metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.29D4, D28.27D4, Dic14.27D4, C4.69(D4×D7), (C2×Q16)⋊10D7, (C14×Q16)⋊2C2, (C2×C8).97D14, C56.C4⋊4C2, C28.190(C2×D4), C7⋊4(D4.5D4), C8.29(C7⋊D4), (C2×Q8).68D14, C28.10D4⋊8C2, (C2×C56).34C22, C2.24(C28⋊2D4), (C2×C28).466C23, D28.2C4.1C2, C28.C23.2C2, C4○D28.48C22, (Q8×C14).95C22, C14.123(C4⋊D4), C4.Dic7.21C22, C22.22(D4⋊2D7), C4.87(C2×C7⋊D4), (C2×C4).128(C22×D7), (C2×C14).160(C4○D4), SmallGroup(448,726)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.29D4
G = < a,b,c | a56=c2=1, b4=a28, bab-1=a-1, cac=a41, cbc=b3 >
Subgroups: 420 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, C2×C8, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×Q8, D4.5D4, C8×D7, C8⋊D7, C4.Dic7, C4.Dic7, Q8⋊D7, C7⋊Q16, C2×C56, C7×Q16, C4○D28, Q8×C14, C56.C4, C28.10D4, D28.2C4, C28.C23, C14×Q16, C56.29D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.5D4, D4×D7, D4⋊2D7, C2×C7⋊D4, C28⋊2D4, 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)
(1 218 15 204 29 190 43 176)(2 217 16 203 30 189 44 175)(3 216 17 202 31 188 45 174)(4 215 18 201 32 187 46 173)(5 214 19 200 33 186 47 172)(6 213 20 199 34 185 48 171)(7 212 21 198 35 184 49 170)(8 211 22 197 36 183 50 169)(9 210 23 196 37 182 51 224)(10 209 24 195 38 181 52 223)(11 208 25 194 39 180 53 222)(12 207 26 193 40 179 54 221)(13 206 27 192 41 178 55 220)(14 205 28 191 42 177 56 219)(57 160 71 146 85 132 99 118)(58 159 72 145 86 131 100 117)(59 158 73 144 87 130 101 116)(60 157 74 143 88 129 102 115)(61 156 75 142 89 128 103 114)(62 155 76 141 90 127 104 113)(63 154 77 140 91 126 105 168)(64 153 78 139 92 125 106 167)(65 152 79 138 93 124 107 166)(66 151 80 137 94 123 108 165)(67 150 81 136 95 122 109 164)(68 149 82 135 96 121 110 163)(69 148 83 134 97 120 111 162)(70 147 84 133 98 119 112 161)
(1 115)(2 156)(3 141)(4 126)(5 167)(6 152)(7 137)(8 122)(9 163)(10 148)(11 133)(12 118)(13 159)(14 144)(15 129)(16 114)(17 155)(18 140)(19 125)(20 166)(21 151)(22 136)(23 121)(24 162)(25 147)(26 132)(27 117)(28 158)(29 143)(30 128)(31 113)(32 154)(33 139)(34 124)(35 165)(36 150)(37 135)(38 120)(39 161)(40 146)(41 131)(42 116)(43 157)(44 142)(45 127)(46 168)(47 153)(48 138)(49 123)(50 164)(51 149)(52 134)(53 119)(54 160)(55 145)(56 130)(57 193)(58 178)(59 219)(60 204)(61 189)(62 174)(63 215)(64 200)(65 185)(66 170)(67 211)(68 196)(69 181)(70 222)(71 207)(72 192)(73 177)(74 218)(75 203)(76 188)(77 173)(78 214)(79 199)(80 184)(81 169)(82 210)(83 195)(84 180)(85 221)(86 206)(87 191)(88 176)(89 217)(90 202)(91 187)(92 172)(93 213)(94 198)(95 183)(96 224)(97 209)(98 194)(99 179)(100 220)(101 205)(102 190)(103 175)(104 216)(105 201)(106 186)(107 171)(108 212)(109 197)(110 182)(111 223)(112 208)
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), (1,218,15,204,29,190,43,176)(2,217,16,203,30,189,44,175)(3,216,17,202,31,188,45,174)(4,215,18,201,32,187,46,173)(5,214,19,200,33,186,47,172)(6,213,20,199,34,185,48,171)(7,212,21,198,35,184,49,170)(8,211,22,197,36,183,50,169)(9,210,23,196,37,182,51,224)(10,209,24,195,38,181,52,223)(11,208,25,194,39,180,53,222)(12,207,26,193,40,179,54,221)(13,206,27,192,41,178,55,220)(14,205,28,191,42,177,56,219)(57,160,71,146,85,132,99,118)(58,159,72,145,86,131,100,117)(59,158,73,144,87,130,101,116)(60,157,74,143,88,129,102,115)(61,156,75,142,89,128,103,114)(62,155,76,141,90,127,104,113)(63,154,77,140,91,126,105,168)(64,153,78,139,92,125,106,167)(65,152,79,138,93,124,107,166)(66,151,80,137,94,123,108,165)(67,150,81,136,95,122,109,164)(68,149,82,135,96,121,110,163)(69,148,83,134,97,120,111,162)(70,147,84,133,98,119,112,161), (1,115)(2,156)(3,141)(4,126)(5,167)(6,152)(7,137)(8,122)(9,163)(10,148)(11,133)(12,118)(13,159)(14,144)(15,129)(16,114)(17,155)(18,140)(19,125)(20,166)(21,151)(22,136)(23,121)(24,162)(25,147)(26,132)(27,117)(28,158)(29,143)(30,128)(31,113)(32,154)(33,139)(34,124)(35,165)(36,150)(37,135)(38,120)(39,161)(40,146)(41,131)(42,116)(43,157)(44,142)(45,127)(46,168)(47,153)(48,138)(49,123)(50,164)(51,149)(52,134)(53,119)(54,160)(55,145)(56,130)(57,193)(58,178)(59,219)(60,204)(61,189)(62,174)(63,215)(64,200)(65,185)(66,170)(67,211)(68,196)(69,181)(70,222)(71,207)(72,192)(73,177)(74,218)(75,203)(76,188)(77,173)(78,214)(79,199)(80,184)(81,169)(82,210)(83,195)(84,180)(85,221)(86,206)(87,191)(88,176)(89,217)(90,202)(91,187)(92,172)(93,213)(94,198)(95,183)(96,224)(97,209)(98,194)(99,179)(100,220)(101,205)(102,190)(103,175)(104,216)(105,201)(106,186)(107,171)(108,212)(109,197)(110,182)(111,223)(112,208)>;
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), (1,218,15,204,29,190,43,176)(2,217,16,203,30,189,44,175)(3,216,17,202,31,188,45,174)(4,215,18,201,32,187,46,173)(5,214,19,200,33,186,47,172)(6,213,20,199,34,185,48,171)(7,212,21,198,35,184,49,170)(8,211,22,197,36,183,50,169)(9,210,23,196,37,182,51,224)(10,209,24,195,38,181,52,223)(11,208,25,194,39,180,53,222)(12,207,26,193,40,179,54,221)(13,206,27,192,41,178,55,220)(14,205,28,191,42,177,56,219)(57,160,71,146,85,132,99,118)(58,159,72,145,86,131,100,117)(59,158,73,144,87,130,101,116)(60,157,74,143,88,129,102,115)(61,156,75,142,89,128,103,114)(62,155,76,141,90,127,104,113)(63,154,77,140,91,126,105,168)(64,153,78,139,92,125,106,167)(65,152,79,138,93,124,107,166)(66,151,80,137,94,123,108,165)(67,150,81,136,95,122,109,164)(68,149,82,135,96,121,110,163)(69,148,83,134,97,120,111,162)(70,147,84,133,98,119,112,161), (1,115)(2,156)(3,141)(4,126)(5,167)(6,152)(7,137)(8,122)(9,163)(10,148)(11,133)(12,118)(13,159)(14,144)(15,129)(16,114)(17,155)(18,140)(19,125)(20,166)(21,151)(22,136)(23,121)(24,162)(25,147)(26,132)(27,117)(28,158)(29,143)(30,128)(31,113)(32,154)(33,139)(34,124)(35,165)(36,150)(37,135)(38,120)(39,161)(40,146)(41,131)(42,116)(43,157)(44,142)(45,127)(46,168)(47,153)(48,138)(49,123)(50,164)(51,149)(52,134)(53,119)(54,160)(55,145)(56,130)(57,193)(58,178)(59,219)(60,204)(61,189)(62,174)(63,215)(64,200)(65,185)(66,170)(67,211)(68,196)(69,181)(70,222)(71,207)(72,192)(73,177)(74,218)(75,203)(76,188)(77,173)(78,214)(79,199)(80,184)(81,169)(82,210)(83,195)(84,180)(85,221)(86,206)(87,191)(88,176)(89,217)(90,202)(91,187)(92,172)(93,213)(94,198)(95,183)(96,224)(97,209)(98,194)(99,179)(100,220)(101,205)(102,190)(103,175)(104,216)(105,201)(106,186)(107,171)(108,212)(109,197)(110,182)(111,223)(112,208) );
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)], [(1,218,15,204,29,190,43,176),(2,217,16,203,30,189,44,175),(3,216,17,202,31,188,45,174),(4,215,18,201,32,187,46,173),(5,214,19,200,33,186,47,172),(6,213,20,199,34,185,48,171),(7,212,21,198,35,184,49,170),(8,211,22,197,36,183,50,169),(9,210,23,196,37,182,51,224),(10,209,24,195,38,181,52,223),(11,208,25,194,39,180,53,222),(12,207,26,193,40,179,54,221),(13,206,27,192,41,178,55,220),(14,205,28,191,42,177,56,219),(57,160,71,146,85,132,99,118),(58,159,72,145,86,131,100,117),(59,158,73,144,87,130,101,116),(60,157,74,143,88,129,102,115),(61,156,75,142,89,128,103,114),(62,155,76,141,90,127,104,113),(63,154,77,140,91,126,105,168),(64,153,78,139,92,125,106,167),(65,152,79,138,93,124,107,166),(66,151,80,137,94,123,108,165),(67,150,81,136,95,122,109,164),(68,149,82,135,96,121,110,163),(69,148,83,134,97,120,111,162),(70,147,84,133,98,119,112,161)], [(1,115),(2,156),(3,141),(4,126),(5,167),(6,152),(7,137),(8,122),(9,163),(10,148),(11,133),(12,118),(13,159),(14,144),(15,129),(16,114),(17,155),(18,140),(19,125),(20,166),(21,151),(22,136),(23,121),(24,162),(25,147),(26,132),(27,117),(28,158),(29,143),(30,128),(31,113),(32,154),(33,139),(34,124),(35,165),(36,150),(37,135),(38,120),(39,161),(40,146),(41,131),(42,116),(43,157),(44,142),(45,127),(46,168),(47,153),(48,138),(49,123),(50,164),(51,149),(52,134),(53,119),(54,160),(55,145),(56,130),(57,193),(58,178),(59,219),(60,204),(61,189),(62,174),(63,215),(64,200),(65,185),(66,170),(67,211),(68,196),(69,181),(70,222),(71,207),(72,192),(73,177),(74,218),(75,203),(76,188),(77,173),(78,214),(79,199),(80,184),(81,169),(82,210),(83,195),(84,180),(85,221),(86,206),(87,191),(88,176),(89,217),(90,202),(91,187),(92,172),(93,213),(94,198),(95,183),(96,224),(97,209),(98,194),(99,179),(100,220),(101,205),(102,190),(103,175),(104,216),(105,201),(106,186),(107,171),(108,212),(109,197),(110,182),(111,223),(112,208)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 2 | 2 | 8 | 8 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 56 | 56 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D4.5D4 | D4×D7 | D4⋊2D7 | C56.29D4 |
| kernel | C56.29D4 | C56.C4 | C28.10D4 | D28.2C4 | C28.C23 | C14×Q16 | C56 | Dic14 | D28 | C2×Q16 | C2×C14 | C2×C8 | C2×Q8 | C8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56.29D4 ►in GL4(𝔽113) generated by
| 0 | 92 | 0 | 92 |
| 59 | 52 | 9 | 9 |
| 0 | 0 | 50 | 50 |
| 0 | 0 | 63 | 50 |
| 73 | 32 | 83 | 73 |
| 0 | 0 | 96 | 7 |
| 44 | 89 | 68 | 68 |
| 55 | 103 | 85 | 85 |
| 109 | 0 | 43 | 43 |
| 0 | 0 | 0 | 1 |
| 89 | 112 | 4 | 4 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,59,0,0,92,52,0,0,0,9,50,63,92,9,50,50],[73,0,44,55,32,0,89,103,83,96,68,85,73,7,68,85],[109,0,89,0,0,0,112,1,43,0,4,0,43,1,4,0] >;
C56.29D4 in GAP, Magma, Sage, TeX
C_{56}._{29}D_4 % in TeX
G:=Group("C56.29D4"); // GroupNames label
G:=SmallGroup(448,726);
// by ID
G=gap.SmallGroup(448,726);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,184,1123,297,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=c^2=1,b^4=a^28,b*a*b^-1=a^-1,c*a*c=a^41,c*b*c=b^3>;
// generators/relations