direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C2.D56, C22.16D56, C23.58D28, (C2×D28)⋊8C4, (C2×C8)⋊32D14, C2.3(C2×D56), (C22×C8)⋊4D7, D28⋊17(C2×C4), (C22×C56)⋊3C2, (C2×C14).23D8, (C2×C4).96D28, C14.16(C2×D8), (C2×C56)⋊41C22, C28.410(C2×D4), (C2×C28).474D4, C14⋊2(D4⋊C4), C4.27(D14⋊C4), C4⋊Dic7⋊47C22, C14.16(C2×SD16), (C2×C14).22SD16, (C22×D28).6C2, C22.53(C2×D28), C28.52(C22⋊C4), (C2×C28).766C23, C28.112(C22×C4), (C22×C4).428D14, (C22×C14).138D4, C22.49(D14⋊C4), (C2×D28).198C22, C22.12(C56⋊C2), (C22×C28).517C22, C4.70(C2×C4×D7), C7⋊3(C2×D4⋊C4), C2.4(C2×C56⋊C2), (C2×C4⋊Dic7)⋊15C2, C2.24(C2×D14⋊C4), (C2×C4).116(C4×D7), C4.103(C2×C7⋊D4), (C2×C28).229(C2×C4), (C2×C14).156(C2×D4), C14.52(C2×C22⋊C4), (C2×C4).254(C7⋊D4), (C2×C4).714(C22×D7), (C2×C14).63(C22⋊C4), SmallGroup(448,646)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C2.D56
G = < a,b,c,d | a2=b2=c56=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 1380 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C56, D28, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×C14, C2×D4⋊C4, C4⋊Dic7, C4⋊Dic7, C2×C56, C2×C56, C2×D28, C2×D28, C22×Dic7, C22×C28, C23×D7, C2.D56, C2×C4⋊Dic7, C22×C56, C22×D28, C2×C2.D56
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, D8, SD16, C22×C4, C2×D4, D14, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C4×D7, D28, C7⋊D4, C22×D7, C2×D4⋊C4, C56⋊C2, D56, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C2.D56, C2×C56⋊C2, C2×D56, C2×D14⋊C4, C2×C2.D56
►On 224 points
Generators in S224
(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 151)(17 152)(18 153)(19 154)(20 155)(21 156)(22 157)(23 158)(24 159)(25 160)(26 161)(27 162)(28 163)(29 164)(30 165)(31 166)(32 167)(33 168)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 217)(58 218)(59 219)(60 220)(61 221)(62 222)(63 223)(64 224)(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)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(105 209)(106 210)(107 211)(108 212)(109 213)(110 214)(111 215)(112 216)
(1 181)(2 182)(3 183)(4 184)(5 185)(6 186)(7 187)(8 188)(9 189)(10 190)(11 191)(12 192)(13 193)(14 194)(15 195)(16 196)(17 197)(18 198)(19 199)(20 200)(21 201)(22 202)(23 203)(24 204)(25 205)(26 206)(27 207)(28 208)(29 209)(30 210)(31 211)(32 212)(33 213)(34 214)(35 215)(36 216)(37 217)(38 218)(39 219)(40 220)(41 221)(42 222)(43 223)(44 224)(45 169)(46 170)(47 171)(48 172)(49 173)(50 174)(51 175)(52 176)(53 177)(54 178)(55 179)(56 180)(57 116)(58 117)(59 118)(60 119)(61 120)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 113)(111 114)(112 115)
(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 76 181 135)(2 134 182 75)(3 74 183 133)(4 132 184 73)(5 72 185 131)(6 130 186 71)(7 70 187 129)(8 128 188 69)(9 68 189 127)(10 126 190 67)(11 66 191 125)(12 124 192 65)(13 64 193 123)(14 122 194 63)(15 62 195 121)(16 120 196 61)(17 60 197 119)(18 118 198 59)(19 58 199 117)(20 116 200 57)(21 112 201 115)(22 114 202 111)(23 110 203 113)(24 168 204 109)(25 108 205 167)(26 166 206 107)(27 106 207 165)(28 164 208 105)(29 104 209 163)(30 162 210 103)(31 102 211 161)(32 160 212 101)(33 100 213 159)(34 158 214 99)(35 98 215 157)(36 156 216 97)(37 96 217 155)(38 154 218 95)(39 94 219 153)(40 152 220 93)(41 92 221 151)(42 150 222 91)(43 90 223 149)(44 148 224 89)(45 88 169 147)(46 146 170 87)(47 86 171 145)(48 144 172 85)(49 84 173 143)(50 142 174 83)(51 82 175 141)(52 140 176 81)(53 80 177 139)(54 138 178 79)(55 78 179 137)(56 136 180 77)
G:=sub<Sym(224)| (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(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)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,209)(106,210)(107,211)(108,212)(109,213)(110,214)(111,215)(112,216), (1,181)(2,182)(3,183)(4,184)(5,185)(6,186)(7,187)(8,188)(9,189)(10,190)(11,191)(12,192)(13,193)(14,194)(15,195)(16,196)(17,197)(18,198)(19,199)(20,200)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,210)(31,211)(32,212)(33,213)(34,214)(35,215)(36,216)(37,217)(38,218)(39,219)(40,220)(41,221)(42,222)(43,223)(44,224)(45,169)(46,170)(47,171)(48,172)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,113)(111,114)(112,115), (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), (1,76,181,135)(2,134,182,75)(3,74,183,133)(4,132,184,73)(5,72,185,131)(6,130,186,71)(7,70,187,129)(8,128,188,69)(9,68,189,127)(10,126,190,67)(11,66,191,125)(12,124,192,65)(13,64,193,123)(14,122,194,63)(15,62,195,121)(16,120,196,61)(17,60,197,119)(18,118,198,59)(19,58,199,117)(20,116,200,57)(21,112,201,115)(22,114,202,111)(23,110,203,113)(24,168,204,109)(25,108,205,167)(26,166,206,107)(27,106,207,165)(28,164,208,105)(29,104,209,163)(30,162,210,103)(31,102,211,161)(32,160,212,101)(33,100,213,159)(34,158,214,99)(35,98,215,157)(36,156,216,97)(37,96,217,155)(38,154,218,95)(39,94,219,153)(40,152,220,93)(41,92,221,151)(42,150,222,91)(43,90,223,149)(44,148,224,89)(45,88,169,147)(46,146,170,87)(47,86,171,145)(48,144,172,85)(49,84,173,143)(50,142,174,83)(51,82,175,141)(52,140,176,81)(53,80,177,139)(54,138,178,79)(55,78,179,137)(56,136,180,77)>;
G:=Group( (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(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)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,209)(106,210)(107,211)(108,212)(109,213)(110,214)(111,215)(112,216), (1,181)(2,182)(3,183)(4,184)(5,185)(6,186)(7,187)(8,188)(9,189)(10,190)(11,191)(12,192)(13,193)(14,194)(15,195)(16,196)(17,197)(18,198)(19,199)(20,200)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,210)(31,211)(32,212)(33,213)(34,214)(35,215)(36,216)(37,217)(38,218)(39,219)(40,220)(41,221)(42,222)(43,223)(44,224)(45,169)(46,170)(47,171)(48,172)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,113)(111,114)(112,115), (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), (1,76,181,135)(2,134,182,75)(3,74,183,133)(4,132,184,73)(5,72,185,131)(6,130,186,71)(7,70,187,129)(8,128,188,69)(9,68,189,127)(10,126,190,67)(11,66,191,125)(12,124,192,65)(13,64,193,123)(14,122,194,63)(15,62,195,121)(16,120,196,61)(17,60,197,119)(18,118,198,59)(19,58,199,117)(20,116,200,57)(21,112,201,115)(22,114,202,111)(23,110,203,113)(24,168,204,109)(25,108,205,167)(26,166,206,107)(27,106,207,165)(28,164,208,105)(29,104,209,163)(30,162,210,103)(31,102,211,161)(32,160,212,101)(33,100,213,159)(34,158,214,99)(35,98,215,157)(36,156,216,97)(37,96,217,155)(38,154,218,95)(39,94,219,153)(40,152,220,93)(41,92,221,151)(42,150,222,91)(43,90,223,149)(44,148,224,89)(45,88,169,147)(46,146,170,87)(47,86,171,145)(48,144,172,85)(49,84,173,143)(50,142,174,83)(51,82,175,141)(52,140,176,81)(53,80,177,139)(54,138,178,79)(55,78,179,137)(56,136,180,77) );
G=PermutationGroup([[(1,136),(2,137),(3,138),(4,139),(5,140),(6,141),(7,142),(8,143),(9,144),(10,145),(11,146),(12,147),(13,148),(14,149),(15,150),(16,151),(17,152),(18,153),(19,154),(20,155),(21,156),(22,157),(23,158),(24,159),(25,160),(26,161),(27,162),(28,163),(29,164),(30,165),(31,166),(32,167),(33,168),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(41,120),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(49,128),(50,129),(51,130),(52,131),(53,132),(54,133),(55,134),(56,135),(57,217),(58,218),(59,219),(60,220),(61,221),(62,222),(63,223),(64,224),(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),(85,189),(86,190),(87,191),(88,192),(89,193),(90,194),(91,195),(92,196),(93,197),(94,198),(95,199),(96,200),(97,201),(98,202),(99,203),(100,204),(101,205),(102,206),(103,207),(104,208),(105,209),(106,210),(107,211),(108,212),(109,213),(110,214),(111,215),(112,216)], [(1,181),(2,182),(3,183),(4,184),(5,185),(6,186),(7,187),(8,188),(9,189),(10,190),(11,191),(12,192),(13,193),(14,194),(15,195),(16,196),(17,197),(18,198),(19,199),(20,200),(21,201),(22,202),(23,203),(24,204),(25,205),(26,206),(27,207),(28,208),(29,209),(30,210),(31,211),(32,212),(33,213),(34,214),(35,215),(36,216),(37,217),(38,218),(39,219),(40,220),(41,221),(42,222),(43,223),(44,224),(45,169),(46,170),(47,171),(48,172),(49,173),(50,174),(51,175),(52,176),(53,177),(54,178),(55,179),(56,180),(57,116),(58,117),(59,118),(60,119),(61,120),(62,121),(63,122),(64,123),(65,124),(66,125),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,168),(110,113),(111,114),(112,115)], [(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)], [(1,76,181,135),(2,134,182,75),(3,74,183,133),(4,132,184,73),(5,72,185,131),(6,130,186,71),(7,70,187,129),(8,128,188,69),(9,68,189,127),(10,126,190,67),(11,66,191,125),(12,124,192,65),(13,64,193,123),(14,122,194,63),(15,62,195,121),(16,120,196,61),(17,60,197,119),(18,118,198,59),(19,58,199,117),(20,116,200,57),(21,112,201,115),(22,114,202,111),(23,110,203,113),(24,168,204,109),(25,108,205,167),(26,166,206,107),(27,106,207,165),(28,164,208,105),(29,104,209,163),(30,162,210,103),(31,102,211,161),(32,160,212,101),(33,100,213,159),(34,158,214,99),(35,98,215,157),(36,156,216,97),(37,96,217,155),(38,154,218,95),(39,94,219,153),(40,152,220,93),(41,92,221,151),(42,150,222,91),(43,90,223,149),(44,148,224,89),(45,88,169,147),(46,146,170,87),(47,86,171,145),(48,144,172,85),(49,84,173,143),(50,142,174,83),(51,82,175,141),(52,140,176,81),(53,80,177,139),(54,138,178,79),(55,78,179,137),(56,136,180,77)]])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D28 | C56⋊C2 | D56 |
| kernel | C2×C2.D56 | C2.D56 | C2×C4⋊Dic7 | C22×C56 | C22×D28 | C2×D28 | C2×C28 | C22×C14 | C22×C8 | C2×C14 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 6 | 12 | 6 | 24 | 24 |
Matrix representation of C2×C2.D56 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 100 | 13 | 0 | 0 |
| 0 | 100 | 100 | 0 | 0 |
| 0 | 0 | 0 | 106 | 84 |
| 0 | 0 | 0 | 29 | 88 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 100 | 13 | 0 | 0 |
| 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 6 | 88 |
| 0 | 0 | 0 | 6 | 107 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112],[98,0,0,0,0,0,100,100,0,0,0,13,100,0,0,0,0,0,106,29,0,0,0,84,88],[98,0,0,0,0,0,100,13,0,0,0,13,13,0,0,0,0,0,6,6,0,0,0,88,107] >;
C2×C2.D56 in GAP, Magma, Sage, TeX
C_2\times C_2.D_{56} % in TeX
G:=Group("C2xC2.D56"); // GroupNames label
G:=SmallGroup(448,646);
// by ID
G=gap.SmallGroup(448,646);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^56=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations