direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C25, D4⋊C50, C8⋊1C50, C200⋊5C2, C40.3C10, C50.14D4, C100.17C22, C5.(C5×D8), (C5×D8).C5, (D4×C25)⋊4C2, C4.1(C2×C50), C2.3(D4×C25), (C5×D4).2C10, C10.14(C5×D4), C20.17(C2×C10), SmallGroup(400,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C25
G = < a,b,c | a25=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×C25
►On 200 points
Generators in S200
(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 91 140 63 49 153 186 114)(2 92 141 64 50 154 187 115)(3 93 142 65 26 155 188 116)(4 94 143 66 27 156 189 117)(5 95 144 67 28 157 190 118)(6 96 145 68 29 158 191 119)(7 97 146 69 30 159 192 120)(8 98 147 70 31 160 193 121)(9 99 148 71 32 161 194 122)(10 100 149 72 33 162 195 123)(11 76 150 73 34 163 196 124)(12 77 126 74 35 164 197 125)(13 78 127 75 36 165 198 101)(14 79 128 51 37 166 199 102)(15 80 129 52 38 167 200 103)(16 81 130 53 39 168 176 104)(17 82 131 54 40 169 177 105)(18 83 132 55 41 170 178 106)(19 84 133 56 42 171 179 107)(20 85 134 57 43 172 180 108)(21 86 135 58 44 173 181 109)(22 87 136 59 45 174 182 110)(23 88 137 60 46 175 183 111)(24 89 138 61 47 151 184 112)(25 90 139 62 48 152 185 113)
(51 166)(52 167)(53 168)(54 169)(55 170)(56 171)(57 172)(58 173)(59 174)(60 175)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 163)(74 164)(75 165)(76 124)(77 125)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(126 197)(127 198)(128 199)(129 200)(130 176)(131 177)(132 178)(133 179)(134 180)(135 181)(136 182)(137 183)(138 184)(139 185)(140 186)(141 187)(142 188)(143 189)(144 190)(145 191)(146 192)(147 193)(148 194)(149 195)(150 196)
G:=sub<Sym(200)| (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), (1,91,140,63,49,153,186,114)(2,92,141,64,50,154,187,115)(3,93,142,65,26,155,188,116)(4,94,143,66,27,156,189,117)(5,95,144,67,28,157,190,118)(6,96,145,68,29,158,191,119)(7,97,146,69,30,159,192,120)(8,98,147,70,31,160,193,121)(9,99,148,71,32,161,194,122)(10,100,149,72,33,162,195,123)(11,76,150,73,34,163,196,124)(12,77,126,74,35,164,197,125)(13,78,127,75,36,165,198,101)(14,79,128,51,37,166,199,102)(15,80,129,52,38,167,200,103)(16,81,130,53,39,168,176,104)(17,82,131,54,40,169,177,105)(18,83,132,55,41,170,178,106)(19,84,133,56,42,171,179,107)(20,85,134,57,43,172,180,108)(21,86,135,58,44,173,181,109)(22,87,136,59,45,174,182,110)(23,88,137,60,46,175,183,111)(24,89,138,61,47,151,184,112)(25,90,139,62,48,152,185,113), (51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,124)(77,125)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(126,197)(127,198)(128,199)(129,200)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184)(139,185)(140,186)(141,187)(142,188)(143,189)(144,190)(145,191)(146,192)(147,193)(148,194)(149,195)(150,196)>;
G:=Group( (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), (1,91,140,63,49,153,186,114)(2,92,141,64,50,154,187,115)(3,93,142,65,26,155,188,116)(4,94,143,66,27,156,189,117)(5,95,144,67,28,157,190,118)(6,96,145,68,29,158,191,119)(7,97,146,69,30,159,192,120)(8,98,147,70,31,160,193,121)(9,99,148,71,32,161,194,122)(10,100,149,72,33,162,195,123)(11,76,150,73,34,163,196,124)(12,77,126,74,35,164,197,125)(13,78,127,75,36,165,198,101)(14,79,128,51,37,166,199,102)(15,80,129,52,38,167,200,103)(16,81,130,53,39,168,176,104)(17,82,131,54,40,169,177,105)(18,83,132,55,41,170,178,106)(19,84,133,56,42,171,179,107)(20,85,134,57,43,172,180,108)(21,86,135,58,44,173,181,109)(22,87,136,59,45,174,182,110)(23,88,137,60,46,175,183,111)(24,89,138,61,47,151,184,112)(25,90,139,62,48,152,185,113), (51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,124)(77,125)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(126,197)(127,198)(128,199)(129,200)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184)(139,185)(140,186)(141,187)(142,188)(143,189)(144,190)(145,191)(146,192)(147,193)(148,194)(149,195)(150,196) );
G=PermutationGroup([[(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)], [(1,91,140,63,49,153,186,114),(2,92,141,64,50,154,187,115),(3,93,142,65,26,155,188,116),(4,94,143,66,27,156,189,117),(5,95,144,67,28,157,190,118),(6,96,145,68,29,158,191,119),(7,97,146,69,30,159,192,120),(8,98,147,70,31,160,193,121),(9,99,148,71,32,161,194,122),(10,100,149,72,33,162,195,123),(11,76,150,73,34,163,196,124),(12,77,126,74,35,164,197,125),(13,78,127,75,36,165,198,101),(14,79,128,51,37,166,199,102),(15,80,129,52,38,167,200,103),(16,81,130,53,39,168,176,104),(17,82,131,54,40,169,177,105),(18,83,132,55,41,170,178,106),(19,84,133,56,42,171,179,107),(20,85,134,57,43,172,180,108),(21,86,135,58,44,173,181,109),(22,87,136,59,45,174,182,110),(23,88,137,60,46,175,183,111),(24,89,138,61,47,151,184,112),(25,90,139,62,48,152,185,113)], [(51,166),(52,167),(53,168),(54,169),(55,170),(56,171),(57,172),(58,173),(59,174),(60,175),(61,151),(62,152),(63,153),(64,154),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,161),(72,162),(73,163),(74,164),(75,165),(76,124),(77,125),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(126,197),(127,198),(128,199),(129,200),(130,176),(131,177),(132,178),(133,179),(134,180),(135,181),(136,182),(137,183),(138,184),(139,185),(140,186),(141,187),(142,188),(143,189),(144,190),(145,191),(146,192),(147,193),(148,194),(149,195),(150,196)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 20A | 20B | 20C | 20D | 25A | ··· | 25T | 40A | ··· | 40H | 50A | ··· | 50T | 50U | ··· | 50BH | 100A | ··· | 100T | 200A | ··· | 200AN |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 40 | ··· | 40 | 50 | ··· | 50 | 50 | ··· | 50 | 100 | ··· | 100 | 200 | ··· | 200 |
| size | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | C25 | C50 | C50 | D4 | D8 | C5×D4 | C5×D8 | D4×C25 | D8×C25 |
| kernel | D8×C25 | C200 | D4×C25 | C5×D8 | C40 | C5×D4 | D8 | C8 | D4 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 20 | 20 | 40 | 1 | 2 | 4 | 8 | 20 | 40 |
Matrix representation of D8×C25 ►in GL2(𝔽401) generated by
| 385 | 0 |
| 0 | 385 |
| 0 | 348 |
| 227 | 348 |
| 1 | 0 |
| 1 | 400 |
G:=sub<GL(2,GF(401))| [385,0,0,385],[0,227,348,348],[1,1,0,400] >;
D8×C25 in GAP, Magma, Sage, TeX
D_8\times C_{25} % in TeX
G:=Group("D8xC25"); // GroupNames label
G:=SmallGroup(400,25);
// by ID
G=gap.SmallGroup(400,25);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-5,-2,265,194,5283,2649,261]);
// Polycyclic
G:=Group<a,b,c|a^25=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export