195,85,134,39,172,63,120)(15,196,86,135,40,173,64,121)(16,197,87,136,41,174,65,122)(17,198,88,137,42,175,66,123)(18,199,89,138,43,151,67,124)(19,200,90,139,44,152,68,125)(20,176,91,140,45,153,69,101)(21,177,92,141,46,154,70,102)(22,178,93,142,47,155,71,103)(23,179,94,143,48,156,72,104)(24,180,95,144,49,157,73,105)(25,181,96,145,50,158,74,106), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(74,75)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(93,100)(94,99)(95,98)(96,97)(101,126)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(151,189)(152,188)(153,187)(154,186)(155,185)(156,184)(157,183)(158,182)(159,181)(160,180)(161,179)(162,178)(163,177)(164,176)(165,200)(166,199)(167,198)(168,197)(169,196)(170,195)(171,194)(172,193)(173,192)(174,191)(175,190)>;
G:=Group( (1,182,97,146,26,159,75,107)(2,183,98,147,27,160,51,108)(3,184,99,148,28,161,52,109)(4,185,100,149,29,162,53,110)(5,186,76,150,30,163,54,111)(6,187,77,126,31,164,55,112)(7,188,78,127,32,165,56,113)(8,189,79,128,33,166,57,114)(9,190,80,129,34,167,58,115)(10,191,81,130,35,168,59,116)(11,192,82,131,36,169,60,117)(12,193,83,132,37,170,61,118)(13,194,84,133,38,171,62,119)(14,195,85,134,39,172,63,120)(15,196,86,135,40,173,64,121)(16,197,87,136,41,174,65,122)(17,198,88,137,42,175,66,123)(18,199,89,138,43,151,67,124)(19,200,90,139,44,152,68,125)(20,176,91,140,45,153,69,101)(21,177,92,141,46,154,70,102)(22,178,93,142,47,155,71,103)(23,179,94,143,48,156,72,104)(24,180,95,144,49,157,73,105)(25,181,96,145,50,158,74,106), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(74,75)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(93,100)(94,99)(95,98)(96,97)(101,126)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(151,189)(152,188)(153,187)(154,186)(155,185)(156,184)(157,183)(158,182)(159,181)(160,180)(161,179)(162,178)(163,177)(164,176)(165,200)(166,199)(167,198)(168,197)(169,196)(170,195)(171,194)(172,193)(173,192)(174,191)(175,190) );
G=PermutationGroup([[(1,182,97,146,26,159,75,107),(2,183,98,147,27,160,51,108),(3,184,99,148,28,161,52,109),(4,185,100,149,29,162,53,110),(5,186,76,150,30,163,54,111),(6,187,77,126,31,164,55,112),(7,188,78,127,32,165,56,113),(8,189,79,128,33,166,57,114),(9,190,80,129,34,167,58,115),(10,191,81,130,35,168,59,116),(11,192,82,131,36,169,60,117),(12,193,83,132,37,170,61,118),(13,194,84,133,38,171,62,119),(14,195,85,134,39,172,63,120),(15,196,86,135,40,173,64,121),(16,197,87,136,41,174,65,122),(17,198,88,137,42,175,66,123),(18,199,89,138,43,151,67,124),(19,200,90,139,44,152,68,125),(20,176,91,140,45,153,69,101),(21,177,92,141,46,154,70,102),(22,178,93,142,47,155,71,103),(23,179,94,143,48,156,72,104),(24,180,95,144,49,157,73,105),(25,181,96,145,50,158,74,106)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(61,63),(74,75),(76,92),(77,91),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85),(93,100),(94,99),(95,98),(96,97),(101,126),(102,150),(103,149),(104,148),(105,147),(106,146),(107,145),(108,144),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,135),(118,134),(119,133),(120,132),(121,131),(122,130),(123,129),(124,128),(125,127),(151,189),(152,188),(153,187),(154,186),(155,185),(156,184),(157,183),(158,182),(159,181),(160,180),(161,179),(162,178),(163,177),(164,176),(165,200),(166,199),(167,198),(168,197),(169,196),(170,195),(171,194),(172,193),(173,192),(174,191),(175,190)]])
106 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 20A | 20B | 20C | 20D | 25A | ··· | 25J | 40A | ··· | 40H | 50A | ··· | 50J | 100A | ··· | 100T | 200A | ··· | 200AN |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 40 | ··· | 40 | 50 | ··· | 50 | 100 | ··· | 100 | 200 | ··· | 200 |
| size | 1 | 1 | 50 | 1 | 1 | 50 | 2 | 2 | 2 | 2 | 50 | 50 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D5 | M4(2) | D10 | C4×D5 | D25 | C8⋊D5 | D50 | C4×D25 | C8⋊D25 |
| kernel | C8⋊D25 | C25⋊2C8 | C200 | C4×D25 | Dic25 | D50 | C40 | C25 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 10 | 8 | 10 | 20 | 40 |
Matrix representation of C8⋊D25 ►in GL2(𝔽401) generated by
| 227 | 185 |
| 348 | 174 |
| 311 | 339 |
| 189 | 99 |
| 329 | 366 |
| 171 | 72 |
G:=sub<GL(2,GF(401))| [227,348,185,174],[311,189,339,99],[329,171,366,72] >;
C8⋊D25 in GAP, Magma, Sage, TeX
C_8\rtimes D_{25} % in TeX
G:=Group("C8:D25"); // GroupNames label
G:=SmallGroup(400,6);
// by ID
G=gap.SmallGroup(400,6);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^8=b^25=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
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