metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D25⋊C8, C20.6F5, C100.3C4, D50.2C4, Dic25.5C22, C25⋊C8⋊3C2, C25⋊1(C2×C8), C5.(D5⋊C8), C4.3(C25⋊C4), C50.1(C2×C4), C10.6(C2×F5), (C4×D25).5C2, C2.1(C2×C25⋊C4), SmallGroup(400,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — D25⋊C8 |
Generators and relations for D25⋊C8
G = < a,b,c | a25=b2=c8=1, bab=a-1, cac-1=a18, cbc-1=a17b >
Smallest permutation representation of D25⋊C8
►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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 28)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(73 75)(76 78)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(101 103)(104 125)(105 124)(106 123)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(127 150)(128 149)(129 148)(130 147)(131 146)(132 145)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(151 172)(152 171)(153 170)(154 169)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(173 175)(176 178)(179 200)(180 199)(181 198)(182 197)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)
(1 190 90 139 40 162 62 115)(2 197 89 132 41 169 61 108)(3 179 88 150 42 151 60 101)(4 186 87 143 43 158 59 119)(5 193 86 136 44 165 58 112)(6 200 85 129 45 172 57 105)(7 182 84 147 46 154 56 123)(8 189 83 140 47 161 55 116)(9 196 82 133 48 168 54 109)(10 178 81 126 49 175 53 102)(11 185 80 144 50 157 52 120)(12 192 79 137 26 164 51 113)(13 199 78 130 27 171 75 106)(14 181 77 148 28 153 74 124)(15 188 76 141 29 160 73 117)(16 195 100 134 30 167 72 110)(17 177 99 127 31 174 71 103)(18 184 98 145 32 156 70 121)(19 191 97 138 33 163 69 114)(20 198 96 131 34 170 68 107)(21 180 95 149 35 152 67 125)(22 187 94 142 36 159 66 118)(23 194 93 135 37 166 65 111)(24 176 92 128 38 173 64 104)(25 183 91 146 39 155 63 122)
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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,28)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(73,75)(76,78)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(101,103)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(151,172)(152,171)(153,170)(154,169)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)(173,175)(176,178)(179,200)(180,199)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190), (1,190,90,139,40,162,62,115)(2,197,89,132,41,169,61,108)(3,179,88,150,42,151,60,101)(4,186,87,143,43,158,59,119)(5,193,86,136,44,165,58,112)(6,200,85,129,45,172,57,105)(7,182,84,147,46,154,56,123)(8,189,83,140,47,161,55,116)(9,196,82,133,48,168,54,109)(10,178,81,126,49,175,53,102)(11,185,80,144,50,157,52,120)(12,192,79,137,26,164,51,113)(13,199,78,130,27,171,75,106)(14,181,77,148,28,153,74,124)(15,188,76,141,29,160,73,117)(16,195,100,134,30,167,72,110)(17,177,99,127,31,174,71,103)(18,184,98,145,32,156,70,121)(19,191,97,138,33,163,69,114)(20,198,96,131,34,170,68,107)(21,180,95,149,35,152,67,125)(22,187,94,142,36,159,66,118)(23,194,93,135,37,166,65,111)(24,176,92,128,38,173,64,104)(25,183,91,146,39,155,63,122)>;
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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,28)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(73,75)(76,78)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(101,103)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(151,172)(152,171)(153,170)(154,169)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)(173,175)(176,178)(179,200)(180,199)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190), (1,190,90,139,40,162,62,115)(2,197,89,132,41,169,61,108)(3,179,88,150,42,151,60,101)(4,186,87,143,43,158,59,119)(5,193,86,136,44,165,58,112)(6,200,85,129,45,172,57,105)(7,182,84,147,46,154,56,123)(8,189,83,140,47,161,55,116)(9,196,82,133,48,168,54,109)(10,178,81,126,49,175,53,102)(11,185,80,144,50,157,52,120)(12,192,79,137,26,164,51,113)(13,199,78,130,27,171,75,106)(14,181,77,148,28,153,74,124)(15,188,76,141,29,160,73,117)(16,195,100,134,30,167,72,110)(17,177,99,127,31,174,71,103)(18,184,98,145,32,156,70,121)(19,191,97,138,33,163,69,114)(20,198,96,131,34,170,68,107)(21,180,95,149,35,152,67,125)(22,187,94,142,36,159,66,118)(23,194,93,135,37,166,65,111)(24,176,92,128,38,173,64,104)(25,183,91,146,39,155,63,122) );
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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,28),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(73,75),(76,78),(79,100),(80,99),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(101,103),(104,125),(105,124),(106,123),(107,122),(108,121),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(127,150),(128,149),(129,148),(130,147),(131,146),(132,145),(133,144),(134,143),(135,142),(136,141),(137,140),(138,139),(151,172),(152,171),(153,170),(154,169),(155,168),(156,167),(157,166),(158,165),(159,164),(160,163),(161,162),(173,175),(176,178),(179,200),(180,199),(181,198),(182,197),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190)], [(1,190,90,139,40,162,62,115),(2,197,89,132,41,169,61,108),(3,179,88,150,42,151,60,101),(4,186,87,143,43,158,59,119),(5,193,86,136,44,165,58,112),(6,200,85,129,45,172,57,105),(7,182,84,147,46,154,56,123),(8,189,83,140,47,161,55,116),(9,196,82,133,48,168,54,109),(10,178,81,126,49,175,53,102),(11,185,80,144,50,157,52,120),(12,192,79,137,26,164,51,113),(13,199,78,130,27,171,75,106),(14,181,77,148,28,153,74,124),(15,188,76,141,29,160,73,117),(16,195,100,134,30,167,72,110),(17,177,99,127,31,174,71,103),(18,184,98,145,32,156,70,121),(19,191,97,138,33,163,69,114),(20,198,96,131,34,170,68,107),(21,180,95,149,35,152,67,125),(22,187,94,142,36,159,66,118),(23,194,93,135,37,166,65,111),(24,176,92,128,38,173,64,104),(25,183,91,146,39,155,63,122)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | ··· | 8H | 10 | 20A | 20B | 25A | ··· | 25E | 50A | ··· | 50E | 100A | ··· | 100J |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 25 | 25 | 4 | 25 | ··· | 25 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | F5 | C2×F5 | D5⋊C8 | C25⋊C4 | C2×C25⋊C4 | D25⋊C8 |
| kernel | D25⋊C8 | C25⋊C8 | C4×D25 | C100 | D50 | D25 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 5 | 5 | 10 |
Matrix representation of D25⋊C8 ►in GL5(𝔽401)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 39 | 272 | 140 | 227 |
| 0 | 174 | 213 | 45 | 314 |
| 0 | 87 | 261 | 300 | 132 |
| 0 | 269 | 356 | 129 | 168 |
| 400 | 0 | 0 | 0 | 0 |
| 0 | 39 | 272 | 140 | 227 |
| 0 | 233 | 101 | 188 | 362 |
| 0 | 269 | 356 | 129 | 168 |
| 0 | 87 | 261 | 300 | 132 |
| 303 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 393 | 393 |
| 0 | 393 | 393 | 0 | 5 |
| 0 | 8 | 13 | 8 | 0 |
| 0 | 396 | 388 | 388 | 396 |
G:=sub<GL(5,GF(401))| [1,0,0,0,0,0,39,174,87,269,0,272,213,261,356,0,140,45,300,129,0,227,314,132,168],[400,0,0,0,0,0,39,233,269,87,0,272,101,356,261,0,140,188,129,300,0,227,362,168,132],[303,0,0,0,0,0,5,393,8,396,0,0,393,13,388,0,393,0,8,388,0,393,5,0,396] >;
D25⋊C8 in GAP, Magma, Sage, TeX
D_{25}\rtimes C_8 % in TeX
G:=Group("D25:C8"); // GroupNames label
G:=SmallGroup(400,28);
// by ID
G=gap.SmallGroup(400,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,50,3364,2896,178,5765,2897]);
// Polycyclic
G:=Group<a,b,c|a^25=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^18,c*b*c^-1=a^17*b>;
// generators/relations
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