direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic25, C50⋊2C4, C2.2D50, C22.D25, C10.9D10, C50.4C22, C10.3Dic5, C25⋊3(C2×C4), (C2×C50).C2, C5.(C2×Dic5), (C2×C10).1D5, SmallGroup(200,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C2×Dic25 |
Generators and relations for C2×Dic25
G = < a,b,c | a2=b50=1, c2=b25, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic25
►Regular action on 200 points
Generators in S200
(1 97)(2 98)(3 99)(4 100)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(101 198)(102 199)(103 200)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(113 160)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)(121 168)(122 169)(123 170)(124 171)(125 172)(126 173)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(133 180)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)(141 188)(142 189)(143 190)(144 191)(145 192)(146 193)(147 194)(148 195)(149 196)(150 197)
(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)
(1 147 26 122)(2 146 27 121)(3 145 28 120)(4 144 29 119)(5 143 30 118)(6 142 31 117)(7 141 32 116)(8 140 33 115)(9 139 34 114)(10 138 35 113)(11 137 36 112)(12 136 37 111)(13 135 38 110)(14 134 39 109)(15 133 40 108)(16 132 41 107)(17 131 42 106)(18 130 43 105)(19 129 44 104)(20 128 45 103)(21 127 46 102)(22 126 47 101)(23 125 48 150)(24 124 49 149)(25 123 50 148)(51 190 76 165)(52 189 77 164)(53 188 78 163)(54 187 79 162)(55 186 80 161)(56 185 81 160)(57 184 82 159)(58 183 83 158)(59 182 84 157)(60 181 85 156)(61 180 86 155)(62 179 87 154)(63 178 88 153)(64 177 89 152)(65 176 90 151)(66 175 91 200)(67 174 92 199)(68 173 93 198)(69 172 94 197)(70 171 95 196)(71 170 96 195)(72 169 97 194)(73 168 98 193)(74 167 99 192)(75 166 100 191)
G:=sub<Sym(200)| (1,97)(2,98)(3,99)(4,100)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(101,198)(102,199)(103,200)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187)(141,188)(142,189)(143,190)(144,191)(145,192)(146,193)(147,194)(148,195)(149,196)(150,197), (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), (1,147,26,122)(2,146,27,121)(3,145,28,120)(4,144,29,119)(5,143,30,118)(6,142,31,117)(7,141,32,116)(8,140,33,115)(9,139,34,114)(10,138,35,113)(11,137,36,112)(12,136,37,111)(13,135,38,110)(14,134,39,109)(15,133,40,108)(16,132,41,107)(17,131,42,106)(18,130,43,105)(19,129,44,104)(20,128,45,103)(21,127,46,102)(22,126,47,101)(23,125,48,150)(24,124,49,149)(25,123,50,148)(51,190,76,165)(52,189,77,164)(53,188,78,163)(54,187,79,162)(55,186,80,161)(56,185,81,160)(57,184,82,159)(58,183,83,158)(59,182,84,157)(60,181,85,156)(61,180,86,155)(62,179,87,154)(63,178,88,153)(64,177,89,152)(65,176,90,151)(66,175,91,200)(67,174,92,199)(68,173,93,198)(69,172,94,197)(70,171,95,196)(71,170,96,195)(72,169,97,194)(73,168,98,193)(74,167,99,192)(75,166,100,191)>;
G:=Group( (1,97)(2,98)(3,99)(4,100)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(101,198)(102,199)(103,200)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187)(141,188)(142,189)(143,190)(144,191)(145,192)(146,193)(147,194)(148,195)(149,196)(150,197), (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), (1,147,26,122)(2,146,27,121)(3,145,28,120)(4,144,29,119)(5,143,30,118)(6,142,31,117)(7,141,32,116)(8,140,33,115)(9,139,34,114)(10,138,35,113)(11,137,36,112)(12,136,37,111)(13,135,38,110)(14,134,39,109)(15,133,40,108)(16,132,41,107)(17,131,42,106)(18,130,43,105)(19,129,44,104)(20,128,45,103)(21,127,46,102)(22,126,47,101)(23,125,48,150)(24,124,49,149)(25,123,50,148)(51,190,76,165)(52,189,77,164)(53,188,78,163)(54,187,79,162)(55,186,80,161)(56,185,81,160)(57,184,82,159)(58,183,83,158)(59,182,84,157)(60,181,85,156)(61,180,86,155)(62,179,87,154)(63,178,88,153)(64,177,89,152)(65,176,90,151)(66,175,91,200)(67,174,92,199)(68,173,93,198)(69,172,94,197)(70,171,95,196)(71,170,96,195)(72,169,97,194)(73,168,98,193)(74,167,99,192)(75,166,100,191) );
G=PermutationGroup([[(1,97),(2,98),(3,99),(4,100),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,83),(38,84),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,95),(50,96),(101,198),(102,199),(103,200),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159),(113,160),(114,161),(115,162),(116,163),(117,164),(118,165),(119,166),(120,167),(121,168),(122,169),(123,170),(124,171),(125,172),(126,173),(127,174),(128,175),(129,176),(130,177),(131,178),(132,179),(133,180),(134,181),(135,182),(136,183),(137,184),(138,185),(139,186),(140,187),(141,188),(142,189),(143,190),(144,191),(145,192),(146,193),(147,194),(148,195),(149,196),(150,197)], [(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)], [(1,147,26,122),(2,146,27,121),(3,145,28,120),(4,144,29,119),(5,143,30,118),(6,142,31,117),(7,141,32,116),(8,140,33,115),(9,139,34,114),(10,138,35,113),(11,137,36,112),(12,136,37,111),(13,135,38,110),(14,134,39,109),(15,133,40,108),(16,132,41,107),(17,131,42,106),(18,130,43,105),(19,129,44,104),(20,128,45,103),(21,127,46,102),(22,126,47,101),(23,125,48,150),(24,124,49,149),(25,123,50,148),(51,190,76,165),(52,189,77,164),(53,188,78,163),(54,187,79,162),(55,186,80,161),(56,185,81,160),(57,184,82,159),(58,183,83,158),(59,182,84,157),(60,181,85,156),(61,180,86,155),(62,179,87,154),(63,178,88,153),(64,177,89,152),(65,176,90,151),(66,175,91,200),(67,174,92,199),(68,173,93,198),(69,172,94,197),(70,171,95,196),(71,170,96,195),(72,169,97,194),(73,168,98,193),(74,167,99,192),(75,166,100,191)]])
C2×Dic25 is a maximal subgroup of
C50.D4 C4⋊Dic25 D50⋊C4 C23.D25 C25⋊M4(2) C2×C4×D25 D4⋊2D25
C2×Dic25 is a maximal quotient of C4.Dic25 C4⋊Dic25 C23.D25
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 10A | ··· | 10F | 25A | ··· | 25J | 50A | ··· | 50AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 1 | 1 | 25 | 25 | 25 | 25 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D5 | Dic5 | D10 | D25 | Dic25 | D50 |
| kernel | C2×Dic25 | Dic25 | C2×C50 | C50 | C2×C10 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 10 | 20 | 10 |
Matrix representation of C2×Dic25 ►in GL3(𝔽101) generated by
| 100 | 0 | 0 |
| 0 | 100 | 0 |
| 0 | 0 | 100 |
| 1 | 0 | 0 |
| 0 | 13 | 39 |
| 0 | 11 | 2 |
| 100 | 0 | 0 |
| 0 | 18 | 10 |
| 0 | 18 | 83 |
G:=sub<GL(3,GF(101))| [100,0,0,0,100,0,0,0,100],[1,0,0,0,13,11,0,39,2],[100,0,0,0,18,18,0,10,83] >;
C2×Dic25 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{25} % in TeX
G:=Group("C2xDic25"); // GroupNames label
G:=SmallGroup(200,7);
// by ID
G=gap.SmallGroup(200,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,1443,418,4004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^50=1,c^2=b^25,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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