metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊2D56, C28⋊1D8, D28⋊8D4, C42.34D14, C4⋊C8⋊3D7, (C2×D56)⋊7C2, C7⋊2(C4⋊D8), C14.7(C2×D8), C2.9(C2×D56), (C4×D28)⋊17C2, C28⋊4D4⋊7C2, C4.130(D4×D7), (C2×C8).21D14, C2.D56⋊7C2, (C2×C4).133D28, (C2×C28).122D4, C28.339(C2×D4), (C4×C28).69C22, (C2×C56).22C22, C28.328(C4○D4), C2.17(C8⋊D14), C14.38(C4⋊D4), C2.11(C4⋊D28), C14.14(C8⋊C22), (C2×C28).753C23, C4.44(Q8⋊2D7), (C2×D28).14C22, C22.116(C2×D28), C4⋊Dic7.273C22, (C7×C4⋊C8)⋊5C2, (C2×C14).136(C2×D4), (C2×C4).698(C22×D7), SmallGroup(448,377)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D56
G = < a,b,c | a4=b56=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 1124 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, 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, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, 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, C2×D28, C2×D28, C2.D56, C7×C4⋊C8, C4×D28, C28⋊4D4, C2×D56, C4⋊D56
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C4⋊D4, C2×D8, C8⋊C22, D28, C22×D7, C4⋊D8, D56, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, C2×D56, C8⋊D14, C4⋊D56
►On 224 points
Generators in S224
(1 69 138 204)(2 205 139 70)(3 71 140 206)(4 207 141 72)(5 73 142 208)(6 209 143 74)(7 75 144 210)(8 211 145 76)(9 77 146 212)(10 213 147 78)(11 79 148 214)(12 215 149 80)(13 81 150 216)(14 217 151 82)(15 83 152 218)(16 219 153 84)(17 85 154 220)(18 221 155 86)(19 87 156 222)(20 223 157 88)(21 89 158 224)(22 169 159 90)(23 91 160 170)(24 171 161 92)(25 93 162 172)(26 173 163 94)(27 95 164 174)(28 175 165 96)(29 97 166 176)(30 177 167 98)(31 99 168 178)(32 179 113 100)(33 101 114 180)(34 181 115 102)(35 103 116 182)(36 183 117 104)(37 105 118 184)(38 185 119 106)(39 107 120 186)(40 187 121 108)(41 109 122 188)(42 189 123 110)(43 111 124 190)(44 191 125 112)(45 57 126 192)(46 193 127 58)(47 59 128 194)(48 195 129 60)(49 61 130 196)(50 197 131 62)(51 63 132 198)(52 199 133 64)(53 65 134 200)(54 201 135 66)(55 67 136 202)(56 203 137 68)
(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)
(1 137)(2 136)(3 135)(4 134)(5 133)(6 132)(7 131)(8 130)(9 129)(10 128)(11 127)(12 126)(13 125)(14 124)(15 123)(16 122)(17 121)(18 120)(19 119)(20 118)(21 117)(22 116)(23 115)(24 114)(25 113)(26 168)(27 167)(28 166)(29 165)(30 164)(31 163)(32 162)(33 161)(34 160)(35 159)(36 158)(37 157)(38 156)(39 155)(40 154)(41 153)(42 152)(43 151)(44 150)(45 149)(46 148)(47 147)(48 146)(49 145)(50 144)(51 143)(52 142)(53 141)(54 140)(55 139)(56 138)(57 215)(58 214)(59 213)(60 212)(61 211)(62 210)(63 209)(64 208)(65 207)(66 206)(67 205)(68 204)(69 203)(70 202)(71 201)(72 200)(73 199)(74 198)(75 197)(76 196)(77 195)(78 194)(79 193)(80 192)(81 191)(82 190)(83 189)(84 188)(85 187)(86 186)(87 185)(88 184)(89 183)(90 182)(91 181)(92 180)(93 179)(94 178)(95 177)(96 176)(97 175)(98 174)(99 173)(100 172)(101 171)(102 170)(103 169)(104 224)(105 223)(106 222)(107 221)(108 220)(109 219)(110 218)(111 217)(112 216)
G:=sub<Sym(224)| (1,69,138,204)(2,205,139,70)(3,71,140,206)(4,207,141,72)(5,73,142,208)(6,209,143,74)(7,75,144,210)(8,211,145,76)(9,77,146,212)(10,213,147,78)(11,79,148,214)(12,215,149,80)(13,81,150,216)(14,217,151,82)(15,83,152,218)(16,219,153,84)(17,85,154,220)(18,221,155,86)(19,87,156,222)(20,223,157,88)(21,89,158,224)(22,169,159,90)(23,91,160,170)(24,171,161,92)(25,93,162,172)(26,173,163,94)(27,95,164,174)(28,175,165,96)(29,97,166,176)(30,177,167,98)(31,99,168,178)(32,179,113,100)(33,101,114,180)(34,181,115,102)(35,103,116,182)(36,183,117,104)(37,105,118,184)(38,185,119,106)(39,107,120,186)(40,187,121,108)(41,109,122,188)(42,189,123,110)(43,111,124,190)(44,191,125,112)(45,57,126,192)(46,193,127,58)(47,59,128,194)(48,195,129,60)(49,61,130,196)(50,197,131,62)(51,63,132,198)(52,199,133,64)(53,65,134,200)(54,201,135,66)(55,67,136,202)(56,203,137,68), (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), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,168)(27,167)(28,166)(29,165)(30,164)(31,163)(32,162)(33,161)(34,160)(35,159)(36,158)(37,157)(38,156)(39,155)(40,154)(41,153)(42,152)(43,151)(44,150)(45,149)(46,148)(47,147)(48,146)(49,145)(50,144)(51,143)(52,142)(53,141)(54,140)(55,139)(56,138)(57,215)(58,214)(59,213)(60,212)(61,211)(62,210)(63,209)(64,208)(65,207)(66,206)(67,205)(68,204)(69,203)(70,202)(71,201)(72,200)(73,199)(74,198)(75,197)(76,196)(77,195)(78,194)(79,193)(80,192)(81,191)(82,190)(83,189)(84,188)(85,187)(86,186)(87,185)(88,184)(89,183)(90,182)(91,181)(92,180)(93,179)(94,178)(95,177)(96,176)(97,175)(98,174)(99,173)(100,172)(101,171)(102,170)(103,169)(104,224)(105,223)(106,222)(107,221)(108,220)(109,219)(110,218)(111,217)(112,216)>;
G:=Group( (1,69,138,204)(2,205,139,70)(3,71,140,206)(4,207,141,72)(5,73,142,208)(6,209,143,74)(7,75,144,210)(8,211,145,76)(9,77,146,212)(10,213,147,78)(11,79,148,214)(12,215,149,80)(13,81,150,216)(14,217,151,82)(15,83,152,218)(16,219,153,84)(17,85,154,220)(18,221,155,86)(19,87,156,222)(20,223,157,88)(21,89,158,224)(22,169,159,90)(23,91,160,170)(24,171,161,92)(25,93,162,172)(26,173,163,94)(27,95,164,174)(28,175,165,96)(29,97,166,176)(30,177,167,98)(31,99,168,178)(32,179,113,100)(33,101,114,180)(34,181,115,102)(35,103,116,182)(36,183,117,104)(37,105,118,184)(38,185,119,106)(39,107,120,186)(40,187,121,108)(41,109,122,188)(42,189,123,110)(43,111,124,190)(44,191,125,112)(45,57,126,192)(46,193,127,58)(47,59,128,194)(48,195,129,60)(49,61,130,196)(50,197,131,62)(51,63,132,198)(52,199,133,64)(53,65,134,200)(54,201,135,66)(55,67,136,202)(56,203,137,68), (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), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,168)(27,167)(28,166)(29,165)(30,164)(31,163)(32,162)(33,161)(34,160)(35,159)(36,158)(37,157)(38,156)(39,155)(40,154)(41,153)(42,152)(43,151)(44,150)(45,149)(46,148)(47,147)(48,146)(49,145)(50,144)(51,143)(52,142)(53,141)(54,140)(55,139)(56,138)(57,215)(58,214)(59,213)(60,212)(61,211)(62,210)(63,209)(64,208)(65,207)(66,206)(67,205)(68,204)(69,203)(70,202)(71,201)(72,200)(73,199)(74,198)(75,197)(76,196)(77,195)(78,194)(79,193)(80,192)(81,191)(82,190)(83,189)(84,188)(85,187)(86,186)(87,185)(88,184)(89,183)(90,182)(91,181)(92,180)(93,179)(94,178)(95,177)(96,176)(97,175)(98,174)(99,173)(100,172)(101,171)(102,170)(103,169)(104,224)(105,223)(106,222)(107,221)(108,220)(109,219)(110,218)(111,217)(112,216) );
G=PermutationGroup([[(1,69,138,204),(2,205,139,70),(3,71,140,206),(4,207,141,72),(5,73,142,208),(6,209,143,74),(7,75,144,210),(8,211,145,76),(9,77,146,212),(10,213,147,78),(11,79,148,214),(12,215,149,80),(13,81,150,216),(14,217,151,82),(15,83,152,218),(16,219,153,84),(17,85,154,220),(18,221,155,86),(19,87,156,222),(20,223,157,88),(21,89,158,224),(22,169,159,90),(23,91,160,170),(24,171,161,92),(25,93,162,172),(26,173,163,94),(27,95,164,174),(28,175,165,96),(29,97,166,176),(30,177,167,98),(31,99,168,178),(32,179,113,100),(33,101,114,180),(34,181,115,102),(35,103,116,182),(36,183,117,104),(37,105,118,184),(38,185,119,106),(39,107,120,186),(40,187,121,108),(41,109,122,188),(42,189,123,110),(43,111,124,190),(44,191,125,112),(45,57,126,192),(46,193,127,58),(47,59,128,194),(48,195,129,60),(49,61,130,196),(50,197,131,62),(51,63,132,198),(52,199,133,64),(53,65,134,200),(54,201,135,66),(55,67,136,202),(56,203,137,68)], [(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)], [(1,137),(2,136),(3,135),(4,134),(5,133),(6,132),(7,131),(8,130),(9,129),(10,128),(11,127),(12,126),(13,125),(14,124),(15,123),(16,122),(17,121),(18,120),(19,119),(20,118),(21,117),(22,116),(23,115),(24,114),(25,113),(26,168),(27,167),(28,166),(29,165),(30,164),(31,163),(32,162),(33,161),(34,160),(35,159),(36,158),(37,157),(38,156),(39,155),(40,154),(41,153),(42,152),(43,151),(44,150),(45,149),(46,148),(47,147),(48,146),(49,145),(50,144),(51,143),(52,142),(53,141),(54,140),(55,139),(56,138),(57,215),(58,214),(59,213),(60,212),(61,211),(62,210),(63,209),(64,208),(65,207),(66,206),(67,205),(68,204),(69,203),(70,202),(71,201),(72,200),(73,199),(74,198),(75,197),(76,196),(77,195),(78,194),(79,193),(80,192),(81,191),(82,190),(83,189),(84,188),(85,187),(86,186),(87,185),(88,184),(89,183),(90,182),(91,181),(92,180),(93,179),(94,178),(95,177),(96,176),(97,175),(98,174),(99,173),(100,172),(101,171),(102,170),(103,169),(104,224),(105,223),(106,222),(107,221),(108,220),(109,219),(110,218),(111,217),(112,216)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | C4○D4 | D14 | D14 | D28 | D56 | C8⋊C22 | D4×D7 | Q8⋊2D7 | C8⋊D14 |
| kernel | C4⋊D56 | C2.D56 | C7×C4⋊C8 | C4×D28 | C28⋊4D4 | C2×D56 | D28 | C2×C28 | C4⋊C8 | C28 | C28 | C42 | C2×C8 | C2×C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of C4⋊D56 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 112 | 112 | 0 | 0 | 0 | 0 |
| 81 | 80 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 105 | 0 | 0 |
| 0 | 0 | 25 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 104 | 79 | 0 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 105 | 0 | 0 |
| 0 | 0 | 28 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 112 | 0 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,0,0,0,0,0,0,15],[112,81,0,0,0,0,112,80,0,0,0,0,0,0,15,25,0,0,0,0,105,47,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[104,9,0,0,0,0,79,9,0,0,0,0,0,0,15,28,0,0,0,0,105,98,0,0,0,0,0,0,0,112,0,0,0,0,112,0] >;
C4⋊D56 in GAP, Magma, Sage, TeX
C_4\rtimes D_{56} % in TeX
G:=Group("C4:D56"); // GroupNames label
G:=SmallGroup(448,377);
// by ID
G=gap.SmallGroup(448,377);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,226,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations