direct product, abelian, monomial, 2-elementary
Aliases: C23×C28, SmallGroup(224,189)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C28 |
| C1 — C23×C28 |
| 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, C4, C22, C7, C2×C4, C23, C14, C14, C22×C4, C24, C28, C2×C14, C23×C4, C2×C28, C22×C14, C22×C28, C23×C14, C23×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C22×C4, C24, C28, C2×C14, C23×C4, C2×C28, C22×C14, C22×C28, C23×C14, C23×C28
►Regular action on 224 points
Generators in S224
(1 177)(2 178)(3 179)(4 180)(5 181)(6 182)(7 183)(8 184)(9 185)(10 186)(11 187)(12 188)(13 189)(14 190)(15 191)(16 192)(17 193)(18 194)(19 195)(20 196)(21 169)(22 170)(23 171)(24 172)(25 173)(26 174)(27 175)(28 176)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 120)(141 207)(142 208)(143 209)(144 210)(145 211)(146 212)(147 213)(148 214)(149 215)(150 216)(151 217)(152 218)(153 219)(154 220)(155 221)(156 222)(157 223)(158 224)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(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)(57 212)(58 213)(59 214)(60 215)(61 216)(62 217)(63 218)(64 219)(65 220)(66 221)(67 222)(68 223)(69 224)(70 197)(71 198)(72 199)(73 200)(74 201)(75 202)(76 203)(77 204)(78 205)(79 206)(80 207)(81 208)(82 209)(83 210)(84 211)(113 177)(114 178)(115 179)(116 180)(117 181)(118 182)(119 183)(120 184)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 193)(130 194)(131 195)(132 196)(133 169)(134 170)(135 171)(136 172)(137 173)(138 174)(139 175)(140 176)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(57 184)(58 185)(59 186)(60 187)(61 188)(62 189)(63 190)(64 191)(65 192)(66 193)(67 194)(68 195)(69 196)(70 169)(71 170)(72 171)(73 172)(74 173)(75 174)(76 175)(77 176)(78 177)(79 178)(80 179)(81 180)(82 181)(83 182)(84 183)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 161)(100 162)(101 163)(102 164)(103 165)(104 166)(105 167)(106 168)(107 141)(108 142)(109 143)(110 144)(111 145)(112 146)(113 205)(114 206)(115 207)(116 208)(117 209)(118 210)(119 211)(120 212)(121 213)(122 214)(123 215)(124 216)(125 217)(126 218)(127 219)(128 220)(129 221)(130 222)(131 223)(132 224)(133 197)(134 198)(135 199)(136 200)(137 201)(138 202)(139 203)(140 204)
(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)| (1,177)(2,178)(3,179)(4,180)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,169)(22,170)(23,171)(24,172)(25,173)(26,174)(27,175)(28,176)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120)(141,207)(142,208)(143,209)(144,210)(145,211)(146,212)(147,213)(148,214)(149,215)(150,216)(151,217)(152,218)(153,219)(154,220)(155,221)(156,222)(157,223)(158,224)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(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)(57,212)(58,213)(59,214)(60,215)(61,216)(62,217)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,197)(71,198)(72,199)(73,200)(74,201)(75,202)(76,203)(77,204)(78,205)(79,206)(80,207)(81,208)(82,209)(83,210)(84,211)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,183)(120,184)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194)(131,195)(132,196)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,196)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,177)(79,178)(80,179)(81,180)(82,181)(83,182)(84,183)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,211)(120,212)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,221)(130,222)(131,223)(132,224)(133,197)(134,198)(135,199)(136,200)(137,201)(138,202)(139,203)(140,204), (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( (1,177)(2,178)(3,179)(4,180)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,169)(22,170)(23,171)(24,172)(25,173)(26,174)(27,175)(28,176)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120)(141,207)(142,208)(143,209)(144,210)(145,211)(146,212)(147,213)(148,214)(149,215)(150,216)(151,217)(152,218)(153,219)(154,220)(155,221)(156,222)(157,223)(158,224)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(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)(57,212)(58,213)(59,214)(60,215)(61,216)(62,217)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,197)(71,198)(72,199)(73,200)(74,201)(75,202)(76,203)(77,204)(78,205)(79,206)(80,207)(81,208)(82,209)(83,210)(84,211)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,183)(120,184)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194)(131,195)(132,196)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,196)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,177)(79,178)(80,179)(81,180)(82,181)(83,182)(84,183)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,211)(120,212)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,221)(130,222)(131,223)(132,224)(133,197)(134,198)(135,199)(136,200)(137,201)(138,202)(139,203)(140,204), (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([[(1,177),(2,178),(3,179),(4,180),(5,181),(6,182),(7,183),(8,184),(9,185),(10,186),(11,187),(12,188),(13,189),(14,190),(15,191),(16,192),(17,193),(18,194),(19,195),(20,196),(21,169),(22,170),(23,171),(24,172),(25,173),(26,174),(27,175),(28,176),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(85,121),(86,122),(87,123),(88,124),(89,125),(90,126),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,113),(106,114),(107,115),(108,116),(109,117),(110,118),(111,119),(112,120),(141,207),(142,208),(143,209),(144,210),(145,211),(146,212),(147,213),(148,214),(149,215),(150,216),(151,217),(152,218),(153,219),(154,220),(155,221),(156,222),(157,223),(158,224),(159,197),(160,198),(161,199),(162,200),(163,201),(164,202),(165,203),(166,204),(167,205),(168,206)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(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),(57,212),(58,213),(59,214),(60,215),(61,216),(62,217),(63,218),(64,219),(65,220),(66,221),(67,222),(68,223),(69,224),(70,197),(71,198),(72,199),(73,200),(74,201),(75,202),(76,203),(77,204),(78,205),(79,206),(80,207),(81,208),(82,209),(83,210),(84,211),(113,177),(114,178),(115,179),(116,180),(117,181),(118,182),(119,183),(120,184),(121,185),(122,186),(123,187),(124,188),(125,189),(126,190),(127,191),(128,192),(129,193),(130,194),(131,195),(132,196),(133,169),(134,170),(135,171),(136,172),(137,173),(138,174),(139,175),(140,176)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(57,184),(58,185),(59,186),(60,187),(61,188),(62,189),(63,190),(64,191),(65,192),(66,193),(67,194),(68,195),(69,196),(70,169),(71,170),(72,171),(73,172),(74,173),(75,174),(76,175),(77,176),(78,177),(79,178),(80,179),(81,180),(82,181),(83,182),(84,183),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,153),(92,154),(93,155),(94,156),(95,157),(96,158),(97,159),(98,160),(99,161),(100,162),(101,163),(102,164),(103,165),(104,166),(105,167),(106,168),(107,141),(108,142),(109,143),(110,144),(111,145),(112,146),(113,205),(114,206),(115,207),(116,208),(117,209),(118,210),(119,211),(120,212),(121,213),(122,214),(123,215),(124,216),(125,217),(126,218),(127,219),(128,220),(129,221),(130,222),(131,223),(132,224),(133,197),(134,198),(135,199),(136,200),(137,201),(138,202),(139,203),(140,204)], [(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