metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.18D4, C8.20D28, D28.20D4, Dic14.20D4, M4(2).9D14, C4.56(C2×D28), (C2×C8).70D14, C8.C4⋊5D7, C4.135(D4×D7), C28.136(C2×D4), C7⋊2(D4.5D4), (C2×Dic28)⋊21C2, C28.47D4⋊3C2, C8.D14.2C2, C14.49(C4⋊D4), C2.22(C28⋊1D4), (C2×C56).102C22, (C2×C28).312C23, D28.2C4.2C2, C4○D28.39C22, C22.6(Q8⋊2D7), (C7×M4(2)).6C22, C4.Dic7.37C22, (C2×Dic14).94C22, (C7×C8.C4)⋊6C2, (C2×C14).3(C4○D4), (C2×C4).113(C22×D7), SmallGroup(448,430)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.18D4
G = < a,b,c | a56=1, b4=c2=a28, bab-1=a15, cac-1=a-1, cbc-1=b3 >
Subgroups: 540 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C7, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], D7, C14, C14, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, Dic7 [×3], C28 [×2], D14, C2×C14, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], C7⋊C8, C56 [×2], C56 [×2], Dic14, Dic14 [×4], C4×D7, D28, C2×Dic7 [×2], C7⋊D4, C2×C28, D4.5D4, C8×D7, C8⋊D7, C56⋊C2 [×2], Dic28 [×4], C4.Dic7, C2×C56, C7×M4(2) [×2], C2×Dic14 [×2], C4○D28, C28.47D4 [×2], C7×C8.C4, D28.2C4, C2×Dic28, C8.D14 [×2], C56.18D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D7, C2×D4 [×2], C4○D4, D14 [×3], C4⋊D4, D28 [×2], C22×D7, D4.5D4, C2×D28, D4×D7, Q8⋊2D7, C28⋊1D4, C56.18D4
►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)
(1 198 15 184 29 170 43 212)(2 213 16 199 30 185 44 171)(3 172 17 214 31 200 45 186)(4 187 18 173 32 215 46 201)(5 202 19 188 33 174 47 216)(6 217 20 203 34 189 48 175)(7 176 21 218 35 204 49 190)(8 191 22 177 36 219 50 205)(9 206 23 192 37 178 51 220)(10 221 24 207 38 193 52 179)(11 180 25 222 39 208 53 194)(12 195 26 181 40 223 54 209)(13 210 27 196 41 182 55 224)(14 169 28 211 42 197 56 183)(57 165 99 123 85 137 71 151)(58 124 100 138 86 152 72 166)(59 139 101 153 87 167 73 125)(60 154 102 168 88 126 74 140)(61 113 103 127 89 141 75 155)(62 128 104 142 90 156 76 114)(63 143 105 157 91 115 77 129)(64 158 106 116 92 130 78 144)(65 117 107 131 93 145 79 159)(66 132 108 146 94 160 80 118)(67 147 109 161 95 119 81 133)(68 162 110 120 96 134 82 148)(69 121 111 135 97 149 83 163)(70 136 112 150 98 164 84 122)
(1 115 29 143)(2 114 30 142)(3 113 31 141)(4 168 32 140)(5 167 33 139)(6 166 34 138)(7 165 35 137)(8 164 36 136)(9 163 37 135)(10 162 38 134)(11 161 39 133)(12 160 40 132)(13 159 41 131)(14 158 42 130)(15 157 43 129)(16 156 44 128)(17 155 45 127)(18 154 46 126)(19 153 47 125)(20 152 48 124)(21 151 49 123)(22 150 50 122)(23 149 51 121)(24 148 52 120)(25 147 53 119)(26 146 54 118)(27 145 55 117)(28 144 56 116)(57 176 85 204)(58 175 86 203)(59 174 87 202)(60 173 88 201)(61 172 89 200)(62 171 90 199)(63 170 91 198)(64 169 92 197)(65 224 93 196)(66 223 94 195)(67 222 95 194)(68 221 96 193)(69 220 97 192)(70 219 98 191)(71 218 99 190)(72 217 100 189)(73 216 101 188)(74 215 102 187)(75 214 103 186)(76 213 104 185)(77 212 105 184)(78 211 106 183)(79 210 107 182)(80 209 108 181)(81 208 109 180)(82 207 110 179)(83 206 111 178)(84 205 112 177)
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), (1,198,15,184,29,170,43,212)(2,213,16,199,30,185,44,171)(3,172,17,214,31,200,45,186)(4,187,18,173,32,215,46,201)(5,202,19,188,33,174,47,216)(6,217,20,203,34,189,48,175)(7,176,21,218,35,204,49,190)(8,191,22,177,36,219,50,205)(9,206,23,192,37,178,51,220)(10,221,24,207,38,193,52,179)(11,180,25,222,39,208,53,194)(12,195,26,181,40,223,54,209)(13,210,27,196,41,182,55,224)(14,169,28,211,42,197,56,183)(57,165,99,123,85,137,71,151)(58,124,100,138,86,152,72,166)(59,139,101,153,87,167,73,125)(60,154,102,168,88,126,74,140)(61,113,103,127,89,141,75,155)(62,128,104,142,90,156,76,114)(63,143,105,157,91,115,77,129)(64,158,106,116,92,130,78,144)(65,117,107,131,93,145,79,159)(66,132,108,146,94,160,80,118)(67,147,109,161,95,119,81,133)(68,162,110,120,96,134,82,148)(69,121,111,135,97,149,83,163)(70,136,112,150,98,164,84,122), (1,115,29,143)(2,114,30,142)(3,113,31,141)(4,168,32,140)(5,167,33,139)(6,166,34,138)(7,165,35,137)(8,164,36,136)(9,163,37,135)(10,162,38,134)(11,161,39,133)(12,160,40,132)(13,159,41,131)(14,158,42,130)(15,157,43,129)(16,156,44,128)(17,155,45,127)(18,154,46,126)(19,153,47,125)(20,152,48,124)(21,151,49,123)(22,150,50,122)(23,149,51,121)(24,148,52,120)(25,147,53,119)(26,146,54,118)(27,145,55,117)(28,144,56,116)(57,176,85,204)(58,175,86,203)(59,174,87,202)(60,173,88,201)(61,172,89,200)(62,171,90,199)(63,170,91,198)(64,169,92,197)(65,224,93,196)(66,223,94,195)(67,222,95,194)(68,221,96,193)(69,220,97,192)(70,219,98,191)(71,218,99,190)(72,217,100,189)(73,216,101,188)(74,215,102,187)(75,214,103,186)(76,213,104,185)(77,212,105,184)(78,211,106,183)(79,210,107,182)(80,209,108,181)(81,208,109,180)(82,207,110,179)(83,206,111,178)(84,205,112,177)>;
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), (1,198,15,184,29,170,43,212)(2,213,16,199,30,185,44,171)(3,172,17,214,31,200,45,186)(4,187,18,173,32,215,46,201)(5,202,19,188,33,174,47,216)(6,217,20,203,34,189,48,175)(7,176,21,218,35,204,49,190)(8,191,22,177,36,219,50,205)(9,206,23,192,37,178,51,220)(10,221,24,207,38,193,52,179)(11,180,25,222,39,208,53,194)(12,195,26,181,40,223,54,209)(13,210,27,196,41,182,55,224)(14,169,28,211,42,197,56,183)(57,165,99,123,85,137,71,151)(58,124,100,138,86,152,72,166)(59,139,101,153,87,167,73,125)(60,154,102,168,88,126,74,140)(61,113,103,127,89,141,75,155)(62,128,104,142,90,156,76,114)(63,143,105,157,91,115,77,129)(64,158,106,116,92,130,78,144)(65,117,107,131,93,145,79,159)(66,132,108,146,94,160,80,118)(67,147,109,161,95,119,81,133)(68,162,110,120,96,134,82,148)(69,121,111,135,97,149,83,163)(70,136,112,150,98,164,84,122), (1,115,29,143)(2,114,30,142)(3,113,31,141)(4,168,32,140)(5,167,33,139)(6,166,34,138)(7,165,35,137)(8,164,36,136)(9,163,37,135)(10,162,38,134)(11,161,39,133)(12,160,40,132)(13,159,41,131)(14,158,42,130)(15,157,43,129)(16,156,44,128)(17,155,45,127)(18,154,46,126)(19,153,47,125)(20,152,48,124)(21,151,49,123)(22,150,50,122)(23,149,51,121)(24,148,52,120)(25,147,53,119)(26,146,54,118)(27,145,55,117)(28,144,56,116)(57,176,85,204)(58,175,86,203)(59,174,87,202)(60,173,88,201)(61,172,89,200)(62,171,90,199)(63,170,91,198)(64,169,92,197)(65,224,93,196)(66,223,94,195)(67,222,95,194)(68,221,96,193)(69,220,97,192)(70,219,98,191)(71,218,99,190)(72,217,100,189)(73,216,101,188)(74,215,102,187)(75,214,103,186)(76,213,104,185)(77,212,105,184)(78,211,106,183)(79,210,107,182)(80,209,108,181)(81,208,109,180)(82,207,110,179)(83,206,111,178)(84,205,112,177) );
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)], [(1,198,15,184,29,170,43,212),(2,213,16,199,30,185,44,171),(3,172,17,214,31,200,45,186),(4,187,18,173,32,215,46,201),(5,202,19,188,33,174,47,216),(6,217,20,203,34,189,48,175),(7,176,21,218,35,204,49,190),(8,191,22,177,36,219,50,205),(9,206,23,192,37,178,51,220),(10,221,24,207,38,193,52,179),(11,180,25,222,39,208,53,194),(12,195,26,181,40,223,54,209),(13,210,27,196,41,182,55,224),(14,169,28,211,42,197,56,183),(57,165,99,123,85,137,71,151),(58,124,100,138,86,152,72,166),(59,139,101,153,87,167,73,125),(60,154,102,168,88,126,74,140),(61,113,103,127,89,141,75,155),(62,128,104,142,90,156,76,114),(63,143,105,157,91,115,77,129),(64,158,106,116,92,130,78,144),(65,117,107,131,93,145,79,159),(66,132,108,146,94,160,80,118),(67,147,109,161,95,119,81,133),(68,162,110,120,96,134,82,148),(69,121,111,135,97,149,83,163),(70,136,112,150,98,164,84,122)], [(1,115,29,143),(2,114,30,142),(3,113,31,141),(4,168,32,140),(5,167,33,139),(6,166,34,138),(7,165,35,137),(8,164,36,136),(9,163,37,135),(10,162,38,134),(11,161,39,133),(12,160,40,132),(13,159,41,131),(14,158,42,130),(15,157,43,129),(16,156,44,128),(17,155,45,127),(18,154,46,126),(19,153,47,125),(20,152,48,124),(21,151,49,123),(22,150,50,122),(23,149,51,121),(24,148,52,120),(25,147,53,119),(26,146,54,118),(27,145,55,117),(28,144,56,116),(57,176,85,204),(58,175,86,203),(59,174,87,202),(60,173,88,201),(61,172,89,200),(62,171,90,199),(63,170,91,198),(64,169,92,197),(65,224,93,196),(66,223,94,195),(67,222,95,194),(68,221,96,193),(69,220,97,192),(70,219,98,191),(71,218,99,190),(72,217,100,189),(73,216,101,188),(74,215,102,187),(75,214,103,186),(76,213,104,185),(77,212,105,184),(78,211,106,183),(79,210,107,182),(80,209,108,181),(81,208,109,180),(82,207,110,179),(83,206,111,178),(84,205,112,177)])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 2 | 2 | 28 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
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 | D28 | D4.5D4 | D4×D7 | Q8⋊2D7 | C56.18D4 |
| kernel | C56.18D4 | C28.47D4 | C7×C8.C4 | D28.2C4 | C2×Dic28 | C8.D14 | C56 | Dic14 | D28 | C8.C4 | C2×C14 | C2×C8 | M4(2) | C8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56.18D4 ►in GL4(𝔽113) generated by
| 105 | 21 | 0 | 0 |
| 96 | 101 | 0 | 0 |
| 94 | 65 | 81 | 41 |
| 93 | 84 | 31 | 45 |
| 102 | 38 | 105 | 20 |
| 71 | 104 | 65 | 22 |
| 88 | 35 | 66 | 75 |
| 52 | 84 | 68 | 67 |
| 46 | 40 | 0 | 0 |
| 29 | 67 | 0 | 0 |
| 62 | 51 | 43 | 89 |
| 42 | 64 | 30 | 70 |
G:=sub<GL(4,GF(113))| [105,96,94,93,21,101,65,84,0,0,81,31,0,0,41,45],[102,71,88,52,38,104,35,84,105,65,66,68,20,22,75,67],[46,29,62,42,40,67,51,64,0,0,43,30,0,0,89,70] >;
C56.18D4 in GAP, Magma, Sage, TeX
C_{56}._{18}D_4 % in TeX
G:=Group("C56.18D4"); // GroupNames label
G:=SmallGroup(448,430);
// by ID
G=gap.SmallGroup(448,430);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,58,1123,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^4=c^2=a^28,b*a*b^-1=a^15,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations