direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C56⋊C2, C56⋊10C23, C28.54C24, C23.60D28, Dic14⋊4C23, D28.20C23, (C2×C8)⋊35D14, C8⋊9(C22×D7), C14⋊1(C2×SD16), (C2×C14)⋊9SD16, C4.44(C2×D28), (C2×C4).99D28, (C22×C8)⋊10D7, C7⋊1(C22×SD16), (C2×C56)⋊46C22, (C22×C56)⋊14C2, C28.289(C2×D4), (C2×C28).390D4, C4.51(C23×D7), (C22×D28).9C2, C14.21(C22×D4), C2.23(C22×D28), C22.69(C2×D28), (C2×C28).786C23, (C22×C14).144D4, (C22×C4).442D14, (C22×Dic14)⋊11C2, (C2×Dic14)⋊56C22, (C2×D28).228C22, (C22×C28).525C22, (C2×C14).177(C2×D4), (C2×C4).735(C22×D7), SmallGroup(448,1192)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1764 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C7, C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×10], Q8 [×10], C23, C23 [×10], D7 [×4], C14, C14 [×6], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×9], C2×Q8 [×9], C24, Dic7 [×4], C28, C28 [×3], D14 [×16], C2×C14 [×7], C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C56 [×4], Dic14 [×4], Dic14 [×6], D28 [×4], D28 [×6], C2×Dic7 [×6], C2×C28 [×6], C22×D7 [×10], C22×C14, C22×SD16, C56⋊C2 [×16], C2×C56 [×6], C2×Dic14 [×6], C2×Dic14 [×3], C2×D28 [×6], C2×D28 [×3], C22×Dic7, C22×C28, C23×D7, C2×C56⋊C2 [×12], C22×C56, C22×Dic14, C22×D28, C22×C56⋊C2
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, SD16 [×4], C2×D4 [×6], C24, D14 [×7], C2×SD16 [×6], C22×D4, D28 [×4], C22×D7 [×7], C22×SD16, C56⋊C2 [×4], C2×D28 [×6], C23×D7, C2×C56⋊C2 [×6], C22×D28, C22×C56⋊C2
Generators and relations
G = < a,b,c,d | a2=b2=c56=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c27 >
►On 224 points
Generators in S224
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 160)(43 161)(44 162)(45 163)(46 164)(47 165)(48 166)(49 167)(50 168)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 181)(7 182)(8 183)(9 184)(10 185)(11 186)(12 187)(13 188)(14 189)(15 190)(16 191)(17 192)(18 193)(19 194)(20 195)(21 196)(22 197)(23 198)(24 199)(25 200)(26 201)(27 202)(28 203)(29 204)(30 205)(31 206)(32 207)(33 208)(34 209)(35 210)(36 211)(37 212)(38 213)(39 214)(40 215)(41 216)(42 217)(43 218)(44 219)(45 220)(46 221)(47 222)(48 223)(49 224)(50 169)(51 170)(52 171)(53 172)(54 173)(55 174)(56 175)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 145)(83 146)(84 147)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 113)(107 114)(108 115)(109 116)(110 117)(111 118)(112 119)
(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)
(1 176)(2 203)(3 174)(4 201)(5 172)(6 199)(7 170)(8 197)(9 224)(10 195)(11 222)(12 193)(13 220)(14 191)(15 218)(16 189)(17 216)(18 187)(19 214)(20 185)(21 212)(22 183)(23 210)(24 181)(25 208)(26 179)(27 206)(28 177)(29 204)(30 175)(31 202)(32 173)(33 200)(34 171)(35 198)(36 169)(37 196)(38 223)(39 194)(40 221)(41 192)(42 219)(43 190)(44 217)(45 188)(46 215)(47 186)(48 213)(49 184)(50 211)(51 182)(52 209)(53 180)(54 207)(55 178)(56 205)(57 146)(58 117)(59 144)(60 115)(61 142)(62 113)(63 140)(64 167)(65 138)(66 165)(67 136)(68 163)(69 134)(70 161)(71 132)(72 159)(73 130)(74 157)(75 128)(76 155)(77 126)(78 153)(79 124)(80 151)(81 122)(82 149)(83 120)(84 147)(85 118)(86 145)(87 116)(88 143)(89 114)(90 141)(91 168)(92 139)(93 166)(94 137)(95 164)(96 135)(97 162)(98 133)(99 160)(100 131)(101 158)(102 129)(103 156)(104 127)(105 154)(106 125)(107 152)(108 123)(109 150)(110 121)(111 148)(112 119)
G:=sub<Sym(224)| (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176), (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,190)(16,191)(17,192)(18,193)(19,194)(20,195)(21,196)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,204)(30,205)(31,206)(32,207)(33,208)(34,209)(35,210)(36,211)(37,212)(38,213)(39,214)(40,215)(41,216)(42,217)(43,218)(44,219)(45,220)(46,221)(47,222)(48,223)(49,224)(50,169)(51,170)(52,171)(53,172)(54,173)(55,174)(56,175)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)(112,119), (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), (1,176)(2,203)(3,174)(4,201)(5,172)(6,199)(7,170)(8,197)(9,224)(10,195)(11,222)(12,193)(13,220)(14,191)(15,218)(16,189)(17,216)(18,187)(19,214)(20,185)(21,212)(22,183)(23,210)(24,181)(25,208)(26,179)(27,206)(28,177)(29,204)(30,175)(31,202)(32,173)(33,200)(34,171)(35,198)(36,169)(37,196)(38,223)(39,194)(40,221)(41,192)(42,219)(43,190)(44,217)(45,188)(46,215)(47,186)(48,213)(49,184)(50,211)(51,182)(52,209)(53,180)(54,207)(55,178)(56,205)(57,146)(58,117)(59,144)(60,115)(61,142)(62,113)(63,140)(64,167)(65,138)(66,165)(67,136)(68,163)(69,134)(70,161)(71,132)(72,159)(73,130)(74,157)(75,128)(76,155)(77,126)(78,153)(79,124)(80,151)(81,122)(82,149)(83,120)(84,147)(85,118)(86,145)(87,116)(88,143)(89,114)(90,141)(91,168)(92,139)(93,166)(94,137)(95,164)(96,135)(97,162)(98,133)(99,160)(100,131)(101,158)(102,129)(103,156)(104,127)(105,154)(106,125)(107,152)(108,123)(109,150)(110,121)(111,148)(112,119)>;
G:=Group( (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176), (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,190)(16,191)(17,192)(18,193)(19,194)(20,195)(21,196)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,204)(30,205)(31,206)(32,207)(33,208)(34,209)(35,210)(36,211)(37,212)(38,213)(39,214)(40,215)(41,216)(42,217)(43,218)(44,219)(45,220)(46,221)(47,222)(48,223)(49,224)(50,169)(51,170)(52,171)(53,172)(54,173)(55,174)(56,175)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)(112,119), (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), (1,176)(2,203)(3,174)(4,201)(5,172)(6,199)(7,170)(8,197)(9,224)(10,195)(11,222)(12,193)(13,220)(14,191)(15,218)(16,189)(17,216)(18,187)(19,214)(20,185)(21,212)(22,183)(23,210)(24,181)(25,208)(26,179)(27,206)(28,177)(29,204)(30,175)(31,202)(32,173)(33,200)(34,171)(35,198)(36,169)(37,196)(38,223)(39,194)(40,221)(41,192)(42,219)(43,190)(44,217)(45,188)(46,215)(47,186)(48,213)(49,184)(50,211)(51,182)(52,209)(53,180)(54,207)(55,178)(56,205)(57,146)(58,117)(59,144)(60,115)(61,142)(62,113)(63,140)(64,167)(65,138)(66,165)(67,136)(68,163)(69,134)(70,161)(71,132)(72,159)(73,130)(74,157)(75,128)(76,155)(77,126)(78,153)(79,124)(80,151)(81,122)(82,149)(83,120)(84,147)(85,118)(86,145)(87,116)(88,143)(89,114)(90,141)(91,168)(92,139)(93,166)(94,137)(95,164)(96,135)(97,162)(98,133)(99,160)(100,131)(101,158)(102,129)(103,156)(104,127)(105,154)(106,125)(107,152)(108,123)(109,150)(110,121)(111,148)(112,119) );
G=PermutationGroup([(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,159),(42,160),(43,161),(44,162),(45,163),(46,164),(47,165),(48,166),(49,167),(50,168),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,169),(106,170),(107,171),(108,172),(109,173),(110,174),(111,175),(112,176)], [(1,176),(2,177),(3,178),(4,179),(5,180),(6,181),(7,182),(8,183),(9,184),(10,185),(11,186),(12,187),(13,188),(14,189),(15,190),(16,191),(17,192),(18,193),(19,194),(20,195),(21,196),(22,197),(23,198),(24,199),(25,200),(26,201),(27,202),(28,203),(29,204),(30,205),(31,206),(32,207),(33,208),(34,209),(35,210),(36,211),(37,212),(38,213),(39,214),(40,215),(41,216),(42,217),(43,218),(44,219),(45,220),(46,221),(47,222),(48,223),(49,224),(50,169),(51,170),(52,171),(53,172),(54,173),(55,174),(56,175),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,144),(82,145),(83,146),(84,147),(85,148),(86,149),(87,150),(88,151),(89,152),(90,153),(91,154),(92,155),(93,156),(94,157),(95,158),(96,159),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,113),(107,114),(108,115),(109,116),(110,117),(111,118),(112,119)], [(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)], [(1,176),(2,203),(3,174),(4,201),(5,172),(6,199),(7,170),(8,197),(9,224),(10,195),(11,222),(12,193),(13,220),(14,191),(15,218),(16,189),(17,216),(18,187),(19,214),(20,185),(21,212),(22,183),(23,210),(24,181),(25,208),(26,179),(27,206),(28,177),(29,204),(30,175),(31,202),(32,173),(33,200),(34,171),(35,198),(36,169),(37,196),(38,223),(39,194),(40,221),(41,192),(42,219),(43,190),(44,217),(45,188),(46,215),(47,186),(48,213),(49,184),(50,211),(51,182),(52,209),(53,180),(54,207),(55,178),(56,205),(57,146),(58,117),(59,144),(60,115),(61,142),(62,113),(63,140),(64,167),(65,138),(66,165),(67,136),(68,163),(69,134),(70,161),(71,132),(72,159),(73,130),(74,157),(75,128),(76,155),(77,126),(78,153),(79,124),(80,151),(81,122),(82,149),(83,120),(84,147),(85,118),(86,145),(87,116),(88,143),(89,114),(90,141),(91,168),(92,139),(93,166),(94,137),(95,164),(96,135),(97,162),(98,133),(99,160),(100,131),(101,158),(102,129),(103,156),(104,127),(105,154),(106,125),(107,152),(108,123),(109,150),(110,121),(111,148),(112,119)])
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 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 105 |
| 0 | 0 | 8 | 104 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 103 | 103 |
| 0 | 0 | 89 | 10 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,70,8,0,0,105,104],[1,0,0,0,0,112,0,0,0,0,103,89,0,0,103,10] >;
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | SD16 | D14 | D14 | D28 | D28 | C56⋊C2 |
| kernel | C22×C56⋊C2 | C2×C56⋊C2 | C22×C56 | C22×Dic14 | C22×D28 | C2×C28 | C22×C14 | C22×C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 3 | 1 | 3 | 8 | 18 | 3 | 18 | 6 | 48 |
In GAP, Magma, Sage, TeX
C_2^2\times C_{56}\rtimes C_2 % in TeX
G:=Group("C2^2xC56:C2"); // GroupNames label
G:=SmallGroup(448,1192);
// by ID
G=gap.SmallGroup(448,1192);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,80,1684,102,18822]);
// 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^27>;
// generators/relations
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