metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.31D4, C7⋊C8.9D4, C56⋊C4⋊5C2, C4.23(D4×D7), (C2×C8).89D14, C7⋊4(C8.2D4), C8.3(C7⋊D4), (C2×D4).70D14, C28.174(C2×D4), (C2×Q8).52D14, (C2×SD16).2D7, (C2×Dic28)⋊25C2, Dic7⋊Q8⋊4C2, (C2×Dic7).70D4, (C14×SD16).2C2, C22.264(D4×D7), C14.29(C4⋊1D4), C2.20(C28⋊D4), (C2×C56).114C22, (C2×C28).444C23, C28.17D4.6C2, (D4×C14).93C22, (Q8×C14).74C22, C2.28(SD16⋊D7), C14.48(C8.C22), (C4×Dic7).51C22, (C2×Dic14).125C22, C4.7(C2×C7⋊D4), (C2×C7⋊Q16)⋊17C2, (C2×D4.D7).9C2, (C2×C14).356(C2×D4), (C2×C7⋊C8).156C22, (C2×C4).533(C22×D7), SmallGroup(448,701)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.31D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=a13, cac-1=a-1, cbc-1=b-1 >
Subgroups: 580 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×SD16, C2×Q16, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C8.2D4, Dic28, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D4.D7, C7⋊Q16, C23.D7, C2×C56, C7×SD16, C2×Dic14, D4×C14, Q8×C14, C56⋊C4, C2×Dic28, C2×D4.D7, C28.17D4, C2×C7⋊Q16, Dic7⋊Q8, C14×SD16, C56.31D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C8.C22, C7⋊D4, C22×D7, C8.2D4, D4×D7, C2×C7⋊D4, SD16⋊D7, C28⋊D4, C56.31D4
►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)
(1 104 210 128)(2 61 211 141)(3 74 212 154)(4 87 213 167)(5 100 214 124)(6 57 215 137)(7 70 216 150)(8 83 217 163)(9 96 218 120)(10 109 219 133)(11 66 220 146)(12 79 221 159)(13 92 222 116)(14 105 223 129)(15 62 224 142)(16 75 169 155)(17 88 170 168)(18 101 171 125)(19 58 172 138)(20 71 173 151)(21 84 174 164)(22 97 175 121)(23 110 176 134)(24 67 177 147)(25 80 178 160)(26 93 179 117)(27 106 180 130)(28 63 181 143)(29 76 182 156)(30 89 183 113)(31 102 184 126)(32 59 185 139)(33 72 186 152)(34 85 187 165)(35 98 188 122)(36 111 189 135)(37 68 190 148)(38 81 191 161)(39 94 192 118)(40 107 193 131)(41 64 194 144)(42 77 195 157)(43 90 196 114)(44 103 197 127)(45 60 198 140)(46 73 199 153)(47 86 200 166)(48 99 201 123)(49 112 202 136)(50 69 203 149)(51 82 204 162)(52 95 205 119)(53 108 206 132)(54 65 207 145)(55 78 208 158)(56 91 209 115)
(1 135 29 163)(2 134 30 162)(3 133 31 161)(4 132 32 160)(5 131 33 159)(6 130 34 158)(7 129 35 157)(8 128 36 156)(9 127 37 155)(10 126 38 154)(11 125 39 153)(12 124 40 152)(13 123 41 151)(14 122 42 150)(15 121 43 149)(16 120 44 148)(17 119 45 147)(18 118 46 146)(19 117 47 145)(20 116 48 144)(21 115 49 143)(22 114 50 142)(23 113 51 141)(24 168 52 140)(25 167 53 139)(26 166 54 138)(27 165 55 137)(28 164 56 136)(57 180 85 208)(58 179 86 207)(59 178 87 206)(60 177 88 205)(61 176 89 204)(62 175 90 203)(63 174 91 202)(64 173 92 201)(65 172 93 200)(66 171 94 199)(67 170 95 198)(68 169 96 197)(69 224 97 196)(70 223 98 195)(71 222 99 194)(72 221 100 193)(73 220 101 192)(74 219 102 191)(75 218 103 190)(76 217 104 189)(77 216 105 188)(78 215 106 187)(79 214 107 186)(80 213 108 185)(81 212 109 184)(82 211 110 183)(83 210 111 182)(84 209 112 181)
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), (1,104,210,128)(2,61,211,141)(3,74,212,154)(4,87,213,167)(5,100,214,124)(6,57,215,137)(7,70,216,150)(8,83,217,163)(9,96,218,120)(10,109,219,133)(11,66,220,146)(12,79,221,159)(13,92,222,116)(14,105,223,129)(15,62,224,142)(16,75,169,155)(17,88,170,168)(18,101,171,125)(19,58,172,138)(20,71,173,151)(21,84,174,164)(22,97,175,121)(23,110,176,134)(24,67,177,147)(25,80,178,160)(26,93,179,117)(27,106,180,130)(28,63,181,143)(29,76,182,156)(30,89,183,113)(31,102,184,126)(32,59,185,139)(33,72,186,152)(34,85,187,165)(35,98,188,122)(36,111,189,135)(37,68,190,148)(38,81,191,161)(39,94,192,118)(40,107,193,131)(41,64,194,144)(42,77,195,157)(43,90,196,114)(44,103,197,127)(45,60,198,140)(46,73,199,153)(47,86,200,166)(48,99,201,123)(49,112,202,136)(50,69,203,149)(51,82,204,162)(52,95,205,119)(53,108,206,132)(54,65,207,145)(55,78,208,158)(56,91,209,115), (1,135,29,163)(2,134,30,162)(3,133,31,161)(4,132,32,160)(5,131,33,159)(6,130,34,158)(7,129,35,157)(8,128,36,156)(9,127,37,155)(10,126,38,154)(11,125,39,153)(12,124,40,152)(13,123,41,151)(14,122,42,150)(15,121,43,149)(16,120,44,148)(17,119,45,147)(18,118,46,146)(19,117,47,145)(20,116,48,144)(21,115,49,143)(22,114,50,142)(23,113,51,141)(24,168,52,140)(25,167,53,139)(26,166,54,138)(27,165,55,137)(28,164,56,136)(57,180,85,208)(58,179,86,207)(59,178,87,206)(60,177,88,205)(61,176,89,204)(62,175,90,203)(63,174,91,202)(64,173,92,201)(65,172,93,200)(66,171,94,199)(67,170,95,198)(68,169,96,197)(69,224,97,196)(70,223,98,195)(71,222,99,194)(72,221,100,193)(73,220,101,192)(74,219,102,191)(75,218,103,190)(76,217,104,189)(77,216,105,188)(78,215,106,187)(79,214,107,186)(80,213,108,185)(81,212,109,184)(82,211,110,183)(83,210,111,182)(84,209,112,181)>;
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), (1,104,210,128)(2,61,211,141)(3,74,212,154)(4,87,213,167)(5,100,214,124)(6,57,215,137)(7,70,216,150)(8,83,217,163)(9,96,218,120)(10,109,219,133)(11,66,220,146)(12,79,221,159)(13,92,222,116)(14,105,223,129)(15,62,224,142)(16,75,169,155)(17,88,170,168)(18,101,171,125)(19,58,172,138)(20,71,173,151)(21,84,174,164)(22,97,175,121)(23,110,176,134)(24,67,177,147)(25,80,178,160)(26,93,179,117)(27,106,180,130)(28,63,181,143)(29,76,182,156)(30,89,183,113)(31,102,184,126)(32,59,185,139)(33,72,186,152)(34,85,187,165)(35,98,188,122)(36,111,189,135)(37,68,190,148)(38,81,191,161)(39,94,192,118)(40,107,193,131)(41,64,194,144)(42,77,195,157)(43,90,196,114)(44,103,197,127)(45,60,198,140)(46,73,199,153)(47,86,200,166)(48,99,201,123)(49,112,202,136)(50,69,203,149)(51,82,204,162)(52,95,205,119)(53,108,206,132)(54,65,207,145)(55,78,208,158)(56,91,209,115), (1,135,29,163)(2,134,30,162)(3,133,31,161)(4,132,32,160)(5,131,33,159)(6,130,34,158)(7,129,35,157)(8,128,36,156)(9,127,37,155)(10,126,38,154)(11,125,39,153)(12,124,40,152)(13,123,41,151)(14,122,42,150)(15,121,43,149)(16,120,44,148)(17,119,45,147)(18,118,46,146)(19,117,47,145)(20,116,48,144)(21,115,49,143)(22,114,50,142)(23,113,51,141)(24,168,52,140)(25,167,53,139)(26,166,54,138)(27,165,55,137)(28,164,56,136)(57,180,85,208)(58,179,86,207)(59,178,87,206)(60,177,88,205)(61,176,89,204)(62,175,90,203)(63,174,91,202)(64,173,92,201)(65,172,93,200)(66,171,94,199)(67,170,95,198)(68,169,96,197)(69,224,97,196)(70,223,98,195)(71,222,99,194)(72,221,100,193)(73,220,101,192)(74,219,102,191)(75,218,103,190)(76,217,104,189)(77,216,105,188)(78,215,106,187)(79,214,107,186)(80,213,108,185)(81,212,109,184)(82,211,110,183)(83,210,111,182)(84,209,112,181) );
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)], [(1,104,210,128),(2,61,211,141),(3,74,212,154),(4,87,213,167),(5,100,214,124),(6,57,215,137),(7,70,216,150),(8,83,217,163),(9,96,218,120),(10,109,219,133),(11,66,220,146),(12,79,221,159),(13,92,222,116),(14,105,223,129),(15,62,224,142),(16,75,169,155),(17,88,170,168),(18,101,171,125),(19,58,172,138),(20,71,173,151),(21,84,174,164),(22,97,175,121),(23,110,176,134),(24,67,177,147),(25,80,178,160),(26,93,179,117),(27,106,180,130),(28,63,181,143),(29,76,182,156),(30,89,183,113),(31,102,184,126),(32,59,185,139),(33,72,186,152),(34,85,187,165),(35,98,188,122),(36,111,189,135),(37,68,190,148),(38,81,191,161),(39,94,192,118),(40,107,193,131),(41,64,194,144),(42,77,195,157),(43,90,196,114),(44,103,197,127),(45,60,198,140),(46,73,199,153),(47,86,200,166),(48,99,201,123),(49,112,202,136),(50,69,203,149),(51,82,204,162),(52,95,205,119),(53,108,206,132),(54,65,207,145),(55,78,208,158),(56,91,209,115)], [(1,135,29,163),(2,134,30,162),(3,133,31,161),(4,132,32,160),(5,131,33,159),(6,130,34,158),(7,129,35,157),(8,128,36,156),(9,127,37,155),(10,126,38,154),(11,125,39,153),(12,124,40,152),(13,123,41,151),(14,122,42,150),(15,121,43,149),(16,120,44,148),(17,119,45,147),(18,118,46,146),(19,117,47,145),(20,116,48,144),(21,115,49,143),(22,114,50,142),(23,113,51,141),(24,168,52,140),(25,167,53,139),(26,166,54,138),(27,165,55,137),(28,164,56,136),(57,180,85,208),(58,179,86,207),(59,178,87,206),(60,177,88,205),(61,176,89,204),(62,175,90,203),(63,174,91,202),(64,173,92,201),(65,172,93,200),(66,171,94,199),(67,170,95,198),(68,169,96,197),(69,224,97,196),(70,223,98,195),(71,222,99,194),(72,221,100,193),(73,220,101,192),(74,219,102,191),(75,218,103,190),(76,217,104,189),(77,216,105,188),(78,215,106,187),(79,214,107,186),(80,213,108,185),(81,212,109,184),(82,211,110,183),(83,210,111,182),(84,209,112,181)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 8 | 2 | 2 | 8 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C8.C22 | D4×D7 | D4×D7 | SD16⋊D7 |
| kernel | C56.31D4 | C56⋊C4 | C2×Dic28 | C2×D4.D7 | C28.17D4 | C2×C7⋊Q16 | Dic7⋊Q8 | C14×SD16 | C7⋊C8 | C56 | C2×Dic7 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56.31D4 ►in GL8(𝔽113)
| 0 | 0 | 63 | 49 | 0 | 0 | 0 | 0 |
| 0 | 30 | 111 | 2 | 0 | 0 | 0 | 0 |
| 112 | 83 | 1 | 112 | 0 | 0 | 0 | 0 |
| 33 | 87 | 31 | 82 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 93 | 106 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 91 |
| 0 | 0 | 0 | 0 | 10 | 60 | 20 | 7 |
| 0 | 0 | 0 | 0 | 112 | 11 | 111 | 22 |
| 0 | 111 | 43 | 24 | 0 | 0 | 0 | 0 |
| 0 | 89 | 23 | 5 | 0 | 0 | 0 | 0 |
| 54 | 24 | 45 | 54 | 0 | 0 | 0 | 0 |
| 68 | 68 | 39 | 92 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 | 18 | 8 |
| 0 | 0 | 0 | 0 | 36 | 104 | 72 | 95 |
| 0 | 0 | 0 | 0 | 104 | 109 | 104 | 109 |
| 0 | 0 | 0 | 0 | 77 | 9 | 77 | 9 |
| 89 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 108 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 59 | 89 | 68 | 59 | 0 | 0 | 0 | 0 |
| 21 | 45 | 4 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 97 |
| 0 | 0 | 0 | 0 | 0 | 0 | 92 | 1 |
| 0 | 0 | 0 | 0 | 56 | 105 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 57 | 0 | 0 |
G:=sub<GL(8,GF(113))| [0,0,112,33,0,0,0,0,0,30,83,87,0,0,0,0,63,111,1,31,0,0,0,0,49,2,112,82,0,0,0,0,0,0,0,0,0,0,10,112,0,0,0,0,0,0,60,11,0,0,0,0,93,2,20,111,0,0,0,0,106,91,7,22],[0,0,54,68,0,0,0,0,111,89,24,68,0,0,0,0,43,23,45,39,0,0,0,0,24,5,54,92,0,0,0,0,0,0,0,0,9,36,104,77,0,0,0,0,4,104,109,9,0,0,0,0,18,72,104,77,0,0,0,0,8,95,109,9],[89,108,59,21,0,0,0,0,2,24,89,45,0,0,0,0,0,0,68,4,0,0,0,0,0,0,59,45,0,0,0,0,0,0,0,0,0,0,56,46,0,0,0,0,0,0,105,57,0,0,0,0,112,92,0,0,0,0,0,0,97,1,0,0] >;
C56.31D4 in GAP, Magma, Sage, TeX
C_{56}._{31}D_4 % in TeX
G:=Group("C56.31D4"); // GroupNames label
G:=SmallGroup(448,701);
// by ID
G=gap.SmallGroup(448,701);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,232,1094,135,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations