direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D56, C56⋊9C23, D28⋊4C23, C28.55C24, C23.61D28, (C2×C14)⋊6D8, C14⋊1(C2×D8), C7⋊1(C22×D8), (C2×C8)⋊33D14, C8⋊8(C22×D7), (C22×C8)⋊7D7, C4.45(C2×D28), (C22×C56)⋊11C2, (C2×C56)⋊44C22, (C2×C28).391D4, C28.290(C2×D4), (C2×C4).100D28, C4.52(C23×D7), (C22×D28)⋊11C2, (C2×D28)⋊48C22, C14.22(C22×D4), C2.24(C22×D28), C22.70(C2×D28), (C2×C28).787C23, (C22×C4).443D14, (C22×C14).145D4, (C22×C28).526C22, (C2×C14).178(C2×D4), (C2×C4).736(C22×D7), SmallGroup(448,1193)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2404 in 338 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×8], C4, C4 [×3], C22 [×7], C22 [×32], C7, C8 [×4], C2×C4 [×6], D4 [×20], C23, C23 [×20], D7 [×8], C14, C14 [×6], C2×C8 [×6], D8 [×16], C22×C4, C2×D4 [×18], C24 [×2], C28, C28 [×3], D14 [×32], C2×C14 [×7], C22×C8, C2×D8 [×12], C22×D4 [×2], C56 [×4], D28 [×8], D28 [×12], C2×C28 [×6], C22×D7 [×20], C22×C14, C22×D8, D56 [×16], C2×C56 [×6], C2×D28 [×12], C2×D28 [×6], C22×C28, C23×D7 [×2], C2×D56 [×12], C22×C56, C22×D28 [×2], C22×D56
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, D8 [×4], C2×D4 [×6], C24, D14 [×7], C2×D8 [×6], C22×D4, D28 [×4], C22×D7 [×7], C22×D8, D56 [×4], C2×D28 [×6], C23×D7, C2×D56 [×6], C22×D28, C22×D56
Generators and relations
G = < a,b,c,d | a2=b2=c56=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 224 points
Generators in S224
(1 201)(2 202)(3 203)(4 204)(5 205)(6 206)(7 207)(8 208)(9 209)(10 210)(11 211)(12 212)(13 213)(14 214)(15 215)(16 216)(17 217)(18 218)(19 219)(20 220)(21 221)(22 222)(23 223)(24 224)(25 169)(26 170)(27 171)(28 172)(29 173)(30 174)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 181)(38 182)(39 183)(40 184)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 191)(48 192)(49 193)(50 194)(51 195)(52 196)(53 197)(54 198)(55 199)(56 200)(57 165)(58 166)(59 167)(60 168)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(105 157)(106 158)(107 159)(108 160)(109 161)(110 162)(111 163)(112 164)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(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 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(113 179)(114 180)(115 181)(116 182)(117 183)(118 184)(119 185)(120 186)(121 187)(122 188)(123 189)(124 190)(125 191)(126 192)(127 193)(128 194)(129 195)(130 196)(131 197)(132 198)(133 199)(134 200)(135 201)(136 202)(137 203)(138 204)(139 205)(140 206)(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)(159 169)(160 170)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)
(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 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 60)(17 59)(18 58)(19 57)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 103)(30 102)(31 101)(32 100)(33 99)(34 98)(35 97)(36 96)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(113 215)(114 214)(115 213)(116 212)(117 211)(118 210)(119 209)(120 208)(121 207)(122 206)(123 205)(124 204)(125 203)(126 202)(127 201)(128 200)(129 199)(130 198)(131 197)(132 196)(133 195)(134 194)(135 193)(136 192)(137 191)(138 190)(139 189)(140 188)(141 187)(142 186)(143 185)(144 184)(145 183)(146 182)(147 181)(148 180)(149 179)(150 178)(151 177)(152 176)(153 175)(154 174)(155 173)(156 172)(157 171)(158 170)(159 169)(160 224)(161 223)(162 222)(163 221)(164 220)(165 219)(166 218)(167 217)(168 216)
G:=sub<Sym(224)| (1,201)(2,202)(3,203)(4,204)(5,205)(6,206)(7,207)(8,208)(9,209)(10,210)(11,211)(12,212)(13,213)(14,214)(15,215)(16,216)(17,217)(18,218)(19,219)(20,220)(21,221)(22,222)(23,223)(24,224)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,165)(58,166)(59,167)(60,168)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132)(81,133)(82,134)(83,135)(84,136)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(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)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(113,179)(114,180)(115,181)(116,182)(117,183)(118,184)(119,185)(120,186)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(139,205)(140,206)(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)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,57)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(113,215)(114,214)(115,213)(116,212)(117,211)(118,210)(119,209)(120,208)(121,207)(122,206)(123,205)(124,204)(125,203)(126,202)(127,201)(128,200)(129,199)(130,198)(131,197)(132,196)(133,195)(134,194)(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,187)(142,186)(143,185)(144,184)(145,183)(146,182)(147,181)(148,180)(149,179)(150,178)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,224)(161,223)(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216)>;
G:=Group( (1,201)(2,202)(3,203)(4,204)(5,205)(6,206)(7,207)(8,208)(9,209)(10,210)(11,211)(12,212)(13,213)(14,214)(15,215)(16,216)(17,217)(18,218)(19,219)(20,220)(21,221)(22,222)(23,223)(24,224)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,165)(58,166)(59,167)(60,168)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132)(81,133)(82,134)(83,135)(84,136)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(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)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(113,179)(114,180)(115,181)(116,182)(117,183)(118,184)(119,185)(120,186)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(139,205)(140,206)(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)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,57)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(113,215)(114,214)(115,213)(116,212)(117,211)(118,210)(119,209)(120,208)(121,207)(122,206)(123,205)(124,204)(125,203)(126,202)(127,201)(128,200)(129,199)(130,198)(131,197)(132,196)(133,195)(134,194)(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,187)(142,186)(143,185)(144,184)(145,183)(146,182)(147,181)(148,180)(149,179)(150,178)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,224)(161,223)(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216) );
G=PermutationGroup([(1,201),(2,202),(3,203),(4,204),(5,205),(6,206),(7,207),(8,208),(9,209),(10,210),(11,211),(12,212),(13,213),(14,214),(15,215),(16,216),(17,217),(18,218),(19,219),(20,220),(21,221),(22,222),(23,223),(24,224),(25,169),(26,170),(27,171),(28,172),(29,173),(30,174),(31,175),(32,176),(33,177),(34,178),(35,179),(36,180),(37,181),(38,182),(39,183),(40,184),(41,185),(42,186),(43,187),(44,188),(45,189),(46,190),(47,191),(48,192),(49,193),(50,194),(51,195),(52,196),(53,197),(54,198),(55,199),(56,200),(57,165),(58,166),(59,167),(60,168),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,121),(70,122),(71,123),(72,124),(73,125),(74,126),(75,127),(76,128),(77,129),(78,130),(79,131),(80,132),(81,133),(82,134),(83,135),(84,136),(85,137),(86,138),(87,139),(88,140),(89,141),(90,142),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,151),(100,152),(101,153),(102,154),(103,155),(104,156),(105,157),(106,158),(107,159),(108,160),(109,161),(110,162),(111,163),(112,164)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,97),(16,98),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(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),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(113,179),(114,180),(115,181),(116,182),(117,183),(118,184),(119,185),(120,186),(121,187),(122,188),(123,189),(124,190),(125,191),(126,192),(127,193),(128,194),(129,195),(130,196),(131,197),(132,198),(133,199),(134,200),(135,201),(136,202),(137,203),(138,204),(139,205),(140,206),(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),(159,169),(160,170),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178)], [(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)], [(1,75),(2,74),(3,73),(4,72),(5,71),(6,70),(7,69),(8,68),(9,67),(10,66),(11,65),(12,64),(13,63),(14,62),(15,61),(16,60),(17,59),(18,58),(19,57),(20,112),(21,111),(22,110),(23,109),(24,108),(25,107),(26,106),(27,105),(28,104),(29,103),(30,102),(31,101),(32,100),(33,99),(34,98),(35,97),(36,96),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,84),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(113,215),(114,214),(115,213),(116,212),(117,211),(118,210),(119,209),(120,208),(121,207),(122,206),(123,205),(124,204),(125,203),(126,202),(127,201),(128,200),(129,199),(130,198),(131,197),(132,196),(133,195),(134,194),(135,193),(136,192),(137,191),(138,190),(139,189),(140,188),(141,187),(142,186),(143,185),(144,184),(145,183),(146,182),(147,181),(148,180),(149,179),(150,178),(151,177),(152,176),(153,175),(154,174),(155,173),(156,172),(157,171),(158,170),(159,169),(160,224),(161,223),(162,222),(163,221),(164,220),(165,219),(166,218),(167,217),(168,216)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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