metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊3D4, C23.18D28, C8⋊2(C7⋊D4), C7⋊4(C8⋊2D4), C8⋊Dic7⋊4C2, (C2×D56)⋊12C2, (C2×C4).52D28, (C2×C8).78D14, C28⋊7D4⋊41C2, C28.421(C2×D4), (C2×C28).298D4, (C2×M4(2))⋊2D7, C2.D56⋊42C2, (C14×M4(2))⋊2C2, (C2×C56).64C22, C4.115(C4○D28), C28.231(C4○D4), C2.22(C28⋊7D4), C14.74(C4⋊D4), C2.23(C8⋊D14), C14.23(C8⋊C22), (C2×C28).776C23, (C2×D28).21C22, (C22×C14).104D4, (C22×C4).143D14, C22.135(C2×D28), C4⋊Dic7.27C22, (C22×C28).305C22, C4.114(C2×C7⋊D4), (C2×C14).166(C2×D4), (C2×C4).725(C22×D7), SmallGroup(448,669)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊3D4
G = < a,b,c | a56=b4=c2=1, bab-1=a27, cac=a-1, cbc=b-1 >
Subgroups: 900 in 130 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C56, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊2D4, D56, C4⋊Dic7, D14⋊C4, C2×C56, C7×M4(2), C2×D28, C2×C7⋊D4, C22×C28, C8⋊Dic7, C2.D56, C2×D56, C28⋊7D4, C14×M4(2), C56⋊3D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, D28, C7⋊D4, C22×D7, C8⋊2D4, C2×D28, C4○D28, C2×C7⋊D4, C8⋊D14, C28⋊7D4, C56⋊3D4
►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 96 159 170)(2 67 160 197)(3 94 161 224)(4 65 162 195)(5 92 163 222)(6 63 164 193)(7 90 165 220)(8 61 166 191)(9 88 167 218)(10 59 168 189)(11 86 113 216)(12 57 114 187)(13 84 115 214)(14 111 116 185)(15 82 117 212)(16 109 118 183)(17 80 119 210)(18 107 120 181)(19 78 121 208)(20 105 122 179)(21 76 123 206)(22 103 124 177)(23 74 125 204)(24 101 126 175)(25 72 127 202)(26 99 128 173)(27 70 129 200)(28 97 130 171)(29 68 131 198)(30 95 132 169)(31 66 133 196)(32 93 134 223)(33 64 135 194)(34 91 136 221)(35 62 137 192)(36 89 138 219)(37 60 139 190)(38 87 140 217)(39 58 141 188)(40 85 142 215)(41 112 143 186)(42 83 144 213)(43 110 145 184)(44 81 146 211)(45 108 147 182)(46 79 148 209)(47 106 149 180)(48 77 150 207)(49 104 151 178)(50 75 152 205)(51 102 153 176)(52 73 154 203)(53 100 155 174)(54 71 156 201)(55 98 157 172)(56 69 158 199)
(1 170)(2 169)(3 224)(4 223)(5 222)(6 221)(7 220)(8 219)(9 218)(10 217)(11 216)(12 215)(13 214)(14 213)(15 212)(16 211)(17 210)(18 209)(19 208)(20 207)(21 206)(22 205)(23 204)(24 203)(25 202)(26 201)(27 200)(28 199)(29 198)(30 197)(31 196)(32 195)(33 194)(34 193)(35 192)(36 191)(37 190)(38 189)(39 188)(40 187)(41 186)(42 185)(43 184)(44 183)(45 182)(46 181)(47 180)(48 179)(49 178)(50 177)(51 176)(52 175)(53 174)(54 173)(55 172)(56 171)(57 142)(58 141)(59 140)(60 139)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)(73 126)(74 125)(75 124)(76 123)(77 122)(78 121)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 168)(88 167)(89 166)(90 165)(91 164)(92 163)(93 162)(94 161)(95 160)(96 159)(97 158)(98 157)(99 156)(100 155)(101 154)(102 153)(103 152)(104 151)(105 150)(106 149)(107 148)(108 147)(109 146)(110 145)(111 144)(112 143)
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), (1,96,159,170)(2,67,160,197)(3,94,161,224)(4,65,162,195)(5,92,163,222)(6,63,164,193)(7,90,165,220)(8,61,166,191)(9,88,167,218)(10,59,168,189)(11,86,113,216)(12,57,114,187)(13,84,115,214)(14,111,116,185)(15,82,117,212)(16,109,118,183)(17,80,119,210)(18,107,120,181)(19,78,121,208)(20,105,122,179)(21,76,123,206)(22,103,124,177)(23,74,125,204)(24,101,126,175)(25,72,127,202)(26,99,128,173)(27,70,129,200)(28,97,130,171)(29,68,131,198)(30,95,132,169)(31,66,133,196)(32,93,134,223)(33,64,135,194)(34,91,136,221)(35,62,137,192)(36,89,138,219)(37,60,139,190)(38,87,140,217)(39,58,141,188)(40,85,142,215)(41,112,143,186)(42,83,144,213)(43,110,145,184)(44,81,146,211)(45,108,147,182)(46,79,148,209)(47,106,149,180)(48,77,150,207)(49,104,151,178)(50,75,152,205)(51,102,153,176)(52,73,154,203)(53,100,155,174)(54,71,156,201)(55,98,157,172)(56,69,158,199), (1,170)(2,169)(3,224)(4,223)(5,222)(6,221)(7,220)(8,219)(9,218)(10,217)(11,216)(12,215)(13,214)(14,213)(15,212)(16,211)(17,210)(18,209)(19,208)(20,207)(21,206)(22,205)(23,204)(24,203)(25,202)(26,201)(27,200)(28,199)(29,198)(30,197)(31,196)(32,195)(33,194)(34,193)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,186)(42,185)(43,184)(44,183)(45,182)(46,181)(47,180)(48,179)(49,178)(50,177)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,168)(88,167)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)>;
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), (1,96,159,170)(2,67,160,197)(3,94,161,224)(4,65,162,195)(5,92,163,222)(6,63,164,193)(7,90,165,220)(8,61,166,191)(9,88,167,218)(10,59,168,189)(11,86,113,216)(12,57,114,187)(13,84,115,214)(14,111,116,185)(15,82,117,212)(16,109,118,183)(17,80,119,210)(18,107,120,181)(19,78,121,208)(20,105,122,179)(21,76,123,206)(22,103,124,177)(23,74,125,204)(24,101,126,175)(25,72,127,202)(26,99,128,173)(27,70,129,200)(28,97,130,171)(29,68,131,198)(30,95,132,169)(31,66,133,196)(32,93,134,223)(33,64,135,194)(34,91,136,221)(35,62,137,192)(36,89,138,219)(37,60,139,190)(38,87,140,217)(39,58,141,188)(40,85,142,215)(41,112,143,186)(42,83,144,213)(43,110,145,184)(44,81,146,211)(45,108,147,182)(46,79,148,209)(47,106,149,180)(48,77,150,207)(49,104,151,178)(50,75,152,205)(51,102,153,176)(52,73,154,203)(53,100,155,174)(54,71,156,201)(55,98,157,172)(56,69,158,199), (1,170)(2,169)(3,224)(4,223)(5,222)(6,221)(7,220)(8,219)(9,218)(10,217)(11,216)(12,215)(13,214)(14,213)(15,212)(16,211)(17,210)(18,209)(19,208)(20,207)(21,206)(22,205)(23,204)(24,203)(25,202)(26,201)(27,200)(28,199)(29,198)(30,197)(31,196)(32,195)(33,194)(34,193)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,186)(42,185)(43,184)(44,183)(45,182)(46,181)(47,180)(48,179)(49,178)(50,177)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,168)(88,167)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143) );
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)], [(1,96,159,170),(2,67,160,197),(3,94,161,224),(4,65,162,195),(5,92,163,222),(6,63,164,193),(7,90,165,220),(8,61,166,191),(9,88,167,218),(10,59,168,189),(11,86,113,216),(12,57,114,187),(13,84,115,214),(14,111,116,185),(15,82,117,212),(16,109,118,183),(17,80,119,210),(18,107,120,181),(19,78,121,208),(20,105,122,179),(21,76,123,206),(22,103,124,177),(23,74,125,204),(24,101,126,175),(25,72,127,202),(26,99,128,173),(27,70,129,200),(28,97,130,171),(29,68,131,198),(30,95,132,169),(31,66,133,196),(32,93,134,223),(33,64,135,194),(34,91,136,221),(35,62,137,192),(36,89,138,219),(37,60,139,190),(38,87,140,217),(39,58,141,188),(40,85,142,215),(41,112,143,186),(42,83,144,213),(43,110,145,184),(44,81,146,211),(45,108,147,182),(46,79,148,209),(47,106,149,180),(48,77,150,207),(49,104,151,178),(50,75,152,205),(51,102,153,176),(52,73,154,203),(53,100,155,174),(54,71,156,201),(55,98,157,172),(56,69,158,199)], [(1,170),(2,169),(3,224),(4,223),(5,222),(6,221),(7,220),(8,219),(9,218),(10,217),(11,216),(12,215),(13,214),(14,213),(15,212),(16,211),(17,210),(18,209),(19,208),(20,207),(21,206),(22,205),(23,204),(24,203),(25,202),(26,201),(27,200),(28,199),(29,198),(30,197),(31,196),(32,195),(33,194),(34,193),(35,192),(36,191),(37,190),(38,189),(39,188),(40,187),(41,186),(42,185),(43,184),(44,183),(45,182),(46,181),(47,180),(48,179),(49,178),(50,177),(51,176),(52,175),(53,174),(54,173),(55,172),(56,171),(57,142),(58,141),(59,140),(60,139),(61,138),(62,137),(63,136),(64,135),(65,134),(66,133),(67,132),(68,131),(69,130),(70,129),(71,128),(72,127),(73,126),(74,125),(75,124),(76,123),(77,122),(78,121),(79,120),(80,119),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,168),(88,167),(89,166),(90,165),(91,164),(92,163),(93,162),(94,161),(95,160),(96,159),(97,158),(98,157),(99,156),(100,155),(101,154),(102,153),(103,152),(104,151),(105,150),(106,149),(107,148),(108,147),(109,146),(110,145),(111,144),(112,143)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 56 | 56 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D28 | D28 | C4○D28 | C8⋊C22 | C8⋊D14 |
| kernel | C56⋊3D4 | C8⋊Dic7 | C2.D56 | C2×D56 | C28⋊7D4 | C14×M4(2) | C56 | C2×C28 | C22×C14 | C2×M4(2) | C28 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C56⋊3D4 ►in GL6(𝔽113)
| 98 | 0 | 0 | 0 | 0 | 0 |
| 44 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 49 | 11 | 26 | 69 |
| 0 | 0 | 40 | 9 | 0 | 2 |
| 0 | 0 | 42 | 110 | 88 | 61 |
| 0 | 0 | 47 | 23 | 45 | 80 |
| 18 | 2 | 0 | 0 | 0 | 0 |
| 7 | 95 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 1 | 22 | 100 |
| 0 | 0 | 38 | 102 | 105 | 64 |
| 0 | 0 | 6 | 16 | 104 | 50 |
| 0 | 0 | 60 | 56 | 48 | 85 |
| 18 | 2 | 0 | 0 | 0 | 0 |
| 8 | 95 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 1 | 22 | 100 |
| 0 | 0 | 38 | 102 | 105 | 64 |
| 0 | 0 | 47 | 72 | 104 | 50 |
| 0 | 0 | 51 | 3 | 48 | 85 |
G:=sub<GL(6,GF(113))| [98,44,0,0,0,0,0,15,0,0,0,0,0,0,49,40,42,47,0,0,11,9,110,23,0,0,26,0,88,45,0,0,69,2,61,80],[18,7,0,0,0,0,2,95,0,0,0,0,0,0,48,38,6,60,0,0,1,102,16,56,0,0,22,105,104,48,0,0,100,64,50,85],[18,8,0,0,0,0,2,95,0,0,0,0,0,0,48,38,47,51,0,0,1,102,72,3,0,0,22,105,104,48,0,0,100,64,50,85] >;
C56⋊3D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_3D_4 % in TeX
G:=Group("C56:3D4"); // GroupNames label
G:=SmallGroup(448,669);
// by ID
G=gap.SmallGroup(448,669);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,387,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^27,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations