metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊29D4, C22⋊1D56, C23.25D28, (C2×C14)⋊5D8, (C2×D56)⋊9C2, C7⋊4(C8⋊7D4), (C22×C8)⋊6D7, C28⋊7D4⋊2C2, C8⋊13(C7⋊D4), C56⋊1C4⋊14C2, C2.17(C2×D56), (C2×C4).68D28, C14.17(C2×D8), C2.D56⋊3C2, (C22×C56)⋊10C2, (C2×C8).308D14, (C2×C28).356D4, C28.413(C2×D4), C14.18(C4○D8), C4.112(C4○D28), C28.228(C4○D4), C14.71(C4⋊D4), C2.19(C28⋊7D4), (C2×C28).769C23, (C2×C56).380C22, (C2×D28).19C22, C22.132(C2×D28), (C22×C14).141D4, (C22×C4).431D14, C4⋊Dic7.24C22, C2.18(D56⋊7C2), (C22×C28).519C22, C4.106(C2×C7⋊D4), (C2×C14).159(C2×D4), (C2×C4).717(C22×D7), SmallGroup(448,649)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊29D4
G = < a,b,c | a56=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 900 in 134 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C56, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊7D4, D56, C4⋊Dic7, D14⋊C4, C2×C56, C2×C56, C2×D28, C2×C7⋊D4, C22×C28, C56⋊1C4, C2.D56, C2×D56, C28⋊7D4, C22×C56, C56⋊29D4
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C4⋊D4, C2×D8, C4○D8, D28, C7⋊D4, C22×D7, C8⋊7D4, D56, C2×D28, C4○D28, C2×C7⋊D4, C2×D56, D56⋊7C2, C28⋊7D4, C56⋊29D4
►On 224 points
Generators in S224
(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)
(1 156 82 197)(2 155 83 196)(3 154 84 195)(4 153 85 194)(5 152 86 193)(6 151 87 192)(7 150 88 191)(8 149 89 190)(9 148 90 189)(10 147 91 188)(11 146 92 187)(12 145 93 186)(13 144 94 185)(14 143 95 184)(15 142 96 183)(16 141 97 182)(17 140 98 181)(18 139 99 180)(19 138 100 179)(20 137 101 178)(21 136 102 177)(22 135 103 176)(23 134 104 175)(24 133 105 174)(25 132 106 173)(26 131 107 172)(27 130 108 171)(28 129 109 170)(29 128 110 169)(30 127 111 224)(31 126 112 223)(32 125 57 222)(33 124 58 221)(34 123 59 220)(35 122 60 219)(36 121 61 218)(37 120 62 217)(38 119 63 216)(39 118 64 215)(40 117 65 214)(41 116 66 213)(42 115 67 212)(43 114 68 211)(44 113 69 210)(45 168 70 209)(46 167 71 208)(47 166 72 207)(48 165 73 206)(49 164 74 205)(50 163 75 204)(51 162 76 203)(52 161 77 202)(53 160 78 201)(54 159 79 200)(55 158 80 199)(56 157 81 198)
(1 197)(2 196)(3 195)(4 194)(5 193)(6 192)(7 191)(8 190)(9 189)(10 188)(11 187)(12 186)(13 185)(14 184)(15 183)(16 182)(17 181)(18 180)(19 179)(20 178)(21 177)(22 176)(23 175)(24 174)(25 173)(26 172)(27 171)(28 170)(29 169)(30 224)(31 223)(32 222)(33 221)(34 220)(35 219)(36 218)(37 217)(38 216)(39 215)(40 214)(41 213)(42 212)(43 211)(44 210)(45 209)(46 208)(47 207)(48 206)(49 205)(50 204)(51 203)(52 202)(53 201)(54 200)(55 199)(56 198)(57 125)(58 124)(59 123)(60 122)(61 121)(62 120)(63 119)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 168)(71 167)(72 166)(73 165)(74 164)(75 163)(76 162)(77 161)(78 160)(79 159)(80 158)(81 157)(82 156)(83 155)(84 154)(85 153)(86 152)(87 151)(88 150)(89 149)(90 148)(91 147)(92 146)(93 145)(94 144)(95 143)(96 142)(97 141)(98 140)(99 139)(100 138)(101 137)(102 136)(103 135)(104 134)(105 133)(106 132)(107 131)(108 130)(109 129)(110 128)(111 127)(112 126)
G:=sub<Sym(224)| (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), (1,156,82,197)(2,155,83,196)(3,154,84,195)(4,153,85,194)(5,152,86,193)(6,151,87,192)(7,150,88,191)(8,149,89,190)(9,148,90,189)(10,147,91,188)(11,146,92,187)(12,145,93,186)(13,144,94,185)(14,143,95,184)(15,142,96,183)(16,141,97,182)(17,140,98,181)(18,139,99,180)(19,138,100,179)(20,137,101,178)(21,136,102,177)(22,135,103,176)(23,134,104,175)(24,133,105,174)(25,132,106,173)(26,131,107,172)(27,130,108,171)(28,129,109,170)(29,128,110,169)(30,127,111,224)(31,126,112,223)(32,125,57,222)(33,124,58,221)(34,123,59,220)(35,122,60,219)(36,121,61,218)(37,120,62,217)(38,119,63,216)(39,118,64,215)(40,117,65,214)(41,116,66,213)(42,115,67,212)(43,114,68,211)(44,113,69,210)(45,168,70,209)(46,167,71,208)(47,166,72,207)(48,165,73,206)(49,164,74,205)(50,163,75,204)(51,162,76,203)(52,161,77,202)(53,160,78,201)(54,159,79,200)(55,158,80,199)(56,157,81,198), (1,197)(2,196)(3,195)(4,194)(5,193)(6,192)(7,191)(8,190)(9,189)(10,188)(11,187)(12,186)(13,185)(14,184)(15,183)(16,182)(17,181)(18,180)(19,179)(20,178)(21,177)(22,176)(23,175)(24,174)(25,173)(26,172)(27,171)(28,170)(29,169)(30,224)(31,223)(32,222)(33,221)(34,220)(35,219)(36,218)(37,217)(38,216)(39,215)(40,214)(41,213)(42,212)(43,211)(44,210)(45,209)(46,208)(47,207)(48,206)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,198)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,119)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,160)(79,159)(80,158)(81,157)(82,156)(83,155)(84,154)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)>;
G:=Group( (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), (1,156,82,197)(2,155,83,196)(3,154,84,195)(4,153,85,194)(5,152,86,193)(6,151,87,192)(7,150,88,191)(8,149,89,190)(9,148,90,189)(10,147,91,188)(11,146,92,187)(12,145,93,186)(13,144,94,185)(14,143,95,184)(15,142,96,183)(16,141,97,182)(17,140,98,181)(18,139,99,180)(19,138,100,179)(20,137,101,178)(21,136,102,177)(22,135,103,176)(23,134,104,175)(24,133,105,174)(25,132,106,173)(26,131,107,172)(27,130,108,171)(28,129,109,170)(29,128,110,169)(30,127,111,224)(31,126,112,223)(32,125,57,222)(33,124,58,221)(34,123,59,220)(35,122,60,219)(36,121,61,218)(37,120,62,217)(38,119,63,216)(39,118,64,215)(40,117,65,214)(41,116,66,213)(42,115,67,212)(43,114,68,211)(44,113,69,210)(45,168,70,209)(46,167,71,208)(47,166,72,207)(48,165,73,206)(49,164,74,205)(50,163,75,204)(51,162,76,203)(52,161,77,202)(53,160,78,201)(54,159,79,200)(55,158,80,199)(56,157,81,198), (1,197)(2,196)(3,195)(4,194)(5,193)(6,192)(7,191)(8,190)(9,189)(10,188)(11,187)(12,186)(13,185)(14,184)(15,183)(16,182)(17,181)(18,180)(19,179)(20,178)(21,177)(22,176)(23,175)(24,174)(25,173)(26,172)(27,171)(28,170)(29,169)(30,224)(31,223)(32,222)(33,221)(34,220)(35,219)(36,218)(37,217)(38,216)(39,215)(40,214)(41,213)(42,212)(43,211)(44,210)(45,209)(46,208)(47,207)(48,206)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,198)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,119)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,160)(79,159)(80,158)(81,157)(82,156)(83,155)(84,154)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126) );
G=PermutationGroup([[(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)], [(1,156,82,197),(2,155,83,196),(3,154,84,195),(4,153,85,194),(5,152,86,193),(6,151,87,192),(7,150,88,191),(8,149,89,190),(9,148,90,189),(10,147,91,188),(11,146,92,187),(12,145,93,186),(13,144,94,185),(14,143,95,184),(15,142,96,183),(16,141,97,182),(17,140,98,181),(18,139,99,180),(19,138,100,179),(20,137,101,178),(21,136,102,177),(22,135,103,176),(23,134,104,175),(24,133,105,174),(25,132,106,173),(26,131,107,172),(27,130,108,171),(28,129,109,170),(29,128,110,169),(30,127,111,224),(31,126,112,223),(32,125,57,222),(33,124,58,221),(34,123,59,220),(35,122,60,219),(36,121,61,218),(37,120,62,217),(38,119,63,216),(39,118,64,215),(40,117,65,214),(41,116,66,213),(42,115,67,212),(43,114,68,211),(44,113,69,210),(45,168,70,209),(46,167,71,208),(47,166,72,207),(48,165,73,206),(49,164,74,205),(50,163,75,204),(51,162,76,203),(52,161,77,202),(53,160,78,201),(54,159,79,200),(55,158,80,199),(56,157,81,198)], [(1,197),(2,196),(3,195),(4,194),(5,193),(6,192),(7,191),(8,190),(9,189),(10,188),(11,187),(12,186),(13,185),(14,184),(15,183),(16,182),(17,181),(18,180),(19,179),(20,178),(21,177),(22,176),(23,175),(24,174),(25,173),(26,172),(27,171),(28,170),(29,169),(30,224),(31,223),(32,222),(33,221),(34,220),(35,219),(36,218),(37,217),(38,216),(39,215),(40,214),(41,213),(42,212),(43,211),(44,210),(45,209),(46,208),(47,207),(48,206),(49,205),(50,204),(51,203),(52,202),(53,201),(54,200),(55,199),(56,198),(57,125),(58,124),(59,123),(60,122),(61,121),(62,120),(63,119),(64,118),(65,117),(66,116),(67,115),(68,114),(69,113),(70,168),(71,167),(72,166),(73,165),(74,164),(75,163),(76,162),(77,161),(78,160),(79,159),(80,158),(81,157),(82,156),(83,155),(84,154),(85,153),(86,152),(87,151),(88,150),(89,149),(90,148),(91,147),(92,146),(93,145),(94,144),(95,143),(96,142),(97,141),(98,140),(99,139),(100,138),(101,137),(102,136),(103,135),(104,134),(105,133),(106,132),(107,131),(108,130),(109,129),(110,128),(111,127),(112,126)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D8 | D14 | D14 | C4○D8 | C7⋊D4 | D28 | D28 | C4○D28 | D56 | D56⋊7C2 |
| kernel | C56⋊29D4 | C56⋊1C4 | C2.D56 | C2×D56 | C28⋊7D4 | C22×C56 | C56 | C2×C28 | C22×C14 | C22×C8 | C28 | C2×C14 | C2×C8 | C22×C4 | C14 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 6 | 3 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C56⋊29D4 ►in GL4(𝔽113) generated by
| 104 | 0 | 0 | 0 |
| 1 | 25 | 0 | 0 |
| 0 | 0 | 44 | 14 |
| 0 | 0 | 0 | 18 |
| 93 | 111 | 0 | 0 |
| 31 | 20 | 0 | 0 |
| 0 | 0 | 72 | 85 |
| 0 | 0 | 60 | 41 |
| 93 | 111 | 0 | 0 |
| 30 | 20 | 0 | 0 |
| 0 | 0 | 72 | 85 |
| 0 | 0 | 60 | 41 |
G:=sub<GL(4,GF(113))| [104,1,0,0,0,25,0,0,0,0,44,0,0,0,14,18],[93,31,0,0,111,20,0,0,0,0,72,60,0,0,85,41],[93,30,0,0,111,20,0,0,0,0,72,60,0,0,85,41] >;
C56⋊29D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{29}D_4 % in TeX
G:=Group("C56:29D4"); // GroupNames label
G:=SmallGroup(448,649);
// by ID
G=gap.SmallGroup(448,649);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations