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## G = C22×D56order 448 = 26·7

### Direct product of C22 and D56

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C22×D56
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C22×D28 — C22×D56
 Lower central C7 — C14 — C28 — C22×D56
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C22×D56
G = < a,b,c,d | a2=b2=c56=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 2404 in 338 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, D4, C23, C23, D7, C14, C14, C2×C8, D8, C22×C4, C2×D4, C24, C28, C28, D14, C2×C14, C22×C8, C2×D8, C22×D4, C56, D28, D28, C2×C28, C22×D7, C22×C14, C22×D8, D56, C2×C56, C2×D28, C2×D28, C22×C28, C23×D7, C2×D56, C22×C56, C22×D28, C22×D56
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C24, D14, C2×D8, C22×D4, D28, C22×D7, C22×D8, D56, C2×D28, C23×D7, C2×D56, C22×D28, C22×D56

Smallest permutation representation of C22×D56
On 224 points
Generators in S224
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(113 186)(114 187)(115 188)(116 189)(117 190)(118 191)(119 192)(120 193)(121 194)(122 195)(123 196)(124 197)(125 198)(126 199)(127 200)(128 201)(129 202)(130 203)(131 204)(132 205)(133 206)(134 207)(135 208)(136 209)(137 210)(138 211)(139 212)(140 213)(141 214)(142 215)(143 216)(144 217)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 224)(152 169)(153 170)(154 171)(155 172)(156 173)(157 174)(158 175)(159 176)(160 177)(161 178)(162 179)(163 180)(164 181)(165 182)(166 183)(167 184)(168 185)
(1 178)(2 179)(3 180)(4 181)(5 182)(6 183)(7 184)(8 185)(9 186)(10 187)(11 188)(12 189)(13 190)(14 191)(15 192)(16 193)(17 194)(18 195)(19 196)(20 197)(21 198)(22 199)(23 200)(24 201)(25 202)(26 203)(27 204)(28 205)(29 206)(30 207)(31 208)(32 209)(33 210)(34 211)(35 212)(36 213)(37 214)(38 215)(39 216)(40 217)(41 218)(42 219)(43 220)(44 221)(45 222)(46 223)(47 224)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)(81 138)(82 139)(83 140)(84 141)(85 142)(86 143)(87 144)(88 145)(89 146)(90 147)(91 148)(92 149)(93 150)(94 151)(95 152)(96 153)(97 154)(98 155)(99 156)(100 157)(101 158)(102 159)(103 160)(104 161)(105 162)(106 163)(107 164)(108 165)(109 166)(110 167)(111 168)(112 113)
(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)
(1 170)(2 169)(3 224)(4 223)(5 222)(6 221)(7 220)(8 219)(9 218)(10 217)(11 216)(12 215)(13 214)(14 213)(15 212)(16 211)(17 210)(18 209)(19 208)(20 207)(21 206)(22 205)(23 204)(24 203)(25 202)(26 201)(27 200)(28 199)(29 198)(30 197)(31 196)(32 195)(33 194)(34 193)(35 192)(36 191)(37 190)(38 189)(39 188)(40 187)(41 186)(42 185)(43 184)(44 183)(45 182)(46 181)(47 180)(48 179)(49 178)(50 177)(51 176)(52 175)(53 174)(54 173)(55 172)(56 171)(57 144)(58 143)(59 142)(60 141)(61 140)(62 139)(63 138)(64 137)(65 136)(66 135)(67 134)(68 133)(69 132)(70 131)(71 130)(72 129)(73 128)(74 127)(75 126)(76 125)(77 124)(78 123)(79 122)(80 121)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 168)(90 167)(91 166)(92 165)(93 164)(94 163)(95 162)(96 161)(97 160)(98 159)(99 158)(100 157)(101 156)(102 155)(103 154)(104 153)(105 152)(106 151)(107 150)(108 149)(109 148)(110 147)(111 146)(112 145)

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

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

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

124 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28X 56A ··· 56AV order 1 2 ··· 2 2 ··· 2 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 ··· 1 28 ··· 28 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D7 D8 D14 D14 D28 D28 D56 kernel C22×D56 C2×D56 C22×C56 C22×D28 C2×C28 C22×C14 C22×C8 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 12 1 2 3 1 3 8 18 3 18 6 48

Matrix representation of C22×D56 in GL4(𝔽113) generated by

 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 1 0 0 0 0 112 0 0 0 0 85 21 0 0 92 48
,
 1 0 0 0 0 1 0 0 0 0 104 1 0 0 33 9
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,112,0,0,0,0,85,92,0,0,21,48],[1,0,0,0,0,1,0,0,0,0,104,33,0,0,1,9] >;

C22×D56 in GAP, Magma, Sage, TeX

C_2^2\times D_{56}
% in TeX

G:=Group("C2^2xD56");
// GroupNames label

G:=SmallGroup(448,1193);
// by ID

G=gap.SmallGroup(448,1193);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,192,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^56=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽