metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D56, C23.39D28, C22⋊C8⋊4D7, C56⋊1C4⋊4C2, C2.8(C2×D56), C14.6(C2×D8), (C2×C14).5D8, (C2×C8).4D14, (C2×C4).34D28, (C2×C28).45D4, C2.D56⋊6C2, C28⋊7D4.3C2, (C2×C56).4C22, (C22×C14).56D4, (C22×C4).86D14, C7⋊1(C22.D8), C28.283(C4○D4), (C2×C28).746C23, (C2×D28).11C22, C22.109(C2×D28), C4.107(D4⋊2D7), C2.13(C8.D14), C14.10(C8.C22), C4⋊Dic7.271C22, (C22×C28).53C22, C2.14(C22.D28), C14.18(C22.D4), (C7×C22⋊C8)⋊6C2, (C2×C4⋊Dic7)⋊6C2, (C2×C14).129(C2×D4), (C2×C4).691(C22×D7), SmallGroup(448,270)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D56
G = < a,b,c,d | a2=b2=c56=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 700 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C22.D8, C4⋊Dic7, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C2×C56, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, C56⋊1C4, C2.D56, C7×C22⋊C8, C2×C4⋊Dic7, C28⋊7D4, C22.D56
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C22.D4, C2×D8, C8.C22, D28, C22×D7, C22.D8, D56, C2×D28, D4⋊2D7, C22.D28, C2×D56, C8.D14, C22.D56
►On 224 points
Generators in S224
(1 29)(2 135)(3 31)(4 137)(5 33)(6 139)(7 35)(8 141)(9 37)(10 143)(11 39)(12 145)(13 41)(14 147)(15 43)(16 149)(17 45)(18 151)(19 47)(20 153)(21 49)(22 155)(23 51)(24 157)(25 53)(26 159)(27 55)(28 161)(30 163)(32 165)(34 167)(36 113)(38 115)(40 117)(42 119)(44 121)(46 123)(48 125)(50 127)(52 129)(54 131)(56 133)(57 85)(58 201)(59 87)(60 203)(61 89)(62 205)(63 91)(64 207)(65 93)(66 209)(67 95)(68 211)(69 97)(70 213)(71 99)(72 215)(73 101)(74 217)(75 103)(76 219)(77 105)(78 221)(79 107)(80 223)(81 109)(82 169)(83 111)(84 171)(86 173)(88 175)(90 177)(92 179)(94 181)(96 183)(98 185)(100 187)(102 189)(104 191)(106 193)(108 195)(110 197)(112 199)(114 142)(116 144)(118 146)(120 148)(122 150)(124 152)(126 154)(128 156)(130 158)(132 160)(134 162)(136 164)(138 166)(140 168)(170 198)(172 200)(174 202)(176 204)(178 206)(180 208)(182 210)(184 212)(186 214)(188 216)(190 218)(192 220)(194 222)(196 224)
(1 162)(2 163)(3 164)(4 165)(5 166)(6 167)(7 168)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 142)(38 143)(39 144)(40 145)(41 146)(42 147)(43 148)(44 149)(45 150)(46 151)(47 152)(48 153)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 172)(58 173)(59 174)(60 175)(61 176)(62 177)(63 178)(64 179)(65 180)(66 181)(67 182)(68 183)(69 184)(70 185)(71 186)(72 187)(73 188)(74 189)(75 190)(76 191)(77 192)(78 193)(79 194)(80 195)(81 196)(82 197)(83 198)(84 199)(85 200)(86 201)(87 202)(88 203)(89 204)(90 205)(91 206)(92 207)(93 208)(94 209)(95 210)(96 211)(97 212)(98 213)(99 214)(100 215)(101 216)(102 217)(103 218)(104 219)(105 220)(106 221)(107 222)(108 223)(109 224)(110 169)(111 170)(112 171)
(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)
(1 182 162 67)(2 66 163 181)(3 180 164 65)(4 64 165 179)(5 178 166 63)(6 62 167 177)(7 176 168 61)(8 60 113 175)(9 174 114 59)(10 58 115 173)(11 172 116 57)(12 112 117 171)(13 170 118 111)(14 110 119 169)(15 224 120 109)(16 108 121 223)(17 222 122 107)(18 106 123 221)(19 220 124 105)(20 104 125 219)(21 218 126 103)(22 102 127 217)(23 216 128 101)(24 100 129 215)(25 214 130 99)(26 98 131 213)(27 212 132 97)(28 96 133 211)(29 210 134 95)(30 94 135 209)(31 208 136 93)(32 92 137 207)(33 206 138 91)(34 90 139 205)(35 204 140 89)(36 88 141 203)(37 202 142 87)(38 86 143 201)(39 200 144 85)(40 84 145 199)(41 198 146 83)(42 82 147 197)(43 196 148 81)(44 80 149 195)(45 194 150 79)(46 78 151 193)(47 192 152 77)(48 76 153 191)(49 190 154 75)(50 74 155 189)(51 188 156 73)(52 72 157 187)(53 186 158 71)(54 70 159 185)(55 184 160 69)(56 68 161 183)
G:=sub<Sym(224)| (1,29)(2,135)(3,31)(4,137)(5,33)(6,139)(7,35)(8,141)(9,37)(10,143)(11,39)(12,145)(13,41)(14,147)(15,43)(16,149)(17,45)(18,151)(19,47)(20,153)(21,49)(22,155)(23,51)(24,157)(25,53)(26,159)(27,55)(28,161)(30,163)(32,165)(34,167)(36,113)(38,115)(40,117)(42,119)(44,121)(46,123)(48,125)(50,127)(52,129)(54,131)(56,133)(57,85)(58,201)(59,87)(60,203)(61,89)(62,205)(63,91)(64,207)(65,93)(66,209)(67,95)(68,211)(69,97)(70,213)(71,99)(72,215)(73,101)(74,217)(75,103)(76,219)(77,105)(78,221)(79,107)(80,223)(81,109)(82,169)(83,111)(84,171)(86,173)(88,175)(90,177)(92,179)(94,181)(96,183)(98,185)(100,187)(102,189)(104,191)(106,193)(108,195)(110,197)(112,199)(114,142)(116,144)(118,146)(120,148)(122,150)(124,152)(126,154)(128,156)(130,158)(132,160)(134,162)(136,164)(138,166)(140,168)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224), (1,162)(2,163)(3,164)(4,165)(5,166)(6,167)(7,168)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,190)(76,191)(77,192)(78,193)(79,194)(80,195)(81,196)(82,197)(83,198)(84,199)(85,200)(86,201)(87,202)(88,203)(89,204)(90,205)(91,206)(92,207)(93,208)(94,209)(95,210)(96,211)(97,212)(98,213)(99,214)(100,215)(101,216)(102,217)(103,218)(104,219)(105,220)(106,221)(107,222)(108,223)(109,224)(110,169)(111,170)(112,171), (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), (1,182,162,67)(2,66,163,181)(3,180,164,65)(4,64,165,179)(5,178,166,63)(6,62,167,177)(7,176,168,61)(8,60,113,175)(9,174,114,59)(10,58,115,173)(11,172,116,57)(12,112,117,171)(13,170,118,111)(14,110,119,169)(15,224,120,109)(16,108,121,223)(17,222,122,107)(18,106,123,221)(19,220,124,105)(20,104,125,219)(21,218,126,103)(22,102,127,217)(23,216,128,101)(24,100,129,215)(25,214,130,99)(26,98,131,213)(27,212,132,97)(28,96,133,211)(29,210,134,95)(30,94,135,209)(31,208,136,93)(32,92,137,207)(33,206,138,91)(34,90,139,205)(35,204,140,89)(36,88,141,203)(37,202,142,87)(38,86,143,201)(39,200,144,85)(40,84,145,199)(41,198,146,83)(42,82,147,197)(43,196,148,81)(44,80,149,195)(45,194,150,79)(46,78,151,193)(47,192,152,77)(48,76,153,191)(49,190,154,75)(50,74,155,189)(51,188,156,73)(52,72,157,187)(53,186,158,71)(54,70,159,185)(55,184,160,69)(56,68,161,183)>;
G:=Group( (1,29)(2,135)(3,31)(4,137)(5,33)(6,139)(7,35)(8,141)(9,37)(10,143)(11,39)(12,145)(13,41)(14,147)(15,43)(16,149)(17,45)(18,151)(19,47)(20,153)(21,49)(22,155)(23,51)(24,157)(25,53)(26,159)(27,55)(28,161)(30,163)(32,165)(34,167)(36,113)(38,115)(40,117)(42,119)(44,121)(46,123)(48,125)(50,127)(52,129)(54,131)(56,133)(57,85)(58,201)(59,87)(60,203)(61,89)(62,205)(63,91)(64,207)(65,93)(66,209)(67,95)(68,211)(69,97)(70,213)(71,99)(72,215)(73,101)(74,217)(75,103)(76,219)(77,105)(78,221)(79,107)(80,223)(81,109)(82,169)(83,111)(84,171)(86,173)(88,175)(90,177)(92,179)(94,181)(96,183)(98,185)(100,187)(102,189)(104,191)(106,193)(108,195)(110,197)(112,199)(114,142)(116,144)(118,146)(120,148)(122,150)(124,152)(126,154)(128,156)(130,158)(132,160)(134,162)(136,164)(138,166)(140,168)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224), (1,162)(2,163)(3,164)(4,165)(5,166)(6,167)(7,168)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,190)(76,191)(77,192)(78,193)(79,194)(80,195)(81,196)(82,197)(83,198)(84,199)(85,200)(86,201)(87,202)(88,203)(89,204)(90,205)(91,206)(92,207)(93,208)(94,209)(95,210)(96,211)(97,212)(98,213)(99,214)(100,215)(101,216)(102,217)(103,218)(104,219)(105,220)(106,221)(107,222)(108,223)(109,224)(110,169)(111,170)(112,171), (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), (1,182,162,67)(2,66,163,181)(3,180,164,65)(4,64,165,179)(5,178,166,63)(6,62,167,177)(7,176,168,61)(8,60,113,175)(9,174,114,59)(10,58,115,173)(11,172,116,57)(12,112,117,171)(13,170,118,111)(14,110,119,169)(15,224,120,109)(16,108,121,223)(17,222,122,107)(18,106,123,221)(19,220,124,105)(20,104,125,219)(21,218,126,103)(22,102,127,217)(23,216,128,101)(24,100,129,215)(25,214,130,99)(26,98,131,213)(27,212,132,97)(28,96,133,211)(29,210,134,95)(30,94,135,209)(31,208,136,93)(32,92,137,207)(33,206,138,91)(34,90,139,205)(35,204,140,89)(36,88,141,203)(37,202,142,87)(38,86,143,201)(39,200,144,85)(40,84,145,199)(41,198,146,83)(42,82,147,197)(43,196,148,81)(44,80,149,195)(45,194,150,79)(46,78,151,193)(47,192,152,77)(48,76,153,191)(49,190,154,75)(50,74,155,189)(51,188,156,73)(52,72,157,187)(53,186,158,71)(54,70,159,185)(55,184,160,69)(56,68,161,183) );
G=PermutationGroup([[(1,29),(2,135),(3,31),(4,137),(5,33),(6,139),(7,35),(8,141),(9,37),(10,143),(11,39),(12,145),(13,41),(14,147),(15,43),(16,149),(17,45),(18,151),(19,47),(20,153),(21,49),(22,155),(23,51),(24,157),(25,53),(26,159),(27,55),(28,161),(30,163),(32,165),(34,167),(36,113),(38,115),(40,117),(42,119),(44,121),(46,123),(48,125),(50,127),(52,129),(54,131),(56,133),(57,85),(58,201),(59,87),(60,203),(61,89),(62,205),(63,91),(64,207),(65,93),(66,209),(67,95),(68,211),(69,97),(70,213),(71,99),(72,215),(73,101),(74,217),(75,103),(76,219),(77,105),(78,221),(79,107),(80,223),(81,109),(82,169),(83,111),(84,171),(86,173),(88,175),(90,177),(92,179),(94,181),(96,183),(98,185),(100,187),(102,189),(104,191),(106,193),(108,195),(110,197),(112,199),(114,142),(116,144),(118,146),(120,148),(122,150),(124,152),(126,154),(128,156),(130,158),(132,160),(134,162),(136,164),(138,166),(140,168),(170,198),(172,200),(174,202),(176,204),(178,206),(180,208),(182,210),(184,212),(186,214),(188,216),(190,218),(192,220),(194,222),(196,224)], [(1,162),(2,163),(3,164),(4,165),(5,166),(6,167),(7,168),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,121),(17,122),(18,123),(19,124),(20,125),(21,126),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,141),(37,142),(38,143),(39,144),(40,145),(41,146),(42,147),(43,148),(44,149),(45,150),(46,151),(47,152),(48,153),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,161),(57,172),(58,173),(59,174),(60,175),(61,176),(62,177),(63,178),(64,179),(65,180),(66,181),(67,182),(68,183),(69,184),(70,185),(71,186),(72,187),(73,188),(74,189),(75,190),(76,191),(77,192),(78,193),(79,194),(80,195),(81,196),(82,197),(83,198),(84,199),(85,200),(86,201),(87,202),(88,203),(89,204),(90,205),(91,206),(92,207),(93,208),(94,209),(95,210),(96,211),(97,212),(98,213),(99,214),(100,215),(101,216),(102,217),(103,218),(104,219),(105,220),(106,221),(107,222),(108,223),(109,224),(110,169),(111,170),(112,171)], [(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)], [(1,182,162,67),(2,66,163,181),(3,180,164,65),(4,64,165,179),(5,178,166,63),(6,62,167,177),(7,176,168,61),(8,60,113,175),(9,174,114,59),(10,58,115,173),(11,172,116,57),(12,112,117,171),(13,170,118,111),(14,110,119,169),(15,224,120,109),(16,108,121,223),(17,222,122,107),(18,106,123,221),(19,220,124,105),(20,104,125,219),(21,218,126,103),(22,102,127,217),(23,216,128,101),(24,100,129,215),(25,214,130,99),(26,98,131,213),(27,212,132,97),(28,96,133,211),(29,210,134,95),(30,94,135,209),(31,208,136,93),(32,92,137,207),(33,206,138,91),(34,90,139,205),(35,204,140,89),(36,88,141,203),(37,202,142,87),(38,86,143,201),(39,200,144,85),(40,84,145,199),(41,198,146,83),(42,82,147,197),(43,196,148,81),(44,80,149,195),(45,194,150,79),(46,78,151,193),(47,192,152,77),(48,76,153,191),(49,190,154,75),(50,74,155,189),(51,188,156,73),(52,72,157,187),(53,186,158,71),(54,70,159,185),(55,184,160,69),(56,68,161,183)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D8 | D14 | D14 | D28 | D28 | D56 | C8.C22 | D4⋊2D7 | C8.D14 |
| kernel | C22.D56 | C56⋊1C4 | C2.D56 | C7×C22⋊C8 | C2×C4⋊Dic7 | C28⋊7D4 | C2×C28 | C22×C14 | C22⋊C8 | C28 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C14 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 3 | 6 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of C22.D56 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 50 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 44 | 0 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 104 | 0 | 0 |
| 0 | 0 | 9 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 112 |
| 0 | 0 | 0 | 0 | 59 | 88 |
| 11 | 53 | 0 | 0 | 0 | 0 |
| 2 | 102 | 0 | 0 | 0 | 0 |
| 0 | 0 | 76 | 60 | 0 | 0 |
| 0 | 0 | 45 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 72 | 98 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,50,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[44,1,0,0,0,0,0,18,0,0,0,0,0,0,67,9,0,0,0,0,104,100,0,0,0,0,0,0,25,59,0,0,0,0,112,88],[11,2,0,0,0,0,53,102,0,0,0,0,0,0,76,45,0,0,0,0,60,37,0,0,0,0,0,0,15,72,0,0,0,0,0,98] >;
C22.D56 in GAP, Magma, Sage, TeX
C_2^2.D_{56} % in TeX
G:=Group("C2^2.D56"); // GroupNames label
G:=SmallGroup(448,270);
// by ID
G=gap.SmallGroup(448,270);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,219,310,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^56=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations