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## G = C23×D25order 400 = 24·52

### Direct product of C23 and D25

Aliases: C23×D25, C25⋊C24, C50⋊C23, C5.(C23×D5), (C22×C50)⋊3C2, (C2×C50)⋊4C22, (C2×C10).29D10, (C22×C10).5D5, C10.29(C22×D5), SmallGroup(400,54)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C23×D25
 Chief series C1 — C5 — C25 — D25 — D50 — C22×D25 — C23×D25
 Lower central C25 — C23×D25
 Upper central C1 — C23

Generators and relations for C23×D25
G = < a,b,c,d,e | a2=b2=c2=d25=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1629 in 201 conjugacy classes, 99 normal (7 characteristic)
C1, C2, C2, C22, C22, C5, C23, C23, D5, C10, C24, D10, C2×C10, C25, C22×D5, C22×C10, D25, C50, C23×D5, D50, C2×C50, C22×D25, C22×C50, C23×D25
Quotients: C1, C2, C22, C23, D5, C24, D10, C22×D5, D25, C23×D5, D50, C22×D25, C23×D25

Smallest permutation representation of C23×D25
On 200 points
Generators in S200
(1 192)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 199)(9 200)(10 176)(11 177)(12 178)(13 179)(14 180)(15 181)(16 182)(17 183)(18 184)(19 185)(20 186)(21 187)(22 188)(23 189)(24 190)(25 191)(26 167)(27 168)(28 169)(29 170)(30 171)(31 172)(32 173)(33 174)(34 175)(35 151)(36 152)(37 153)(38 154)(39 155)(40 156)(41 157)(42 158)(43 159)(44 160)(45 161)(46 162)(47 163)(48 164)(49 165)(50 166)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 133)(73 134)(74 135)(75 136)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)(99 109)(100 110)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 51)(101 194)(102 195)(103 196)(104 197)(105 198)(106 199)(107 200)(108 176)(109 177)(110 178)(111 179)(112 180)(113 181)(114 182)(115 183)(116 184)(117 185)(118 186)(119 187)(120 188)(121 189)(122 190)(123 191)(124 192)(125 193)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 170)(142 171)(143 172)(144 173)(145 174)(146 175)(147 151)(148 152)(149 153)(150 154)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(25 32)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 76)(72 77)(73 78)(74 79)(75 80)(101 147)(102 148)(103 149)(104 150)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 60)(25 59)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 100)(47 99)(48 98)(49 97)(50 96)(101 171)(102 170)(103 169)(104 168)(105 167)(106 166)(107 165)(108 164)(109 163)(110 162)(111 161)(112 160)(113 159)(114 158)(115 157)(116 156)(117 155)(118 154)(119 153)(120 152)(121 151)(122 175)(123 174)(124 173)(125 172)(126 185)(127 184)(128 183)(129 182)(130 181)(131 180)(132 179)(133 178)(134 177)(135 176)(136 200)(137 199)(138 198)(139 197)(140 196)(141 195)(142 194)(143 193)(144 192)(145 191)(146 190)(147 189)(148 188)(149 187)(150 186)

G:=sub<Sym(200)| (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,176)(11,177)(12,178)(13,179)(14,180)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,151)(36,152)(37,153)(38,154)(39,155)(40,156)(41,157)(42,158)(43,159)(44,160)(45,161)(46,162)(47,163)(48,164)(49,165)(50,166)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,133)(73,134)(74,135)(75,136)(76,111)(77,112)(78,113)(79,114)(80,115)(81,116)(82,117)(83,118)(84,119)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108)(99,109)(100,110), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,51)(101,194)(102,195)(103,196)(104,197)(105,198)(106,199)(107,200)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187)(120,188)(121,189)(122,190)(123,191)(124,192)(125,193)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,151)(148,152)(149,153)(150,154), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(25,32)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,76)(72,77)(73,78)(74,79)(75,80)(101,147)(102,148)(103,149)(104,150)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,100)(47,99)(48,98)(49,97)(50,96)(101,171)(102,170)(103,169)(104,168)(105,167)(106,166)(107,165)(108,164)(109,163)(110,162)(111,161)(112,160)(113,159)(114,158)(115,157)(116,156)(117,155)(118,154)(119,153)(120,152)(121,151)(122,175)(123,174)(124,173)(125,172)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,200)(137,199)(138,198)(139,197)(140,196)(141,195)(142,194)(143,193)(144,192)(145,191)(146,190)(147,189)(148,188)(149,187)(150,186)>;

G:=Group( (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,176)(11,177)(12,178)(13,179)(14,180)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,151)(36,152)(37,153)(38,154)(39,155)(40,156)(41,157)(42,158)(43,159)(44,160)(45,161)(46,162)(47,163)(48,164)(49,165)(50,166)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,133)(73,134)(74,135)(75,136)(76,111)(77,112)(78,113)(79,114)(80,115)(81,116)(82,117)(83,118)(84,119)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108)(99,109)(100,110), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,51)(101,194)(102,195)(103,196)(104,197)(105,198)(106,199)(107,200)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187)(120,188)(121,189)(122,190)(123,191)(124,192)(125,193)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,151)(148,152)(149,153)(150,154), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(25,32)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,76)(72,77)(73,78)(74,79)(75,80)(101,147)(102,148)(103,149)(104,150)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,100)(47,99)(48,98)(49,97)(50,96)(101,171)(102,170)(103,169)(104,168)(105,167)(106,166)(107,165)(108,164)(109,163)(110,162)(111,161)(112,160)(113,159)(114,158)(115,157)(116,156)(117,155)(118,154)(119,153)(120,152)(121,151)(122,175)(123,174)(124,173)(125,172)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,200)(137,199)(138,198)(139,197)(140,196)(141,195)(142,194)(143,193)(144,192)(145,191)(146,190)(147,189)(148,188)(149,187)(150,186) );

G=PermutationGroup([[(1,192),(2,193),(3,194),(4,195),(5,196),(6,197),(7,198),(8,199),(9,200),(10,176),(11,177),(12,178),(13,179),(14,180),(15,181),(16,182),(17,183),(18,184),(19,185),(20,186),(21,187),(22,188),(23,189),(24,190),(25,191),(26,167),(27,168),(28,169),(29,170),(30,171),(31,172),(32,173),(33,174),(34,175),(35,151),(36,152),(37,153),(38,154),(39,155),(40,156),(41,157),(42,158),(43,159),(44,160),(45,161),(46,162),(47,163),(48,164),(49,165),(50,166),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(61,147),(62,148),(63,149),(64,150),(65,126),(66,127),(67,128),(68,129),(69,130),(70,131),(71,132),(72,133),(73,134),(74,135),(75,136),(76,111),(77,112),(78,113),(79,114),(80,115),(81,116),(82,117),(83,118),(84,119),(85,120),(86,121),(87,122),(88,123),(89,124),(90,125),(91,101),(92,102),(93,103),(94,104),(95,105),(96,106),(97,107),(98,108),(99,109),(100,110)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,73),(48,74),(49,75),(50,51),(101,194),(102,195),(103,196),(104,197),(105,198),(106,199),(107,200),(108,176),(109,177),(110,178),(111,179),(112,180),(113,181),(114,182),(115,183),(116,184),(117,185),(118,186),(119,187),(120,188),(121,189),(122,190),(123,191),(124,192),(125,193),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(141,170),(142,171),(143,172),(144,173),(145,174),(146,175),(147,151),(148,152),(149,153),(150,154)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31),(25,32),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,76),(72,77),(73,78),(74,79),(75,80),(101,147),(102,148),(103,149),(104,150),(105,126),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,141),(121,142),(122,143),(123,144),(124,145),(125,146),(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)], [(1,58),(2,57),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,75),(10,74),(11,73),(12,72),(13,71),(14,70),(15,69),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,60),(25,59),(26,95),(27,94),(28,93),(29,92),(30,91),(31,90),(32,89),(33,88),(34,87),(35,86),(36,85),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,100),(47,99),(48,98),(49,97),(50,96),(101,171),(102,170),(103,169),(104,168),(105,167),(106,166),(107,165),(108,164),(109,163),(110,162),(111,161),(112,160),(113,159),(114,158),(115,157),(116,156),(117,155),(118,154),(119,153),(120,152),(121,151),(122,175),(123,174),(124,173),(125,172),(126,185),(127,184),(128,183),(129,182),(130,181),(131,180),(132,179),(133,178),(134,177),(135,176),(136,200),(137,199),(138,198),(139,197),(140,196),(141,195),(142,194),(143,193),(144,192),(145,191),(146,190),(147,189),(148,188),(149,187),(150,186)]])

112 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 5A 5B 10A ··· 10N 25A ··· 25J 50A ··· 50BR order 1 2 ··· 2 2 ··· 2 5 5 10 ··· 10 25 ··· 25 50 ··· 50 size 1 1 ··· 1 25 ··· 25 2 2 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 D5 D10 D25 D50 kernel C23×D25 C22×D25 C22×C50 C22×C10 C2×C10 C23 C22 # reps 1 14 1 2 14 10 70

Matrix representation of C23×D25 in GL5(𝔽101)

 1 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100
,
 1 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 1 0 0 0 0 0 1
,
 100 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 100 0 0 0 0 0 100
,
 1 0 0 0 0 0 0 1 0 0 0 100 78 0 0 0 0 0 40 69 0 0 0 61 80
,
 1 0 0 0 0 0 0 100 0 0 0 100 0 0 0 0 0 0 51 93 0 0 0 22 50

G:=sub<GL(5,GF(101))| [1,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100],[1,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,1,0,0,0,0,0,1],[100,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,100,0,0,0,0,0,100],[1,0,0,0,0,0,0,100,0,0,0,1,78,0,0,0,0,0,40,61,0,0,0,69,80],[1,0,0,0,0,0,0,100,0,0,0,100,0,0,0,0,0,0,51,22,0,0,0,93,50] >;

C23×D25 in GAP, Magma, Sage, TeX

C_2^3\times D_{25}
% in TeX

G:=Group("C2^3xD25");
// GroupNames label

G:=SmallGroup(400,54);
// by ID

G=gap.SmallGroup(400,54);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,4324,628,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^25=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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