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