metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.50D8, D4⋊3Dic14, C42.44D14, (C7×D4)⋊3Q8, (C4×D4).2D7, C7⋊4(D4⋊Q8), (D4×C28).2C2, C28⋊C8⋊17C2, C14.50(C2×D8), (C2×C28).57D4, C28.25(C2×Q8), C4.23(D4⋊D7), C4⋊C4.238D14, C28⋊2Q8⋊14C2, C4.9(C2×Dic14), (C2×D4).185D14, C4.59(C4○D28), C28.45(C4○D4), C28.Q8⋊30C2, (C4×C28).78C22, D4⋊Dic7.7C2, (C2×C28).332C23, C14.61(C22⋊Q8), C2.7(D4.9D14), (D4×C14).227C22, C4⋊Dic7.136C22, C2.12(C28.48D4), C14.107(C8.C22), C2.6(C2×D4⋊D7), (C2×C7⋊C8).89C22, (C2×C14).463(C2×D4), (C2×C4).243(C7⋊D4), (C7×C4⋊C4).269C22, (C2×C4).432(C22×D7), C22.146(C2×C7⋊D4), SmallGroup(448,541)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.50D8
G = < a,b,c | a28=b8=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a14b-1 >
Subgroups: 436 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D4⋊Q8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×C28, D4×C14, C28⋊C8, C28.Q8, D4⋊Dic7, C28⋊2Q8, D4×C28, C28.50D8
Quotients: C1, C2, C22, D4, Q8, C23, D7, D8, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×D8, C8.C22, Dic14, C7⋊D4, C22×D7, D4⋊Q8, D4⋊D7, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4, C2×D4⋊D7, D4.9D14, C28.50D8
►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)
(1 149 107 76 188 35 204 113)(2 148 108 75 189 34 205 140)(3 147 109 74 190 33 206 139)(4 146 110 73 191 32 207 138)(5 145 111 72 192 31 208 137)(6 144 112 71 193 30 209 136)(7 143 85 70 194 29 210 135)(8 142 86 69 195 56 211 134)(9 141 87 68 196 55 212 133)(10 168 88 67 169 54 213 132)(11 167 89 66 170 53 214 131)(12 166 90 65 171 52 215 130)(13 165 91 64 172 51 216 129)(14 164 92 63 173 50 217 128)(15 163 93 62 174 49 218 127)(16 162 94 61 175 48 219 126)(17 161 95 60 176 47 220 125)(18 160 96 59 177 46 221 124)(19 159 97 58 178 45 222 123)(20 158 98 57 179 44 223 122)(21 157 99 84 180 43 224 121)(22 156 100 83 181 42 197 120)(23 155 101 82 182 41 198 119)(24 154 102 81 183 40 199 118)(25 153 103 80 184 39 200 117)(26 152 104 79 185 38 201 116)(27 151 105 78 186 37 202 115)(28 150 106 77 187 36 203 114)
(1 113 15 127)(2 140 16 126)(3 139 17 125)(4 138 18 124)(5 137 19 123)(6 136 20 122)(7 135 21 121)(8 134 22 120)(9 133 23 119)(10 132 24 118)(11 131 25 117)(12 130 26 116)(13 129 27 115)(14 128 28 114)(29 99 43 85)(30 98 44 112)(31 97 45 111)(32 96 46 110)(33 95 47 109)(34 94 48 108)(35 93 49 107)(36 92 50 106)(37 91 51 105)(38 90 52 104)(39 89 53 103)(40 88 54 102)(41 87 55 101)(42 86 56 100)(57 193 71 179)(58 192 72 178)(59 191 73 177)(60 190 74 176)(61 189 75 175)(62 188 76 174)(63 187 77 173)(64 186 78 172)(65 185 79 171)(66 184 80 170)(67 183 81 169)(68 182 82 196)(69 181 83 195)(70 180 84 194)(141 198 155 212)(142 197 156 211)(143 224 157 210)(144 223 158 209)(145 222 159 208)(146 221 160 207)(147 220 161 206)(148 219 162 205)(149 218 163 204)(150 217 164 203)(151 216 165 202)(152 215 166 201)(153 214 167 200)(154 213 168 199)
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), (1,149,107,76,188,35,204,113)(2,148,108,75,189,34,205,140)(3,147,109,74,190,33,206,139)(4,146,110,73,191,32,207,138)(5,145,111,72,192,31,208,137)(6,144,112,71,193,30,209,136)(7,143,85,70,194,29,210,135)(8,142,86,69,195,56,211,134)(9,141,87,68,196,55,212,133)(10,168,88,67,169,54,213,132)(11,167,89,66,170,53,214,131)(12,166,90,65,171,52,215,130)(13,165,91,64,172,51,216,129)(14,164,92,63,173,50,217,128)(15,163,93,62,174,49,218,127)(16,162,94,61,175,48,219,126)(17,161,95,60,176,47,220,125)(18,160,96,59,177,46,221,124)(19,159,97,58,178,45,222,123)(20,158,98,57,179,44,223,122)(21,157,99,84,180,43,224,121)(22,156,100,83,181,42,197,120)(23,155,101,82,182,41,198,119)(24,154,102,81,183,40,199,118)(25,153,103,80,184,39,200,117)(26,152,104,79,185,38,201,116)(27,151,105,78,186,37,202,115)(28,150,106,77,187,36,203,114), (1,113,15,127)(2,140,16,126)(3,139,17,125)(4,138,18,124)(5,137,19,123)(6,136,20,122)(7,135,21,121)(8,134,22,120)(9,133,23,119)(10,132,24,118)(11,131,25,117)(12,130,26,116)(13,129,27,115)(14,128,28,114)(29,99,43,85)(30,98,44,112)(31,97,45,111)(32,96,46,110)(33,95,47,109)(34,94,48,108)(35,93,49,107)(36,92,50,106)(37,91,51,105)(38,90,52,104)(39,89,53,103)(40,88,54,102)(41,87,55,101)(42,86,56,100)(57,193,71,179)(58,192,72,178)(59,191,73,177)(60,190,74,176)(61,189,75,175)(62,188,76,174)(63,187,77,173)(64,186,78,172)(65,185,79,171)(66,184,80,170)(67,183,81,169)(68,182,82,196)(69,181,83,195)(70,180,84,194)(141,198,155,212)(142,197,156,211)(143,224,157,210)(144,223,158,209)(145,222,159,208)(146,221,160,207)(147,220,161,206)(148,219,162,205)(149,218,163,204)(150,217,164,203)(151,216,165,202)(152,215,166,201)(153,214,167,200)(154,213,168,199)>;
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), (1,149,107,76,188,35,204,113)(2,148,108,75,189,34,205,140)(3,147,109,74,190,33,206,139)(4,146,110,73,191,32,207,138)(5,145,111,72,192,31,208,137)(6,144,112,71,193,30,209,136)(7,143,85,70,194,29,210,135)(8,142,86,69,195,56,211,134)(9,141,87,68,196,55,212,133)(10,168,88,67,169,54,213,132)(11,167,89,66,170,53,214,131)(12,166,90,65,171,52,215,130)(13,165,91,64,172,51,216,129)(14,164,92,63,173,50,217,128)(15,163,93,62,174,49,218,127)(16,162,94,61,175,48,219,126)(17,161,95,60,176,47,220,125)(18,160,96,59,177,46,221,124)(19,159,97,58,178,45,222,123)(20,158,98,57,179,44,223,122)(21,157,99,84,180,43,224,121)(22,156,100,83,181,42,197,120)(23,155,101,82,182,41,198,119)(24,154,102,81,183,40,199,118)(25,153,103,80,184,39,200,117)(26,152,104,79,185,38,201,116)(27,151,105,78,186,37,202,115)(28,150,106,77,187,36,203,114), (1,113,15,127)(2,140,16,126)(3,139,17,125)(4,138,18,124)(5,137,19,123)(6,136,20,122)(7,135,21,121)(8,134,22,120)(9,133,23,119)(10,132,24,118)(11,131,25,117)(12,130,26,116)(13,129,27,115)(14,128,28,114)(29,99,43,85)(30,98,44,112)(31,97,45,111)(32,96,46,110)(33,95,47,109)(34,94,48,108)(35,93,49,107)(36,92,50,106)(37,91,51,105)(38,90,52,104)(39,89,53,103)(40,88,54,102)(41,87,55,101)(42,86,56,100)(57,193,71,179)(58,192,72,178)(59,191,73,177)(60,190,74,176)(61,189,75,175)(62,188,76,174)(63,187,77,173)(64,186,78,172)(65,185,79,171)(66,184,80,170)(67,183,81,169)(68,182,82,196)(69,181,83,195)(70,180,84,194)(141,198,155,212)(142,197,156,211)(143,224,157,210)(144,223,158,209)(145,222,159,208)(146,221,160,207)(147,220,161,206)(148,219,162,205)(149,218,163,204)(150,217,164,203)(151,216,165,202)(152,215,166,201)(153,214,167,200)(154,213,168,199) );
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)], [(1,149,107,76,188,35,204,113),(2,148,108,75,189,34,205,140),(3,147,109,74,190,33,206,139),(4,146,110,73,191,32,207,138),(5,145,111,72,192,31,208,137),(6,144,112,71,193,30,209,136),(7,143,85,70,194,29,210,135),(8,142,86,69,195,56,211,134),(9,141,87,68,196,55,212,133),(10,168,88,67,169,54,213,132),(11,167,89,66,170,53,214,131),(12,166,90,65,171,52,215,130),(13,165,91,64,172,51,216,129),(14,164,92,63,173,50,217,128),(15,163,93,62,174,49,218,127),(16,162,94,61,175,48,219,126),(17,161,95,60,176,47,220,125),(18,160,96,59,177,46,221,124),(19,159,97,58,178,45,222,123),(20,158,98,57,179,44,223,122),(21,157,99,84,180,43,224,121),(22,156,100,83,181,42,197,120),(23,155,101,82,182,41,198,119),(24,154,102,81,183,40,199,118),(25,153,103,80,184,39,200,117),(26,152,104,79,185,38,201,116),(27,151,105,78,186,37,202,115),(28,150,106,77,187,36,203,114)], [(1,113,15,127),(2,140,16,126),(3,139,17,125),(4,138,18,124),(5,137,19,123),(6,136,20,122),(7,135,21,121),(8,134,22,120),(9,133,23,119),(10,132,24,118),(11,131,25,117),(12,130,26,116),(13,129,27,115),(14,128,28,114),(29,99,43,85),(30,98,44,112),(31,97,45,111),(32,96,46,110),(33,95,47,109),(34,94,48,108),(35,93,49,107),(36,92,50,106),(37,91,51,105),(38,90,52,104),(39,89,53,103),(40,88,54,102),(41,87,55,101),(42,86,56,100),(57,193,71,179),(58,192,72,178),(59,191,73,177),(60,190,74,176),(61,189,75,175),(62,188,76,174),(63,187,77,173),(64,186,78,172),(65,185,79,171),(66,184,80,170),(67,183,81,169),(68,182,82,196),(69,181,83,195),(70,180,84,194),(141,198,155,212),(142,197,156,211),(143,224,157,210),(144,223,158,209),(145,222,159,208),(146,221,160,207),(147,220,161,206),(148,219,162,205),(149,218,163,204),(150,217,164,203),(151,216,165,202),(152,215,166,201),(153,214,167,200),(154,213,168,199)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | D8 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | Dic14 | C4○D28 | C8.C22 | D4⋊D7 | D4.9D14 |
| kernel | C28.50D8 | C28⋊C8 | C28.Q8 | D4⋊Dic7 | C28⋊2Q8 | D4×C28 | C2×C28 | C7×D4 | C4×D4 | C28 | C28 | C42 | C4⋊C4 | C2×D4 | C2×C4 | D4 | C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of C28.50D8 ►in GL4(𝔽113) generated by
| 69 | 100 | 0 | 0 |
| 89 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 94 | 3 | 0 | 0 |
| 30 | 19 | 0 | 0 |
| 0 | 0 | 51 | 51 |
| 0 | 0 | 31 | 0 |
| 19 | 110 | 0 | 0 |
| 83 | 94 | 0 | 0 |
| 0 | 0 | 51 | 51 |
| 0 | 0 | 31 | 62 |
G:=sub<GL(4,GF(113))| [69,89,0,0,100,16,0,0,0,0,1,0,0,0,0,1],[94,30,0,0,3,19,0,0,0,0,51,31,0,0,51,0],[19,83,0,0,110,94,0,0,0,0,51,31,0,0,51,62] >;
C28.50D8 in GAP, Magma, Sage, TeX
C_{28}._{50}D_8 % in TeX
G:=Group("C28.50D8"); // GroupNames label
G:=SmallGroup(448,541);
// by ID
G=gap.SmallGroup(448,541);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^14*b^-1>;
// generators/relations