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## G = C23×C28order 224 = 25·7

### Abelian group of type [2,2,2,28]

Aliases: C23×C28, SmallGroup(224,189)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C28
 Chief series C1 — C2 — C14 — C28 — C2×C28 — C22×C28 — C23×C28
 Lower central C1 — C23×C28
 Upper central C1 — C23×C28

Generators and relations for C23×C28
G = < a,b,c,d | a2=b2=c2=d28=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 236, all normal (8 characteristic)
C1, C2, C2 [×14], C4 [×8], C22 [×35], C7, C2×C4 [×28], C23 [×15], C14, C14 [×14], C22×C4 [×14], C24, C28 [×8], C2×C14 [×35], C23×C4, C2×C28 [×28], C22×C14 [×15], C22×C28 [×14], C23×C14, C23×C28
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C7, C2×C4 [×28], C23 [×15], C14 [×15], C22×C4 [×14], C24, C28 [×8], C2×C14 [×35], C23×C4, C2×C28 [×28], C22×C14 [×15], C22×C28 [×14], C23×C14, C23×C28

Smallest permutation representation of C23×C28
Regular action on 224 points
Generators in S224
(1 138)(2 139)(3 140)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 121)(13 122)(14 123)(15 124)(16 125)(17 126)(18 127)(19 128)(20 129)(21 130)(22 131)(23 132)(24 133)(25 134)(26 135)(27 136)(28 137)(29 152)(30 153)(31 154)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(43 166)(44 167)(45 168)(46 141)(47 142)(48 143)(49 144)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 196)(65 169)(66 170)(67 171)(68 172)(69 173)(70 174)(71 175)(72 176)(73 177)(74 178)(75 179)(76 180)(77 181)(78 182)(79 183)(80 184)(81 185)(82 186)(83 187)(84 188)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 29)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(113 163)(114 164)(115 165)(116 166)(117 167)(118 168)(119 141)(120 142)(121 143)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 153)(132 154)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)(169 215)(170 216)(171 217)(172 218)(173 219)(174 220)(175 221)(176 222)(177 223)(178 224)(179 197)(180 198)(181 199)(182 200)(183 201)(184 202)(185 203)(186 204)(187 205)(188 206)(189 207)(190 208)(191 209)(192 210)(193 211)(194 212)(195 213)(196 214)
(1 193)(2 194)(3 195)(4 196)(5 169)(6 170)(7 171)(8 172)(9 173)(10 174)(11 175)(12 176)(13 177)(14 178)(15 179)(16 180)(17 181)(18 182)(19 183)(20 184)(21 185)(22 186)(23 187)(24 188)(25 189)(26 190)(27 191)(28 192)(29 203)(30 204)(31 205)(32 206)(33 207)(34 208)(35 209)(36 210)(37 211)(38 212)(39 213)(40 214)(41 215)(42 216)(43 217)(44 218)(45 219)(46 220)(47 221)(48 222)(49 223)(50 224)(51 197)(52 198)(53 199)(54 200)(55 201)(56 202)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 113)(65 114)(66 115)(67 116)(68 117)(69 118)(70 119)(71 120)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)(97 158)(98 159)(99 160)(100 161)(101 162)(102 163)(103 164)(104 165)(105 166)(106 167)(107 168)(108 141)(109 142)(110 143)(111 144)(112 145)
(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)

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

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

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

C23×C28 is a maximal subgroup of   C24.4Dic7  C24.62D14  C24.63D14  C23.27D28  C23.28D28  C24.72D14

224 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4P 7A ··· 7F 14A ··· 14CL 28A ··· 28CR order 1 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

224 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C7 C14 C14 C28 kernel C23×C28 C22×C28 C23×C14 C22×C14 C23×C4 C22×C4 C24 C23 # reps 1 14 1 16 6 84 6 96

Matrix representation of C23×C28 in GL4(𝔽29) generated by

 28 0 0 0 0 1 0 0 0 0 1 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 28
,
 19 0 0 0 0 19 0 0 0 0 20 0 0 0 0 5
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[19,0,0,0,0,19,0,0,0,0,20,0,0,0,0,5] >;

C23×C28 in GAP, Magma, Sage, TeX

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

G:=Group("C2^3xC28");
// GroupNames label

G:=SmallGroup(224,189);
// by ID

G=gap.SmallGroup(224,189);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-7,-2,672]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^28=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

׿
×
𝔽