metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.D25, C50.11D4, C22⋊Dic25, C22.7D50, (C2×C50)⋊4C4, C25⋊3(C22⋊C4), C50.16(C2×C4), C5.(C23.D5), (C2×Dic25)⋊2C2, (C2×C10).22D10, C2.3(C25⋊D4), (C2×C50).7C22, (C22×C50).2C2, C2.5(C2×Dic25), (C2×C10).6Dic5, (C22×C10).2D5, C10.18(C5⋊D4), C10.17(C2×Dic5), SmallGroup(400,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D25
G = < a,b,c,d,e | a2=b2=c2=d25=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Smallest permutation representation of C23.D25
►On 200 points
Generators in S200
(101 148)(102 149)(103 150)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 141)(120 142)(121 143)(122 144)(123 145)(124 146)(125 147)(151 194)(152 195)(153 196)(154 197)(155 198)(156 199)(157 200)(158 176)(159 177)(160 178)(161 179)(162 180)(163 181)(164 182)(165 183)(166 184)(167 185)(168 186)(169 187)(170 188)(171 189)(172 190)(173 191)(174 192)(175 193)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(101 171)(102 172)(103 173)(104 174)(105 175)(106 151)(107 152)(108 153)(109 154)(110 155)(111 156)(112 157)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 166)(122 167)(123 168)(124 169)(125 170)(126 192)(127 193)(128 194)(129 195)(130 196)(131 197)(132 198)(133 199)(134 200)(135 176)(136 177)(137 178)(138 179)(139 180)(140 181)(141 182)(142 183)(143 184)(144 185)(145 186)(146 187)(147 188)(148 189)(149 190)(150 191)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 26)(23 27)(24 28)(25 29)(51 96)(52 97)(53 98)(54 99)(55 100)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(101 148)(102 149)(103 150)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 141)(120 142)(121 143)(122 144)(123 145)(124 146)(125 147)(151 194)(152 195)(153 196)(154 197)(155 198)(156 199)(157 200)(158 176)(159 177)(160 178)(161 179)(162 180)(163 181)(164 182)(165 183)(166 184)(167 185)(168 186)(169 187)(170 188)(171 189)(172 190)(173 191)(174 192)(175 193)
(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)
(1 160 51 115)(2 159 52 114)(3 158 53 113)(4 157 54 112)(5 156 55 111)(6 155 56 110)(7 154 57 109)(8 153 58 108)(9 152 59 107)(10 151 60 106)(11 175 61 105)(12 174 62 104)(13 173 63 103)(14 172 64 102)(15 171 65 101)(16 170 66 125)(17 169 67 124)(18 168 68 123)(19 167 69 122)(20 166 70 121)(21 165 71 120)(22 164 72 119)(23 163 73 118)(24 162 74 117)(25 161 75 116)(26 182 92 141)(27 181 93 140)(28 180 94 139)(29 179 95 138)(30 178 96 137)(31 177 97 136)(32 176 98 135)(33 200 99 134)(34 199 100 133)(35 198 76 132)(36 197 77 131)(37 196 78 130)(38 195 79 129)(39 194 80 128)(40 193 81 127)(41 192 82 126)(42 191 83 150)(43 190 84 149)(44 189 85 148)(45 188 86 147)(46 187 87 146)(47 186 88 145)(48 185 89 144)(49 184 90 143)(50 183 91 142)
G:=sub<Sym(200)| (101,148)(102,149)(103,150)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199)(157,200)(158,176)(159,177)(160,178)(161,179)(162,180)(163,181)(164,182)(165,183)(166,184)(167,185)(168,186)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,193), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,92)(27,93)(28,94)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(101,171)(102,172)(103,173)(104,174)(105,175)(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,169)(125,170)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,176)(136,177)(137,178)(138,179)(139,180)(140,181)(141,182)(142,183)(143,184)(144,185)(145,186)(146,187)(147,188)(148,189)(149,190)(150,191), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,26)(23,27)(24,28)(25,29)(51,96)(52,97)(53,98)(54,99)(55,100)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(101,148)(102,149)(103,150)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199)(157,200)(158,176)(159,177)(160,178)(161,179)(162,180)(163,181)(164,182)(165,183)(166,184)(167,185)(168,186)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,193), (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), (1,160,51,115)(2,159,52,114)(3,158,53,113)(4,157,54,112)(5,156,55,111)(6,155,56,110)(7,154,57,109)(8,153,58,108)(9,152,59,107)(10,151,60,106)(11,175,61,105)(12,174,62,104)(13,173,63,103)(14,172,64,102)(15,171,65,101)(16,170,66,125)(17,169,67,124)(18,168,68,123)(19,167,69,122)(20,166,70,121)(21,165,71,120)(22,164,72,119)(23,163,73,118)(24,162,74,117)(25,161,75,116)(26,182,92,141)(27,181,93,140)(28,180,94,139)(29,179,95,138)(30,178,96,137)(31,177,97,136)(32,176,98,135)(33,200,99,134)(34,199,100,133)(35,198,76,132)(36,197,77,131)(37,196,78,130)(38,195,79,129)(39,194,80,128)(40,193,81,127)(41,192,82,126)(42,191,83,150)(43,190,84,149)(44,189,85,148)(45,188,86,147)(46,187,87,146)(47,186,88,145)(48,185,89,144)(49,184,90,143)(50,183,91,142)>;
G:=Group( (101,148)(102,149)(103,150)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199)(157,200)(158,176)(159,177)(160,178)(161,179)(162,180)(163,181)(164,182)(165,183)(166,184)(167,185)(168,186)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,193), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,92)(27,93)(28,94)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(101,171)(102,172)(103,173)(104,174)(105,175)(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,169)(125,170)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,176)(136,177)(137,178)(138,179)(139,180)(140,181)(141,182)(142,183)(143,184)(144,185)(145,186)(146,187)(147,188)(148,189)(149,190)(150,191), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,26)(23,27)(24,28)(25,29)(51,96)(52,97)(53,98)(54,99)(55,100)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(101,148)(102,149)(103,150)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199)(157,200)(158,176)(159,177)(160,178)(161,179)(162,180)(163,181)(164,182)(165,183)(166,184)(167,185)(168,186)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,193), (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), (1,160,51,115)(2,159,52,114)(3,158,53,113)(4,157,54,112)(5,156,55,111)(6,155,56,110)(7,154,57,109)(8,153,58,108)(9,152,59,107)(10,151,60,106)(11,175,61,105)(12,174,62,104)(13,173,63,103)(14,172,64,102)(15,171,65,101)(16,170,66,125)(17,169,67,124)(18,168,68,123)(19,167,69,122)(20,166,70,121)(21,165,71,120)(22,164,72,119)(23,163,73,118)(24,162,74,117)(25,161,75,116)(26,182,92,141)(27,181,93,140)(28,180,94,139)(29,179,95,138)(30,178,96,137)(31,177,97,136)(32,176,98,135)(33,200,99,134)(34,199,100,133)(35,198,76,132)(36,197,77,131)(37,196,78,130)(38,195,79,129)(39,194,80,128)(40,193,81,127)(41,192,82,126)(42,191,83,150)(43,190,84,149)(44,189,85,148)(45,188,86,147)(46,187,87,146)(47,186,88,145)(48,185,89,144)(49,184,90,143)(50,183,91,142) );
G=PermutationGroup([[(101,148),(102,149),(103,150),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(113,135),(114,136),(115,137),(116,138),(117,139),(118,140),(119,141),(120,142),(121,143),(122,144),(123,145),(124,146),(125,147),(151,194),(152,195),(153,196),(154,197),(155,198),(156,199),(157,200),(158,176),(159,177),(160,178),(161,179),(162,180),(163,181),(164,182),(165,183),(166,184),(167,185),(168,186),(169,187),(170,188),(171,189),(172,190),(173,191),(174,192),(175,193)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,92),(27,93),(28,94),(29,95),(30,96),(31,97),(32,98),(33,99),(34,100),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(50,91),(101,171),(102,172),(103,173),(104,174),(105,175),(106,151),(107,152),(108,153),(109,154),(110,155),(111,156),(112,157),(113,158),(114,159),(115,160),(116,161),(117,162),(118,163),(119,164),(120,165),(121,166),(122,167),(123,168),(124,169),(125,170),(126,192),(127,193),(128,194),(129,195),(130,196),(131,197),(132,198),(133,199),(134,200),(135,176),(136,177),(137,178),(138,179),(139,180),(140,181),(141,182),(142,183),(143,184),(144,185),(145,186),(146,187),(147,188),(148,189),(149,190),(150,191)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,26),(23,27),(24,28),(25,29),(51,96),(52,97),(53,98),(54,99),(55,100),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(101,148),(102,149),(103,150),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(113,135),(114,136),(115,137),(116,138),(117,139),(118,140),(119,141),(120,142),(121,143),(122,144),(123,145),(124,146),(125,147),(151,194),(152,195),(153,196),(154,197),(155,198),(156,199),(157,200),(158,176),(159,177),(160,178),(161,179),(162,180),(163,181),(164,182),(165,183),(166,184),(167,185),(168,186),(169,187),(170,188),(171,189),(172,190),(173,191),(174,192),(175,193)], [(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)], [(1,160,51,115),(2,159,52,114),(3,158,53,113),(4,157,54,112),(5,156,55,111),(6,155,56,110),(7,154,57,109),(8,153,58,108),(9,152,59,107),(10,151,60,106),(11,175,61,105),(12,174,62,104),(13,173,63,103),(14,172,64,102),(15,171,65,101),(16,170,66,125),(17,169,67,124),(18,168,68,123),(19,167,69,122),(20,166,70,121),(21,165,71,120),(22,164,72,119),(23,163,73,118),(24,162,74,117),(25,161,75,116),(26,182,92,141),(27,181,93,140),(28,180,94,139),(29,179,95,138),(30,178,96,137),(31,177,97,136),(32,176,98,135),(33,200,99,134),(34,199,100,133),(35,198,76,132),(36,197,77,131),(37,196,78,130),(38,195,79,129),(39,194,80,128),(40,193,81,127),(41,192,82,126),(42,191,83,150),(43,190,84,149),(44,189,85,148),(45,188,86,147),(46,187,87,146),(47,186,88,145),(48,185,89,144),(49,184,90,143),(50,183,91,142)]])
106 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 10A | ··· | 10N | 25A | ··· | 25J | 50A | ··· | 50BR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 50 | 50 | 50 | 50 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | D4 | D5 | Dic5 | D10 | C5⋊D4 | D25 | Dic25 | D50 | C25⋊D4 |
| kernel | C23.D25 | C2×Dic25 | C22×C50 | C2×C50 | C50 | C22×C10 | C2×C10 | C2×C10 | C10 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 8 | 10 | 20 | 10 | 40 |
Matrix representation of C23.D25 ►in GL3(𝔽101) generated by
| 100 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 100 |
| 100 | 0 | 0 |
| 0 | 100 | 0 |
| 0 | 0 | 100 |
| 1 | 0 | 0 |
| 0 | 100 | 0 |
| 0 | 0 | 100 |
| 1 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 80 |
| 10 | 0 | 0 |
| 0 | 0 | 80 |
| 0 | 77 | 0 |
G:=sub<GL(3,GF(101))| [100,0,0,0,1,0,0,0,100],[100,0,0,0,100,0,0,0,100],[1,0,0,0,100,0,0,0,100],[1,0,0,0,24,0,0,0,80],[10,0,0,0,0,77,0,80,0] >;
C23.D25 in GAP, Magma, Sage, TeX
C_2^3.D_{25} % in TeX
G:=Group("C2^3.D25"); // GroupNames label
G:=SmallGroup(400,19);
// by ID
G=gap.SmallGroup(400,19);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^25=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export