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## G = C23×D28order 448 = 26·7

### Direct product of C23 and D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C23×D28
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — D7×C24 — C23×D28
 Lower central C7 — C14 — C23×D28
 Upper central C1 — C24 — C23×C4

Generators and relations for C23×D28
G = < a,b,c,d,e | a2=b2=c2=d28=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 6788 in 1362 conjugacy classes, 543 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C23, D7, C14, C14, C22×C4, C2×D4, C24, C24, C28, D14, D14, C2×C14, C23×C4, C22×D4, C25, D28, C2×C28, C22×D7, C22×D7, C22×C14, D4×C23, C2×D28, C22×C28, C23×D7, C23×D7, C23×C14, C22×D28, C23×C28, D7×C24, C23×D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C25, D28, C22×D7, D4×C23, C2×D28, C23×D7, C22×D28, D7×C24, C23×D28

Smallest permutation representation of C23×D28
On 224 points
Generators in S224
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 29)(24 30)(25 31)(26 32)(27 33)(28 34)(57 150)(58 151)(59 152)(60 153)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 160)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 141)(77 142)(78 143)(79 144)(80 145)(81 146)(82 147)(83 148)(84 149)(85 195)(86 196)(87 169)(88 170)(89 171)(90 172)(91 173)(92 174)(93 175)(94 176)(95 177)(96 178)(97 179)(98 180)(99 181)(100 182)(101 183)(102 184)(103 185)(104 186)(105 187)(106 188)(107 189)(108 190)(109 191)(110 192)(111 193)(112 194)(113 222)(114 223)(115 224)(116 197)(117 198)(118 199)(119 200)(120 201)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 217)(137 218)(138 219)(139 220)(140 221)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 153)(30 154)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 161)(38 162)(39 163)(40 164)(41 165)(42 166)(43 167)(44 168)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(85 139)(86 140)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(169 222)(170 223)(171 224)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)(181 206)(182 207)(183 208)(184 209)(185 210)(186 211)(187 212)(188 213)(189 214)(190 215)(191 216)(192 217)(193 218)(194 219)(195 220)(196 221)
(1 213)(2 214)(3 215)(4 216)(5 217)(6 218)(7 219)(8 220)(9 221)(10 222)(11 223)(12 224)(13 197)(14 198)(15 199)(16 200)(17 201)(18 202)(19 203)(20 204)(21 205)(22 206)(23 207)(24 208)(25 209)(26 210)(27 211)(28 212)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 138)(42 139)(43 140)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 121)(53 122)(54 123)(55 124)(56 125)(57 179)(58 180)(59 181)(60 182)(61 183)(62 184)(63 185)(64 186)(65 187)(66 188)(67 189)(68 190)(69 191)(70 192)(71 193)(72 194)(73 195)(74 196)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 177)(84 178)(85 166)(86 167)(87 168)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 161)(109 162)(110 163)(111 164)(112 165)
(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)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(57 146)(58 145)(59 144)(60 143)(61 142)(62 141)(63 168)(64 167)(65 166)(66 165)(67 164)(68 163)(69 162)(70 161)(71 160)(72 159)(73 158)(74 157)(75 156)(76 155)(77 154)(78 153)(79 152)(80 151)(81 150)(82 149)(83 148)(84 147)(85 187)(86 186)(87 185)(88 184)(89 183)(90 182)(91 181)(92 180)(93 179)(94 178)(95 177)(96 176)(97 175)(98 174)(99 173)(100 172)(101 171)(102 170)(103 169)(104 196)(105 195)(106 194)(107 193)(108 192)(109 191)(110 190)(111 189)(112 188)(113 210)(114 209)(115 208)(116 207)(117 206)(118 205)(119 204)(120 203)(121 202)(122 201)(123 200)(124 199)(125 198)(126 197)(127 224)(128 223)(129 222)(130 221)(131 220)(132 219)(133 218)(134 217)(135 216)(136 215)(137 214)(138 213)(139 212)(140 211)

G:=sub<Sym(224)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,29)(24,30)(25,31)(26,32)(27,33)(28,34)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,161)(69,162)(70,163)(71,164)(72,165)(73,166)(74,167)(75,168)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,195)(86,196)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194)(113,222)(114,223)(115,224)(116,197)(117,198)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,221), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(85,139)(86,140)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(169,222)(170,223)(171,224)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221), (1,213)(2,214)(3,215)(4,216)(5,217)(6,218)(7,219)(8,220)(9,221)(10,222)(11,223)(12,224)(13,197)(14,198)(15,199)(16,200)(17,201)(18,202)(19,203)(20,204)(21,205)(22,206)(23,207)(24,208)(25,209)(26,210)(27,211)(28,212)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,179)(58,180)(59,181)(60,182)(61,183)(62,184)(63,185)(64,186)(65,187)(66,188)(67,189)(68,190)(69,191)(70,192)(71,193)(72,194)(73,195)(74,196)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,166)(86,167)(87,168)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165), (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)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,168)(64,167)(65,166)(66,165)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,156)(76,155)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,187)(86,186)(87,185)(88,184)(89,183)(90,182)(91,181)(92,180)(93,179)(94,178)(95,177)(96,176)(97,175)(98,174)(99,173)(100,172)(101,171)(102,170)(103,169)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188)(113,210)(114,209)(115,208)(116,207)(117,206)(118,205)(119,204)(120,203)(121,202)(122,201)(123,200)(124,199)(125,198)(126,197)(127,224)(128,223)(129,222)(130,221)(131,220)(132,219)(133,218)(134,217)(135,216)(136,215)(137,214)(138,213)(139,212)(140,211)>;

G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,29)(24,30)(25,31)(26,32)(27,33)(28,34)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,161)(69,162)(70,163)(71,164)(72,165)(73,166)(74,167)(75,168)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,195)(86,196)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194)(113,222)(114,223)(115,224)(116,197)(117,198)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,221), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(85,139)(86,140)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(169,222)(170,223)(171,224)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221), (1,213)(2,214)(3,215)(4,216)(5,217)(6,218)(7,219)(8,220)(9,221)(10,222)(11,223)(12,224)(13,197)(14,198)(15,199)(16,200)(17,201)(18,202)(19,203)(20,204)(21,205)(22,206)(23,207)(24,208)(25,209)(26,210)(27,211)(28,212)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,179)(58,180)(59,181)(60,182)(61,183)(62,184)(63,185)(64,186)(65,187)(66,188)(67,189)(68,190)(69,191)(70,192)(71,193)(72,194)(73,195)(74,196)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,166)(86,167)(87,168)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165), (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)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,168)(64,167)(65,166)(66,165)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,156)(76,155)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,187)(86,186)(87,185)(88,184)(89,183)(90,182)(91,181)(92,180)(93,179)(94,178)(95,177)(96,176)(97,175)(98,174)(99,173)(100,172)(101,171)(102,170)(103,169)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188)(113,210)(114,209)(115,208)(116,207)(117,206)(118,205)(119,204)(120,203)(121,202)(122,201)(123,200)(124,199)(125,198)(126,197)(127,224)(128,223)(129,222)(130,221)(131,220)(132,219)(133,218)(134,217)(135,216)(136,215)(137,214)(138,213)(139,212)(140,211) );

G=PermutationGroup([[(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,29),(24,30),(25,31),(26,32),(27,33),(28,34),(57,150),(58,151),(59,152),(60,153),(61,154),(62,155),(63,156),(64,157),(65,158),(66,159),(67,160),(68,161),(69,162),(70,163),(71,164),(72,165),(73,166),(74,167),(75,168),(76,141),(77,142),(78,143),(79,144),(80,145),(81,146),(82,147),(83,148),(84,149),(85,195),(86,196),(87,169),(88,170),(89,171),(90,172),(91,173),(92,174),(93,175),(94,176),(95,177),(96,178),(97,179),(98,180),(99,181),(100,182),(101,183),(102,184),(103,185),(104,186),(105,187),(106,188),(107,189),(108,190),(109,191),(110,192),(111,193),(112,194),(113,222),(114,223),(115,224),(116,197),(117,198),(118,199),(119,200),(120,201),(121,202),(122,203),(123,204),(124,205),(125,206),(126,207),(127,208),(128,209),(129,210),(130,211),(131,212),(132,213),(133,214),(134,215),(135,216),(136,217),(137,218),(138,219),(139,220),(140,221)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,153),(30,154),(31,155),(32,156),(33,157),(34,158),(35,159),(36,160),(37,161),(38,162),(39,163),(40,164),(41,165),(42,166),(43,167),(44,168),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(85,139),(86,140),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(169,222),(170,223),(171,224),(172,197),(173,198),(174,199),(175,200),(176,201),(177,202),(178,203),(179,204),(180,205),(181,206),(182,207),(183,208),(184,209),(185,210),(186,211),(187,212),(188,213),(189,214),(190,215),(191,216),(192,217),(193,218),(194,219),(195,220),(196,221)], [(1,213),(2,214),(3,215),(4,216),(5,217),(6,218),(7,219),(8,220),(9,221),(10,222),(11,223),(12,224),(13,197),(14,198),(15,199),(16,200),(17,201),(18,202),(19,203),(20,204),(21,205),(22,206),(23,207),(24,208),(25,209),(26,210),(27,211),(28,212),(29,126),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,138),(42,139),(43,140),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,121),(53,122),(54,123),(55,124),(56,125),(57,179),(58,180),(59,181),(60,182),(61,183),(62,184),(63,185),(64,186),(65,187),(66,188),(67,189),(68,190),(69,191),(70,192),(71,193),(72,194),(73,195),(74,196),(75,169),(76,170),(77,171),(78,172),(79,173),(80,174),(81,175),(82,176),(83,177),(84,178),(85,166),(86,167),(87,168),(88,141),(89,142),(90,143),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,151),(99,152),(100,153),(101,154),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,161),(109,162),(110,163),(111,164),(112,165)], [(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),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(57,146),(58,145),(59,144),(60,143),(61,142),(62,141),(63,168),(64,167),(65,166),(66,165),(67,164),(68,163),(69,162),(70,161),(71,160),(72,159),(73,158),(74,157),(75,156),(76,155),(77,154),(78,153),(79,152),(80,151),(81,150),(82,149),(83,148),(84,147),(85,187),(86,186),(87,185),(88,184),(89,183),(90,182),(91,181),(92,180),(93,179),(94,178),(95,177),(96,176),(97,175),(98,174),(99,173),(100,172),(101,171),(102,170),(103,169),(104,196),(105,195),(106,194),(107,193),(108,192),(109,191),(110,190),(111,189),(112,188),(113,210),(114,209),(115,208),(116,207),(117,206),(118,205),(119,204),(120,203),(121,202),(122,201),(123,200),(124,199),(125,198),(126,197),(127,224),(128,223),(129,222),(130,221),(131,220),(132,219),(133,218),(134,217),(135,216),(136,215),(137,214),(138,213),(139,212),(140,211)]])

136 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 4A ··· 4H 7A 7B 7C 14A ··· 14AS 28A ··· 28AV order 1 2 ··· 2 2 ··· 2 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 14 ··· 14 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D7 D14 D14 D28 kernel C23×D28 C22×D28 C23×C28 D7×C24 C22×C14 C23×C4 C22×C4 C24 C23 # reps 1 28 1 2 8 3 42 3 48

Matrix representation of C23×D28 in GL5(𝔽29)

 28 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 22 0 0 0 24 9
,
 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 0 25 0 0 0 7 0

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,24,0,0,0,22,9],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,25,0] >;

C23×D28 in GAP, Magma, Sage, TeX

C_2^3\times D_{28}
% in TeX

G:=Group("C2^3xD28");
// GroupNames label

G:=SmallGroup(448,1367);
// by ID

G=gap.SmallGroup(448,1367);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=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

׿
×
𝔽