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## G = C2×C2.D56order 448 = 26·7

### Direct product of C2 and C2.D56

Series: Derived Chief Lower central Upper central

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

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

Smallest permutation representation of 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

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