metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.9D8, C28.8SD16, C42.9D14, (D4×C14).3C4, C28⋊C8⋊10C2, C4⋊1D4.1D7, C7⋊2(C4.D8), C4.12(D4⋊D7), (C2×C28).106D4, C4.6(D4.D7), (C2×D4).3Dic7, (C4×C28).47C22, C2.4(C28.D4), C14.9(C4.D4), C2.4(D4⋊Dic7), C14.24(D4⋊C4), C22.41(C23.D7), (C7×C4⋊1D4).1C2, (C2×C28).171(C2×C4), (C2×C4).11(C2×Dic7), (C2×C4).176(C7⋊D4), (C2×C14).102(C22⋊C4), SmallGroup(448,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.9D8
G = < a,b,c | a28=b8=1, c2=a21, bab-1=a-1, cac-1=a13, cbc-1=a21b-1 >
Subgroups: 300 in 84 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C14, C14, C14, C42, C2×C8, C2×D4, C2×D4, C28, C28, C2×C14, C2×C14, C4⋊C8, C4⋊1D4, C7⋊C8, C2×C28, C2×C28, C7×D4, C22×C14, C4.D8, C2×C7⋊C8, C4×C28, D4×C14, D4×C14, C28⋊C8, C7×C4⋊1D4, C28.9D8
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, C4.D4, D4⋊C4, C2×Dic7, C7⋊D4, C4.D8, D4⋊D7, D4.D7, C23.D7, D4⋊Dic7, C28.D4, C28.9D8
►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)
(1 142 186 133 99 69 40 202)(2 141 187 132 100 68 41 201)(3 168 188 131 101 67 42 200)(4 167 189 130 102 66 43 199)(5 166 190 129 103 65 44 198)(6 165 191 128 104 64 45 197)(7 164 192 127 105 63 46 224)(8 163 193 126 106 62 47 223)(9 162 194 125 107 61 48 222)(10 161 195 124 108 60 49 221)(11 160 196 123 109 59 50 220)(12 159 169 122 110 58 51 219)(13 158 170 121 111 57 52 218)(14 157 171 120 112 84 53 217)(15 156 172 119 85 83 54 216)(16 155 173 118 86 82 55 215)(17 154 174 117 87 81 56 214)(18 153 175 116 88 80 29 213)(19 152 176 115 89 79 30 212)(20 151 177 114 90 78 31 211)(21 150 178 113 91 77 32 210)(22 149 179 140 92 76 33 209)(23 148 180 139 93 75 34 208)(24 147 181 138 94 74 35 207)(25 146 182 137 95 73 36 206)(26 145 183 136 96 72 37 205)(27 144 184 135 97 71 38 204)(28 143 185 134 98 70 39 203)
(1 62 22 83 15 76 8 69)(2 75 23 68 16 61 9 82)(3 60 24 81 17 74 10 67)(4 73 25 66 18 59 11 80)(5 58 26 79 19 72 12 65)(6 71 27 64 20 57 13 78)(7 84 28 77 21 70 14 63)(29 220 50 213 43 206 36 199)(30 205 51 198 44 219 37 212)(31 218 52 211 45 204 38 197)(32 203 53 224 46 217 39 210)(33 216 54 209 47 202 40 223)(34 201 55 222 48 215 41 208)(35 214 56 207 49 200 42 221)(85 149 106 142 99 163 92 156)(86 162 107 155 100 148 93 141)(87 147 108 168 101 161 94 154)(88 160 109 153 102 146 95 167)(89 145 110 166 103 159 96 152)(90 158 111 151 104 144 97 165)(91 143 112 164 105 157 98 150)(113 178 134 171 127 192 120 185)(114 191 135 184 128 177 121 170)(115 176 136 169 129 190 122 183)(116 189 137 182 130 175 123 196)(117 174 138 195 131 188 124 181)(118 187 139 180 132 173 125 194)(119 172 140 193 133 186 126 179)
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), (1,142,186,133,99,69,40,202)(2,141,187,132,100,68,41,201)(3,168,188,131,101,67,42,200)(4,167,189,130,102,66,43,199)(5,166,190,129,103,65,44,198)(6,165,191,128,104,64,45,197)(7,164,192,127,105,63,46,224)(8,163,193,126,106,62,47,223)(9,162,194,125,107,61,48,222)(10,161,195,124,108,60,49,221)(11,160,196,123,109,59,50,220)(12,159,169,122,110,58,51,219)(13,158,170,121,111,57,52,218)(14,157,171,120,112,84,53,217)(15,156,172,119,85,83,54,216)(16,155,173,118,86,82,55,215)(17,154,174,117,87,81,56,214)(18,153,175,116,88,80,29,213)(19,152,176,115,89,79,30,212)(20,151,177,114,90,78,31,211)(21,150,178,113,91,77,32,210)(22,149,179,140,92,76,33,209)(23,148,180,139,93,75,34,208)(24,147,181,138,94,74,35,207)(25,146,182,137,95,73,36,206)(26,145,183,136,96,72,37,205)(27,144,184,135,97,71,38,204)(28,143,185,134,98,70,39,203), (1,62,22,83,15,76,8,69)(2,75,23,68,16,61,9,82)(3,60,24,81,17,74,10,67)(4,73,25,66,18,59,11,80)(5,58,26,79,19,72,12,65)(6,71,27,64,20,57,13,78)(7,84,28,77,21,70,14,63)(29,220,50,213,43,206,36,199)(30,205,51,198,44,219,37,212)(31,218,52,211,45,204,38,197)(32,203,53,224,46,217,39,210)(33,216,54,209,47,202,40,223)(34,201,55,222,48,215,41,208)(35,214,56,207,49,200,42,221)(85,149,106,142,99,163,92,156)(86,162,107,155,100,148,93,141)(87,147,108,168,101,161,94,154)(88,160,109,153,102,146,95,167)(89,145,110,166,103,159,96,152)(90,158,111,151,104,144,97,165)(91,143,112,164,105,157,98,150)(113,178,134,171,127,192,120,185)(114,191,135,184,128,177,121,170)(115,176,136,169,129,190,122,183)(116,189,137,182,130,175,123,196)(117,174,138,195,131,188,124,181)(118,187,139,180,132,173,125,194)(119,172,140,193,133,186,126,179)>;
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), (1,142,186,133,99,69,40,202)(2,141,187,132,100,68,41,201)(3,168,188,131,101,67,42,200)(4,167,189,130,102,66,43,199)(5,166,190,129,103,65,44,198)(6,165,191,128,104,64,45,197)(7,164,192,127,105,63,46,224)(8,163,193,126,106,62,47,223)(9,162,194,125,107,61,48,222)(10,161,195,124,108,60,49,221)(11,160,196,123,109,59,50,220)(12,159,169,122,110,58,51,219)(13,158,170,121,111,57,52,218)(14,157,171,120,112,84,53,217)(15,156,172,119,85,83,54,216)(16,155,173,118,86,82,55,215)(17,154,174,117,87,81,56,214)(18,153,175,116,88,80,29,213)(19,152,176,115,89,79,30,212)(20,151,177,114,90,78,31,211)(21,150,178,113,91,77,32,210)(22,149,179,140,92,76,33,209)(23,148,180,139,93,75,34,208)(24,147,181,138,94,74,35,207)(25,146,182,137,95,73,36,206)(26,145,183,136,96,72,37,205)(27,144,184,135,97,71,38,204)(28,143,185,134,98,70,39,203), (1,62,22,83,15,76,8,69)(2,75,23,68,16,61,9,82)(3,60,24,81,17,74,10,67)(4,73,25,66,18,59,11,80)(5,58,26,79,19,72,12,65)(6,71,27,64,20,57,13,78)(7,84,28,77,21,70,14,63)(29,220,50,213,43,206,36,199)(30,205,51,198,44,219,37,212)(31,218,52,211,45,204,38,197)(32,203,53,224,46,217,39,210)(33,216,54,209,47,202,40,223)(34,201,55,222,48,215,41,208)(35,214,56,207,49,200,42,221)(85,149,106,142,99,163,92,156)(86,162,107,155,100,148,93,141)(87,147,108,168,101,161,94,154)(88,160,109,153,102,146,95,167)(89,145,110,166,103,159,96,152)(90,158,111,151,104,144,97,165)(91,143,112,164,105,157,98,150)(113,178,134,171,127,192,120,185)(114,191,135,184,128,177,121,170)(115,176,136,169,129,190,122,183)(116,189,137,182,130,175,123,196)(117,174,138,195,131,188,124,181)(118,187,139,180,132,173,125,194)(119,172,140,193,133,186,126,179) );
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)], [(1,142,186,133,99,69,40,202),(2,141,187,132,100,68,41,201),(3,168,188,131,101,67,42,200),(4,167,189,130,102,66,43,199),(5,166,190,129,103,65,44,198),(6,165,191,128,104,64,45,197),(7,164,192,127,105,63,46,224),(8,163,193,126,106,62,47,223),(9,162,194,125,107,61,48,222),(10,161,195,124,108,60,49,221),(11,160,196,123,109,59,50,220),(12,159,169,122,110,58,51,219),(13,158,170,121,111,57,52,218),(14,157,171,120,112,84,53,217),(15,156,172,119,85,83,54,216),(16,155,173,118,86,82,55,215),(17,154,174,117,87,81,56,214),(18,153,175,116,88,80,29,213),(19,152,176,115,89,79,30,212),(20,151,177,114,90,78,31,211),(21,150,178,113,91,77,32,210),(22,149,179,140,92,76,33,209),(23,148,180,139,93,75,34,208),(24,147,181,138,94,74,35,207),(25,146,182,137,95,73,36,206),(26,145,183,136,96,72,37,205),(27,144,184,135,97,71,38,204),(28,143,185,134,98,70,39,203)], [(1,62,22,83,15,76,8,69),(2,75,23,68,16,61,9,82),(3,60,24,81,17,74,10,67),(4,73,25,66,18,59,11,80),(5,58,26,79,19,72,12,65),(6,71,27,64,20,57,13,78),(7,84,28,77,21,70,14,63),(29,220,50,213,43,206,36,199),(30,205,51,198,44,219,37,212),(31,218,52,211,45,204,38,197),(32,203,53,224,46,217,39,210),(33,216,54,209,47,202,40,223),(34,201,55,222,48,215,41,208),(35,214,56,207,49,200,42,221),(85,149,106,142,99,163,92,156),(86,162,107,155,100,148,93,141),(87,147,108,168,101,161,94,154),(88,160,109,153,102,146,95,167),(89,145,110,166,103,159,96,152),(90,158,111,151,104,144,97,165),(91,143,112,164,105,157,98,150),(113,178,134,171,127,192,120,185),(114,191,135,184,128,177,121,170),(115,176,136,169,129,190,122,183),(116,189,137,182,130,175,123,196),(117,174,138,195,131,188,124,181),(118,187,139,180,132,173,125,194),(119,172,140,193,133,186,126,179)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | D4 | D7 | D8 | SD16 | D14 | Dic7 | C7⋊D4 | C4.D4 | D4⋊D7 | D4.D7 | C28.D4 |
| kernel | C28.9D8 | C28⋊C8 | C7×C4⋊1D4 | D4×C14 | C2×C28 | C4⋊1D4 | C28 | C28 | C42 | C2×D4 | C2×C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 3 | 4 | 4 | 3 | 6 | 12 | 1 | 6 | 6 | 6 |
Matrix representation of C28.9D8 ►in GL6(𝔽113)
| 1 | 44 | 0 | 0 | 0 | 0 |
| 77 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 0 | 0 | 0 |
| 0 | 0 | 99 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 51 | 105 | 0 | 0 | 0 | 0 |
| 99 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 29 | 0 | 0 |
| 0 | 0 | 31 | 111 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 100 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 51 | 105 | 0 | 0 | 0 | 0 |
| 99 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 111 | 84 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 13 |
| 0 | 0 | 0 | 0 | 13 | 13 |
G:=sub<GL(6,GF(113))| [1,77,0,0,0,0,44,112,0,0,0,0,0,0,106,99,0,0,0,0,0,16,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[51,99,0,0,0,0,105,62,0,0,0,0,0,0,2,31,0,0,0,0,29,111,0,0,0,0,0,0,100,13,0,0,0,0,100,100],[51,99,0,0,0,0,105,0,0,0,0,0,0,0,111,4,0,0,0,0,84,2,0,0,0,0,0,0,100,13,0,0,0,0,13,13] >;
C28.9D8 in GAP, Magma, Sage, TeX
C_{28}._9D_8 % in TeX
G:=Group("C28.9D8"); // GroupNames label
G:=SmallGroup(448,101);
// by ID
G=gap.SmallGroup(448,101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,100,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^21,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^21*b^-1>;
// generators/relations