direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D56, C28⋊5D8, C42.259D14, C7⋊1(C4×D8), (C4×C8)⋊7D7, C8⋊10(C4×D7), C56⋊22(C2×C4), (C4×C56)⋊12C2, (C4×D28)⋊1C2, D28⋊8(C2×C4), C14.7(C4×D4), C2.1(C2×D56), C14.2(C2×D8), C56⋊1C4⋊28C2, C2.10(C4×D28), (C2×C4).61D28, (C2×D56).14C2, C14.3(C4○D8), (C2×C8).286D14, (C2×C28).351D4, C2.D56⋊43C2, C22.28(C2×D28), C28.217(C4○D4), C4.101(C4○D28), C2.2(D56⋊7C2), (C2×C56).346C22, (C2×C28).721C23, C28.102(C22×C4), (C4×C28).326C22, (C2×D28).188C22, C4⋊Dic7.263C22, C4.60(C2×C4×D7), (C2×C14).104(C2×D4), (C2×C4).664(C22×D7), SmallGroup(448,226)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D56
G = < a,b,c | a4=b56=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 868 in 134 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, Dic7, C28, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C56, C56, C4×D7, D28, D28, C2×Dic7, C2×C28, C22×D7, C4×D8, D56, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C56⋊1C4, C2.D56, C4×C56, C4×D28, C2×D56, C4×D56
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, D8, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×D8, C4○D8, C4×D7, D28, C22×D7, C4×D8, D56, C2×C4×D7, C2×D28, C4○D28, C4×D28, C2×D56, D56⋊7C2, C4×D56
►On 224 points
Generators in S224
(1 206 94 130)(2 207 95 131)(3 208 96 132)(4 209 97 133)(5 210 98 134)(6 211 99 135)(7 212 100 136)(8 213 101 137)(9 214 102 138)(10 215 103 139)(11 216 104 140)(12 217 105 141)(13 218 106 142)(14 219 107 143)(15 220 108 144)(16 221 109 145)(17 222 110 146)(18 223 111 147)(19 224 112 148)(20 169 57 149)(21 170 58 150)(22 171 59 151)(23 172 60 152)(24 173 61 153)(25 174 62 154)(26 175 63 155)(27 176 64 156)(28 177 65 157)(29 178 66 158)(30 179 67 159)(31 180 68 160)(32 181 69 161)(33 182 70 162)(34 183 71 163)(35 184 72 164)(36 185 73 165)(37 186 74 166)(38 187 75 167)(39 188 76 168)(40 189 77 113)(41 190 78 114)(42 191 79 115)(43 192 80 116)(44 193 81 117)(45 194 82 118)(46 195 83 119)(47 196 84 120)(48 197 85 121)(49 198 86 122)(50 199 87 123)(51 200 88 124)(52 201 89 125)(53 202 90 126)(54 203 91 127)(55 204 92 128)(56 205 93 129)
(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 93)(2 92)(3 91)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 84)(11 83)(12 82)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 68)(27 67)(28 66)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(113 222)(114 221)(115 220)(116 219)(117 218)(118 217)(119 216)(120 215)(121 214)(122 213)(123 212)(124 211)(125 210)(126 209)(127 208)(128 207)(129 206)(130 205)(131 204)(132 203)(133 202)(134 201)(135 200)(136 199)(137 198)(138 197)(139 196)(140 195)(141 194)(142 193)(143 192)(144 191)(145 190)(146 189)(147 188)(148 187)(149 186)(150 185)(151 184)(152 183)(153 182)(154 181)(155 180)(156 179)(157 178)(158 177)(159 176)(160 175)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 224)(168 223)
G:=sub<Sym(224)| (1,206,94,130)(2,207,95,131)(3,208,96,132)(4,209,97,133)(5,210,98,134)(6,211,99,135)(7,212,100,136)(8,213,101,137)(9,214,102,138)(10,215,103,139)(11,216,104,140)(12,217,105,141)(13,218,106,142)(14,219,107,143)(15,220,108,144)(16,221,109,145)(17,222,110,146)(18,223,111,147)(19,224,112,148)(20,169,57,149)(21,170,58,150)(22,171,59,151)(23,172,60,152)(24,173,61,153)(25,174,62,154)(26,175,63,155)(27,176,64,156)(28,177,65,157)(29,178,66,158)(30,179,67,159)(31,180,68,160)(32,181,69,161)(33,182,70,162)(34,183,71,163)(35,184,72,164)(36,185,73,165)(37,186,74,166)(38,187,75,167)(39,188,76,168)(40,189,77,113)(41,190,78,114)(42,191,79,115)(43,192,80,116)(44,193,81,117)(45,194,82,118)(46,195,83,119)(47,196,84,120)(48,197,85,121)(49,198,86,122)(50,199,87,123)(51,200,88,124)(52,201,89,125)(53,202,90,126)(54,203,91,127)(55,204,92,128)(56,205,93,129), (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), (1,93)(2,92)(3,91)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,66)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(113,222)(114,221)(115,220)(116,219)(117,218)(118,217)(119,216)(120,215)(121,214)(122,213)(123,212)(124,211)(125,210)(126,209)(127,208)(128,207)(129,206)(130,205)(131,204)(132,203)(133,202)(134,201)(135,200)(136,199)(137,198)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,224)(168,223)>;
G:=Group( (1,206,94,130)(2,207,95,131)(3,208,96,132)(4,209,97,133)(5,210,98,134)(6,211,99,135)(7,212,100,136)(8,213,101,137)(9,214,102,138)(10,215,103,139)(11,216,104,140)(12,217,105,141)(13,218,106,142)(14,219,107,143)(15,220,108,144)(16,221,109,145)(17,222,110,146)(18,223,111,147)(19,224,112,148)(20,169,57,149)(21,170,58,150)(22,171,59,151)(23,172,60,152)(24,173,61,153)(25,174,62,154)(26,175,63,155)(27,176,64,156)(28,177,65,157)(29,178,66,158)(30,179,67,159)(31,180,68,160)(32,181,69,161)(33,182,70,162)(34,183,71,163)(35,184,72,164)(36,185,73,165)(37,186,74,166)(38,187,75,167)(39,188,76,168)(40,189,77,113)(41,190,78,114)(42,191,79,115)(43,192,80,116)(44,193,81,117)(45,194,82,118)(46,195,83,119)(47,196,84,120)(48,197,85,121)(49,198,86,122)(50,199,87,123)(51,200,88,124)(52,201,89,125)(53,202,90,126)(54,203,91,127)(55,204,92,128)(56,205,93,129), (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), (1,93)(2,92)(3,91)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,66)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(113,222)(114,221)(115,220)(116,219)(117,218)(118,217)(119,216)(120,215)(121,214)(122,213)(123,212)(124,211)(125,210)(126,209)(127,208)(128,207)(129,206)(130,205)(131,204)(132,203)(133,202)(134,201)(135,200)(136,199)(137,198)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,224)(168,223) );
G=PermutationGroup([[(1,206,94,130),(2,207,95,131),(3,208,96,132),(4,209,97,133),(5,210,98,134),(6,211,99,135),(7,212,100,136),(8,213,101,137),(9,214,102,138),(10,215,103,139),(11,216,104,140),(12,217,105,141),(13,218,106,142),(14,219,107,143),(15,220,108,144),(16,221,109,145),(17,222,110,146),(18,223,111,147),(19,224,112,148),(20,169,57,149),(21,170,58,150),(22,171,59,151),(23,172,60,152),(24,173,61,153),(25,174,62,154),(26,175,63,155),(27,176,64,156),(28,177,65,157),(29,178,66,158),(30,179,67,159),(31,180,68,160),(32,181,69,161),(33,182,70,162),(34,183,71,163),(35,184,72,164),(36,185,73,165),(37,186,74,166),(38,187,75,167),(39,188,76,168),(40,189,77,113),(41,190,78,114),(42,191,79,115),(43,192,80,116),(44,193,81,117),(45,194,82,118),(46,195,83,119),(47,196,84,120),(48,197,85,121),(49,198,86,122),(50,199,87,123),(51,200,88,124),(52,201,89,125),(53,202,90,126),(54,203,91,127),(55,204,92,128),(56,205,93,129)], [(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)], [(1,93),(2,92),(3,91),(4,90),(5,89),(6,88),(7,87),(8,86),(9,85),(10,84),(11,83),(12,82),(13,81),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,73),(22,72),(23,71),(24,70),(25,69),(26,68),(27,67),(28,66),(29,65),(30,64),(31,63),(32,62),(33,61),(34,60),(35,59),(36,58),(37,57),(38,112),(39,111),(40,110),(41,109),(42,108),(43,107),(44,106),(45,105),(46,104),(47,103),(48,102),(49,101),(50,100),(51,99),(52,98),(53,97),(54,96),(55,95),(56,94),(113,222),(114,221),(115,220),(116,219),(117,218),(118,217),(119,216),(120,215),(121,214),(122,213),(123,212),(124,211),(125,210),(126,209),(127,208),(128,207),(129,206),(130,205),(131,204),(132,203),(133,202),(134,201),(135,200),(136,199),(137,198),(138,197),(139,196),(140,195),(141,194),(142,193),(143,192),(144,191),(145,190),(146,189),(147,188),(148,187),(149,186),(150,185),(151,184),(152,183),(153,182),(154,181),(155,180),(156,179),(157,178),(158,177),(159,176),(160,175),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,224),(168,223)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 28 | 28 | 1 | 1 | 1 | 1 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | D8 | C4○D4 | D14 | D14 | C4○D8 | C4×D7 | D28 | D56 | C4○D28 | D56⋊7C2 |
| kernel | C4×D56 | C56⋊1C4 | C2.D56 | C4×C56 | C4×D28 | C2×D56 | D56 | C2×C28 | C4×C8 | C28 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 12 | 24 | 12 | 24 |
Matrix representation of C4×D56 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 33 | 106 |
| 0 | 7 | 91 |
| 1 | 0 | 0 |
| 0 | 33 | 106 |
| 0 | 107 | 80 |
G:=sub<GL(3,GF(113))| [98,0,0,0,1,0,0,0,1],[1,0,0,0,33,7,0,106,91],[1,0,0,0,33,107,0,106,80] >;
C4×D56 in GAP, Magma, Sage, TeX
C_4\times D_{56} % in TeX
G:=Group("C4xD56"); // GroupNames label
G:=SmallGroup(448,226);
// by ID
G=gap.SmallGroup(448,226);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,58,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations