metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.2D28, C56.30D4, C4.Q8⋊4D7, C4⋊C4.42D14, C4.52(C2×D28), (C2×C8).62D14, C7⋊2(C8.D4), C28.132(C2×D4), (C2×Dic28)⋊24C2, C28.33(C4○D4), C14.Q16⋊18C2, D14⋊2Q8.6C2, C4.5(Q8⋊2D7), (C2×Dic7).43D4, (C22×D7).27D4, C22.220(D4×D7), C14.45(C4⋊D4), C2.18(C4⋊D28), (C2×C56).111C22, (C2×C28).284C23, C2.25(SD16⋊D7), C14.44(C8.C22), (C2×Dic14).85C22, (C7×C4.Q8)⋊4C2, (C2×C8⋊D7).3C2, (C2×C7⋊C8).61C22, (C2×C4×D7).36C22, (C2×C14).289(C2×D4), (C7×C4⋊C4).77C22, (C2×C4).387(C22×D7), SmallGroup(448,402)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.2D28
G = < a,b,c | a8=b28=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 588 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic7, C28, C28, D14, C2×C14, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C8.D4, C8⋊D7, Dic28, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C14.Q16, C7×C4.Q8, D14⋊2Q8, C2×C8⋊D7, C2×Dic28, C8.2D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8.C22, D28, C22×D7, C8.D4, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, SD16⋊D7, C8.2D28
►On 224 points
Generators in S224
(1 43 62 127 213 184 105 144)(2 128 106 44 214 145 63 185)(3 45 64 129 215 186 107 146)(4 130 108 46 216 147 65 187)(5 47 66 131 217 188 109 148)(6 132 110 48 218 149 67 189)(7 49 68 133 219 190 111 150)(8 134 112 50 220 151 69 191)(9 51 70 135 221 192 85 152)(10 136 86 52 222 153 71 193)(11 53 72 137 223 194 87 154)(12 138 88 54 224 155 73 195)(13 55 74 139 197 196 89 156)(14 140 90 56 198 157 75 169)(15 29 76 113 199 170 91 158)(16 114 92 30 200 159 77 171)(17 31 78 115 201 172 93 160)(18 116 94 32 202 161 79 173)(19 33 80 117 203 174 95 162)(20 118 96 34 204 163 81 175)(21 35 82 119 205 176 97 164)(22 120 98 36 206 165 83 177)(23 37 84 121 207 178 99 166)(24 122 100 38 208 167 57 179)(25 39 58 123 209 180 101 168)(26 124 102 40 210 141 59 181)(27 41 60 125 211 182 103 142)(28 126 104 42 212 143 61 183)
(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)
(1 212 213 28)(2 27 214 211)(3 210 215 26)(4 25 216 209)(5 208 217 24)(6 23 218 207)(7 206 219 22)(8 21 220 205)(9 204 221 20)(10 19 222 203)(11 202 223 18)(12 17 224 201)(13 200 197 16)(14 15 198 199)(29 56 170 169)(30 196 171 55)(31 54 172 195)(32 194 173 53)(33 52 174 193)(34 192 175 51)(35 50 176 191)(36 190 177 49)(37 48 178 189)(38 188 179 47)(39 46 180 187)(40 186 181 45)(41 44 182 185)(42 184 183 43)(57 66 100 109)(58 108 101 65)(59 64 102 107)(60 106 103 63)(61 62 104 105)(67 84 110 99)(68 98 111 83)(69 82 112 97)(70 96 85 81)(71 80 86 95)(72 94 87 79)(73 78 88 93)(74 92 89 77)(75 76 90 91)(113 140 158 157)(114 156 159 139)(115 138 160 155)(116 154 161 137)(117 136 162 153)(118 152 163 135)(119 134 164 151)(120 150 165 133)(121 132 166 149)(122 148 167 131)(123 130 168 147)(124 146 141 129)(125 128 142 145)(126 144 143 127)
G:=sub<Sym(224)| (1,43,62,127,213,184,105,144)(2,128,106,44,214,145,63,185)(3,45,64,129,215,186,107,146)(4,130,108,46,216,147,65,187)(5,47,66,131,217,188,109,148)(6,132,110,48,218,149,67,189)(7,49,68,133,219,190,111,150)(8,134,112,50,220,151,69,191)(9,51,70,135,221,192,85,152)(10,136,86,52,222,153,71,193)(11,53,72,137,223,194,87,154)(12,138,88,54,224,155,73,195)(13,55,74,139,197,196,89,156)(14,140,90,56,198,157,75,169)(15,29,76,113,199,170,91,158)(16,114,92,30,200,159,77,171)(17,31,78,115,201,172,93,160)(18,116,94,32,202,161,79,173)(19,33,80,117,203,174,95,162)(20,118,96,34,204,163,81,175)(21,35,82,119,205,176,97,164)(22,120,98,36,206,165,83,177)(23,37,84,121,207,178,99,166)(24,122,100,38,208,167,57,179)(25,39,58,123,209,180,101,168)(26,124,102,40,210,141,59,181)(27,41,60,125,211,182,103,142)(28,126,104,42,212,143,61,183), (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), (1,212,213,28)(2,27,214,211)(3,210,215,26)(4,25,216,209)(5,208,217,24)(6,23,218,207)(7,206,219,22)(8,21,220,205)(9,204,221,20)(10,19,222,203)(11,202,223,18)(12,17,224,201)(13,200,197,16)(14,15,198,199)(29,56,170,169)(30,196,171,55)(31,54,172,195)(32,194,173,53)(33,52,174,193)(34,192,175,51)(35,50,176,191)(36,190,177,49)(37,48,178,189)(38,188,179,47)(39,46,180,187)(40,186,181,45)(41,44,182,185)(42,184,183,43)(57,66,100,109)(58,108,101,65)(59,64,102,107)(60,106,103,63)(61,62,104,105)(67,84,110,99)(68,98,111,83)(69,82,112,97)(70,96,85,81)(71,80,86,95)(72,94,87,79)(73,78,88,93)(74,92,89,77)(75,76,90,91)(113,140,158,157)(114,156,159,139)(115,138,160,155)(116,154,161,137)(117,136,162,153)(118,152,163,135)(119,134,164,151)(120,150,165,133)(121,132,166,149)(122,148,167,131)(123,130,168,147)(124,146,141,129)(125,128,142,145)(126,144,143,127)>;
G:=Group( (1,43,62,127,213,184,105,144)(2,128,106,44,214,145,63,185)(3,45,64,129,215,186,107,146)(4,130,108,46,216,147,65,187)(5,47,66,131,217,188,109,148)(6,132,110,48,218,149,67,189)(7,49,68,133,219,190,111,150)(8,134,112,50,220,151,69,191)(9,51,70,135,221,192,85,152)(10,136,86,52,222,153,71,193)(11,53,72,137,223,194,87,154)(12,138,88,54,224,155,73,195)(13,55,74,139,197,196,89,156)(14,140,90,56,198,157,75,169)(15,29,76,113,199,170,91,158)(16,114,92,30,200,159,77,171)(17,31,78,115,201,172,93,160)(18,116,94,32,202,161,79,173)(19,33,80,117,203,174,95,162)(20,118,96,34,204,163,81,175)(21,35,82,119,205,176,97,164)(22,120,98,36,206,165,83,177)(23,37,84,121,207,178,99,166)(24,122,100,38,208,167,57,179)(25,39,58,123,209,180,101,168)(26,124,102,40,210,141,59,181)(27,41,60,125,211,182,103,142)(28,126,104,42,212,143,61,183), (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), (1,212,213,28)(2,27,214,211)(3,210,215,26)(4,25,216,209)(5,208,217,24)(6,23,218,207)(7,206,219,22)(8,21,220,205)(9,204,221,20)(10,19,222,203)(11,202,223,18)(12,17,224,201)(13,200,197,16)(14,15,198,199)(29,56,170,169)(30,196,171,55)(31,54,172,195)(32,194,173,53)(33,52,174,193)(34,192,175,51)(35,50,176,191)(36,190,177,49)(37,48,178,189)(38,188,179,47)(39,46,180,187)(40,186,181,45)(41,44,182,185)(42,184,183,43)(57,66,100,109)(58,108,101,65)(59,64,102,107)(60,106,103,63)(61,62,104,105)(67,84,110,99)(68,98,111,83)(69,82,112,97)(70,96,85,81)(71,80,86,95)(72,94,87,79)(73,78,88,93)(74,92,89,77)(75,76,90,91)(113,140,158,157)(114,156,159,139)(115,138,160,155)(116,154,161,137)(117,136,162,153)(118,152,163,135)(119,134,164,151)(120,150,165,133)(121,132,166,149)(122,148,167,131)(123,130,168,147)(124,146,141,129)(125,128,142,145)(126,144,143,127) );
G=PermutationGroup([[(1,43,62,127,213,184,105,144),(2,128,106,44,214,145,63,185),(3,45,64,129,215,186,107,146),(4,130,108,46,216,147,65,187),(5,47,66,131,217,188,109,148),(6,132,110,48,218,149,67,189),(7,49,68,133,219,190,111,150),(8,134,112,50,220,151,69,191),(9,51,70,135,221,192,85,152),(10,136,86,52,222,153,71,193),(11,53,72,137,223,194,87,154),(12,138,88,54,224,155,73,195),(13,55,74,139,197,196,89,156),(14,140,90,56,198,157,75,169),(15,29,76,113,199,170,91,158),(16,114,92,30,200,159,77,171),(17,31,78,115,201,172,93,160),(18,116,94,32,202,161,79,173),(19,33,80,117,203,174,95,162),(20,118,96,34,204,163,81,175),(21,35,82,119,205,176,97,164),(22,120,98,36,206,165,83,177),(23,37,84,121,207,178,99,166),(24,122,100,38,208,167,57,179),(25,39,58,123,209,180,101,168),(26,124,102,40,210,141,59,181),(27,41,60,125,211,182,103,142),(28,126,104,42,212,143,61,183)], [(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)], [(1,212,213,28),(2,27,214,211),(3,210,215,26),(4,25,216,209),(5,208,217,24),(6,23,218,207),(7,206,219,22),(8,21,220,205),(9,204,221,20),(10,19,222,203),(11,202,223,18),(12,17,224,201),(13,200,197,16),(14,15,198,199),(29,56,170,169),(30,196,171,55),(31,54,172,195),(32,194,173,53),(33,52,174,193),(34,192,175,51),(35,50,176,191),(36,190,177,49),(37,48,178,189),(38,188,179,47),(39,46,180,187),(40,186,181,45),(41,44,182,185),(42,184,183,43),(57,66,100,109),(58,108,101,65),(59,64,102,107),(60,106,103,63),(61,62,104,105),(67,84,110,99),(68,98,111,83),(69,82,112,97),(70,96,85,81),(71,80,86,95),(72,94,87,79),(73,78,88,93),(74,92,89,77),(75,76,90,91),(113,140,158,157),(114,156,159,139),(115,138,160,155),(116,154,161,137),(117,136,162,153),(118,152,163,135),(119,134,164,151),(120,150,165,133),(121,132,166,149),(122,148,167,131),(123,130,168,147),(124,146,141,129),(125,128,142,145),(126,144,143,127)]])
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 | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 2 | 2 | 8 | 8 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 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 | D28 | C8.C22 | Q8⋊2D7 | D4×D7 | SD16⋊D7 |
| kernel | C8.2D28 | C14.Q16 | C7×C4.Q8 | D14⋊2Q8 | C2×C8⋊D7 | C2×Dic28 | C56 | C2×Dic7 | C22×D7 | C4.Q8 | C28 | C4⋊C4 | C2×C8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C8.2D28 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 44 |
| 0 | 0 | 0 | 0 | 69 | 76 |
| 0 | 0 | 38 | 91 | 37 | 44 |
| 0 | 0 | 22 | 75 | 69 | 76 |
| 1 | 81 | 0 | 0 | 0 | 0 |
| 106 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 108 | 12 | 4 | 13 |
| 0 | 0 | 101 | 57 | 100 | 90 |
| 0 | 0 | 110 | 75 | 5 | 101 |
| 0 | 0 | 38 | 102 | 12 | 56 |
| 1 | 81 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 108 | 12 | 4 | 13 |
| 0 | 0 | 5 | 5 | 109 | 109 |
| 0 | 0 | 110 | 75 | 5 | 101 |
| 0 | 0 | 3 | 3 | 108 | 108 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,38,22,0,0,0,0,91,75,0,0,37,69,37,69,0,0,44,76,44,76],[1,106,0,0,0,0,81,112,0,0,0,0,0,0,108,101,110,38,0,0,12,57,75,102,0,0,4,100,5,12,0,0,13,90,101,56],[1,0,0,0,0,0,81,112,0,0,0,0,0,0,108,5,110,3,0,0,12,5,75,3,0,0,4,109,5,108,0,0,13,109,101,108] >;
C8.2D28 in GAP, Magma, Sage, TeX
C_8._2D_{28} % in TeX
G:=Group("C8.2D28"); // GroupNames label
G:=SmallGroup(448,402);
// by ID
G=gap.SmallGroup(448,402);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,254,555,226,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations