direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D25, C200⋊3C2, C40.6D5, D50.4C4, C4.12D50, C20.51D10, Dic25.4C4, C100.12C22, C5.(C8×D5), C25⋊3(C2×C8), C25⋊2C8⋊6C2, C50.8(C2×C4), C2.1(C4×D25), (C4×D25).7C2, C10.12(C4×D5), SmallGroup(400,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C8×D25 |
Generators and relations for C8×D25
G = < a,b,c | a8=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D25
►On 200 points
Generators in S200
(1 189 76 135 49 164 72 103)(2 190 77 136 50 165 73 104)(3 191 78 137 26 166 74 105)(4 192 79 138 27 167 75 106)(5 193 80 139 28 168 51 107)(6 194 81 140 29 169 52 108)(7 195 82 141 30 170 53 109)(8 196 83 142 31 171 54 110)(9 197 84 143 32 172 55 111)(10 198 85 144 33 173 56 112)(11 199 86 145 34 174 57 113)(12 200 87 146 35 175 58 114)(13 176 88 147 36 151 59 115)(14 177 89 148 37 152 60 116)(15 178 90 149 38 153 61 117)(16 179 91 150 39 154 62 118)(17 180 92 126 40 155 63 119)(18 181 93 127 41 156 64 120)(19 182 94 128 42 157 65 121)(20 183 95 129 43 158 66 122)(21 184 96 130 44 159 67 123)(22 185 97 131 45 160 68 124)(23 186 98 132 46 161 69 125)(24 187 99 133 47 162 70 101)(25 188 100 134 48 163 71 102)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 50)(25 49)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 100)(73 99)(74 98)(75 97)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 150)(113 149)(114 148)(115 147)(116 146)(117 145)(118 144)(119 143)(120 142)(121 141)(122 140)(123 139)(124 138)(125 137)(151 176)(152 200)(153 199)(154 198)(155 197)(156 196)(157 195)(158 194)(159 193)(160 192)(161 191)(162 190)(163 189)(164 188)(165 187)(166 186)(167 185)(168 184)(169 183)(170 182)(171 181)(172 180)(173 179)(174 178)(175 177)
G:=sub<Sym(200)| (1,189,76,135,49,164,72,103)(2,190,77,136,50,165,73,104)(3,191,78,137,26,166,74,105)(4,192,79,138,27,167,75,106)(5,193,80,139,28,168,51,107)(6,194,81,140,29,169,52,108)(7,195,82,141,30,170,53,109)(8,196,83,142,31,171,54,110)(9,197,84,143,32,172,55,111)(10,198,85,144,33,173,56,112)(11,199,86,145,34,174,57,113)(12,200,87,146,35,175,58,114)(13,176,88,147,36,151,59,115)(14,177,89,148,37,152,60,116)(15,178,90,149,38,153,61,117)(16,179,91,150,39,154,62,118)(17,180,92,126,40,155,63,119)(18,181,93,127,41,156,64,120)(19,182,94,128,42,157,65,121)(20,183,95,129,43,158,66,122)(21,184,96,130,44,159,67,123)(22,185,97,131,45,160,68,124)(23,186,98,132,46,161,69,125)(24,187,99,133,47,162,70,101)(25,188,100,134,48,163,71,102), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,50)(25,49)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,100)(73,99)(74,98)(75,97)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(151,176)(152,200)(153,199)(154,198)(155,197)(156,196)(157,195)(158,194)(159,193)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(169,183)(170,182)(171,181)(172,180)(173,179)(174,178)(175,177)>;
G:=Group( (1,189,76,135,49,164,72,103)(2,190,77,136,50,165,73,104)(3,191,78,137,26,166,74,105)(4,192,79,138,27,167,75,106)(5,193,80,139,28,168,51,107)(6,194,81,140,29,169,52,108)(7,195,82,141,30,170,53,109)(8,196,83,142,31,171,54,110)(9,197,84,143,32,172,55,111)(10,198,85,144,33,173,56,112)(11,199,86,145,34,174,57,113)(12,200,87,146,35,175,58,114)(13,176,88,147,36,151,59,115)(14,177,89,148,37,152,60,116)(15,178,90,149,38,153,61,117)(16,179,91,150,39,154,62,118)(17,180,92,126,40,155,63,119)(18,181,93,127,41,156,64,120)(19,182,94,128,42,157,65,121)(20,183,95,129,43,158,66,122)(21,184,96,130,44,159,67,123)(22,185,97,131,45,160,68,124)(23,186,98,132,46,161,69,125)(24,187,99,133,47,162,70,101)(25,188,100,134,48,163,71,102), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,50)(25,49)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,100)(73,99)(74,98)(75,97)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(151,176)(152,200)(153,199)(154,198)(155,197)(156,196)(157,195)(158,194)(159,193)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(169,183)(170,182)(171,181)(172,180)(173,179)(174,178)(175,177) );
G=PermutationGroup([[(1,189,76,135,49,164,72,103),(2,190,77,136,50,165,73,104),(3,191,78,137,26,166,74,105),(4,192,79,138,27,167,75,106),(5,193,80,139,28,168,51,107),(6,194,81,140,29,169,52,108),(7,195,82,141,30,170,53,109),(8,196,83,142,31,171,54,110),(9,197,84,143,32,172,55,111),(10,198,85,144,33,173,56,112),(11,199,86,145,34,174,57,113),(12,200,87,146,35,175,58,114),(13,176,88,147,36,151,59,115),(14,177,89,148,37,152,60,116),(15,178,90,149,38,153,61,117),(16,179,91,150,39,154,62,118),(17,180,92,126,40,155,63,119),(18,181,93,127,41,156,64,120),(19,182,94,128,42,157,65,121),(20,183,95,129,43,158,66,122),(21,184,96,130,44,159,67,123),(22,185,97,131,45,160,68,124),(23,186,98,132,46,161,69,125),(24,187,99,133,47,162,70,101),(25,188,100,134,48,163,71,102)], [(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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,50),(25,49),(51,96),(52,95),(53,94),(54,93),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,100),(73,99),(74,98),(75,97),(101,136),(102,135),(103,134),(104,133),(105,132),(106,131),(107,130),(108,129),(109,128),(110,127),(111,126),(112,150),(113,149),(114,148),(115,147),(116,146),(117,145),(118,144),(119,143),(120,142),(121,141),(122,140),(123,139),(124,138),(125,137),(151,176),(152,200),(153,199),(154,198),(155,197),(156,196),(157,195),(158,194),(159,193),(160,192),(161,191),(162,190),(163,189),(164,188),(165,187),(166,186),(167,185),(168,184),(169,183),(170,182),(171,181),(172,180),(173,179),(174,178),(175,177)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 20A | 20B | 20C | 20D | 25A | ··· | 25J | 40A | ··· | 40H | 50A | ··· | 50J | 100A | ··· | 100T | 200A | ··· | 200AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 40 | ··· | 40 | 50 | ··· | 50 | 100 | ··· | 100 | 200 | ··· | 200 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 25 | 25 | 2 | 2 | 1 | 1 | 1 | 1 | 25 | 25 | 25 | 25 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D5 | D10 | C4×D5 | D25 | C8×D5 | D50 | C4×D25 | C8×D25 |
| kernel | C8×D25 | C25⋊2C8 | C200 | C4×D25 | Dic25 | D50 | D25 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 10 | 8 | 10 | 20 | 40 |
Matrix representation of C8×D25 ►in GL3(𝔽401) generated by
| 356 | 0 | 0 |
| 0 | 381 | 0 |
| 0 | 0 | 381 |
| 1 | 0 | 0 |
| 0 | 62 | 373 |
| 0 | 28 | 162 |
| 1 | 0 | 0 |
| 0 | 62 | 373 |
| 0 | 37 | 339 |
G:=sub<GL(3,GF(401))| [356,0,0,0,381,0,0,0,381],[1,0,0,0,62,28,0,373,162],[1,0,0,0,62,37,0,373,339] >;
C8×D25 in GAP, Magma, Sage, TeX
C_8\times D_{25} % in TeX
G:=Group("C8xD25"); // GroupNames label
G:=SmallGroup(400,5);
// by ID
G=gap.SmallGroup(400,5);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,31,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^8=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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