metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.19D8, C28.19SD16, C42.224D14, C4⋊Q8⋊6D7, C4⋊C4.84D14, C4.7(D4⋊D7), C14.61(C2×D8), C7⋊4(C4.4D8), C4.5(Q8⋊D7), (C2×C28).158D4, C28⋊4D4.9C2, C14.D8⋊42C2, C28.83(C4○D4), C14.78(C2×SD16), (C4×C28).135C22, (C2×C28).406C23, C4.16(Q8⋊2D7), C14.58(C4.4D4), (C2×D28).109C22, C2.11(C28.23D4), (C4×C7⋊C8)⋊18C2, (C7×C4⋊Q8)⋊6C2, C2.16(C2×D4⋊D7), C2.16(C2×Q8⋊D7), (C2×C14).537(C2×D4), (C2×C7⋊C8).263C22, (C2×C4).137(C7⋊D4), (C7×C4⋊C4).131C22, (C2×C4).503(C22×D7), C22.209(C2×C7⋊D4), SmallGroup(448,622)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.D8
G = < a,b,c | a28=b8=c2=1, bab-1=a13, cac=a-1, cbc=a14b-1 >
Subgroups: 780 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C7⋊C8, D28, C2×C28, C2×C28, C7×Q8, C22×D7, C4.4D8, C2×C7⋊C8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×D28, C2×D28, Q8×C14, C4×C7⋊C8, C14.D8, C28⋊4D4, C7×C4⋊Q8, C28.D8
Quotients: C1, C2, C22, D4, C23, D7, D8, SD16, C2×D4, C4○D4, D14, C4.4D4, C2×D8, C2×SD16, C7⋊D4, C22×D7, C4.4D8, D4⋊D7, Q8⋊D7, Q8⋊2D7, C2×C7⋊D4, C2×D4⋊D7, C2×Q8⋊D7, C28.23D4, C28.D8
►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 39 125 70 172 141 103 217)(2 52 126 83 173 154 104 202)(3 37 127 68 174 167 105 215)(4 50 128 81 175 152 106 200)(5 35 129 66 176 165 107 213)(6 48 130 79 177 150 108 198)(7 33 131 64 178 163 109 211)(8 46 132 77 179 148 110 224)(9 31 133 62 180 161 111 209)(10 44 134 75 181 146 112 222)(11 29 135 60 182 159 85 207)(12 42 136 73 183 144 86 220)(13 55 137 58 184 157 87 205)(14 40 138 71 185 142 88 218)(15 53 139 84 186 155 89 203)(16 38 140 69 187 168 90 216)(17 51 113 82 188 153 91 201)(18 36 114 67 189 166 92 214)(19 49 115 80 190 151 93 199)(20 34 116 65 191 164 94 212)(21 47 117 78 192 149 95 197)(22 32 118 63 193 162 96 210)(23 45 119 76 194 147 97 223)(24 30 120 61 195 160 98 208)(25 43 121 74 196 145 99 221)(26 56 122 59 169 158 100 206)(27 41 123 72 170 143 101 219)(28 54 124 57 171 156 102 204)
(1 179)(2 178)(3 177)(4 176)(5 175)(6 174)(7 173)(8 172)(9 171)(10 170)(11 169)(12 196)(13 195)(14 194)(15 193)(16 192)(17 191)(18 190)(19 189)(20 188)(21 187)(22 186)(23 185)(24 184)(25 183)(26 182)(27 181)(28 180)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 57)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 116)(114 115)(117 140)(118 139)(119 138)(120 137)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(141 210)(142 209)(143 208)(144 207)(145 206)(146 205)(147 204)(148 203)(149 202)(150 201)(151 200)(152 199)(153 198)(154 197)(155 224)(156 223)(157 222)(158 221)(159 220)(160 219)(161 218)(162 217)(163 216)(164 215)(165 214)(166 213)(167 212)(168 211)
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), (1,39,125,70,172,141,103,217)(2,52,126,83,173,154,104,202)(3,37,127,68,174,167,105,215)(4,50,128,81,175,152,106,200)(5,35,129,66,176,165,107,213)(6,48,130,79,177,150,108,198)(7,33,131,64,178,163,109,211)(8,46,132,77,179,148,110,224)(9,31,133,62,180,161,111,209)(10,44,134,75,181,146,112,222)(11,29,135,60,182,159,85,207)(12,42,136,73,183,144,86,220)(13,55,137,58,184,157,87,205)(14,40,138,71,185,142,88,218)(15,53,139,84,186,155,89,203)(16,38,140,69,187,168,90,216)(17,51,113,82,188,153,91,201)(18,36,114,67,189,166,92,214)(19,49,115,80,190,151,93,199)(20,34,116,65,191,164,94,212)(21,47,117,78,192,149,95,197)(22,32,118,63,193,162,96,210)(23,45,119,76,194,147,97,223)(24,30,120,61,195,160,98,208)(25,43,121,74,196,145,99,221)(26,56,122,59,169,158,100,206)(27,41,123,72,170,143,101,219)(28,54,124,57,171,156,102,204), (1,179)(2,178)(3,177)(4,176)(5,175)(6,174)(7,173)(8,172)(9,171)(10,170)(11,169)(12,196)(13,195)(14,194)(15,193)(16,192)(17,191)(18,190)(19,189)(20,188)(21,187)(22,186)(23,185)(24,184)(25,183)(26,182)(27,181)(28,180)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,116)(114,115)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(141,210)(142,209)(143,208)(144,207)(145,206)(146,205)(147,204)(148,203)(149,202)(150,201)(151,200)(152,199)(153,198)(154,197)(155,224)(156,223)(157,222)(158,221)(159,220)(160,219)(161,218)(162,217)(163,216)(164,215)(165,214)(166,213)(167,212)(168,211)>;
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), (1,39,125,70,172,141,103,217)(2,52,126,83,173,154,104,202)(3,37,127,68,174,167,105,215)(4,50,128,81,175,152,106,200)(5,35,129,66,176,165,107,213)(6,48,130,79,177,150,108,198)(7,33,131,64,178,163,109,211)(8,46,132,77,179,148,110,224)(9,31,133,62,180,161,111,209)(10,44,134,75,181,146,112,222)(11,29,135,60,182,159,85,207)(12,42,136,73,183,144,86,220)(13,55,137,58,184,157,87,205)(14,40,138,71,185,142,88,218)(15,53,139,84,186,155,89,203)(16,38,140,69,187,168,90,216)(17,51,113,82,188,153,91,201)(18,36,114,67,189,166,92,214)(19,49,115,80,190,151,93,199)(20,34,116,65,191,164,94,212)(21,47,117,78,192,149,95,197)(22,32,118,63,193,162,96,210)(23,45,119,76,194,147,97,223)(24,30,120,61,195,160,98,208)(25,43,121,74,196,145,99,221)(26,56,122,59,169,158,100,206)(27,41,123,72,170,143,101,219)(28,54,124,57,171,156,102,204), (1,179)(2,178)(3,177)(4,176)(5,175)(6,174)(7,173)(8,172)(9,171)(10,170)(11,169)(12,196)(13,195)(14,194)(15,193)(16,192)(17,191)(18,190)(19,189)(20,188)(21,187)(22,186)(23,185)(24,184)(25,183)(26,182)(27,181)(28,180)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,116)(114,115)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(141,210)(142,209)(143,208)(144,207)(145,206)(146,205)(147,204)(148,203)(149,202)(150,201)(151,200)(152,199)(153,198)(154,197)(155,224)(156,223)(157,222)(158,221)(159,220)(160,219)(161,218)(162,217)(163,216)(164,215)(165,214)(166,213)(167,212)(168,211) );
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)], [(1,39,125,70,172,141,103,217),(2,52,126,83,173,154,104,202),(3,37,127,68,174,167,105,215),(4,50,128,81,175,152,106,200),(5,35,129,66,176,165,107,213),(6,48,130,79,177,150,108,198),(7,33,131,64,178,163,109,211),(8,46,132,77,179,148,110,224),(9,31,133,62,180,161,111,209),(10,44,134,75,181,146,112,222),(11,29,135,60,182,159,85,207),(12,42,136,73,183,144,86,220),(13,55,137,58,184,157,87,205),(14,40,138,71,185,142,88,218),(15,53,139,84,186,155,89,203),(16,38,140,69,187,168,90,216),(17,51,113,82,188,153,91,201),(18,36,114,67,189,166,92,214),(19,49,115,80,190,151,93,199),(20,34,116,65,191,164,94,212),(21,47,117,78,192,149,95,197),(22,32,118,63,193,162,96,210),(23,45,119,76,194,147,97,223),(24,30,120,61,195,160,98,208),(25,43,121,74,196,145,99,221),(26,56,122,59,169,158,100,206),(27,41,123,72,170,143,101,219),(28,54,124,57,171,156,102,204)], [(1,179),(2,178),(3,177),(4,176),(5,175),(6,174),(7,173),(8,172),(9,171),(10,170),(11,169),(12,196),(13,195),(14,194),(15,193),(16,192),(17,191),(18,190),(19,189),(20,188),(21,187),(22,186),(23,185),(24,184),(25,183),(26,182),(27,181),(28,180),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,61),(42,60),(43,59),(44,58),(45,57),(46,84),(47,83),(48,82),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,116),(114,115),(117,140),(118,139),(119,138),(120,137),(121,136),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(141,210),(142,209),(143,208),(144,207),(145,206),(146,205),(147,204),(148,203),(149,202),(150,201),(151,200),(152,199),(153,198),(154,197),(155,224),(156,223),(157,222),(158,221),(159,220),(160,219),(161,218),(162,217),(163,216),(164,215),(165,214),(166,213),(167,212),(168,211)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 56 | 56 | 2 | ··· | 2 | 8 | 8 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | D8 | SD16 | C4○D4 | D14 | D14 | C7⋊D4 | D4⋊D7 | Q8⋊D7 | Q8⋊2D7 |
| kernel | C28.D8 | C4×C7⋊C8 | C14.D8 | C28⋊4D4 | C7×C4⋊Q8 | C2×C28 | C4⋊Q8 | C28 | C28 | C28 | C42 | C4⋊C4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 4 | 4 | 3 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of C28.D8 ►in GL6(𝔽113)
| 88 | 24 | 0 | 0 | 0 | 0 |
| 54 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 18 |
| 0 | 0 | 0 | 0 | 91 | 13 |
| 59 | 29 | 0 | 0 | 0 | 0 |
| 28 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 62 | 0 | 0 |
| 0 | 0 | 71 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
| 88 | 1 | 0 | 0 | 0 | 0 |
| 54 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 110 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 18 |
| 0 | 0 | 0 | 0 | 66 | 13 |
G:=sub<GL(6,GF(113))| [88,54,0,0,0,0,24,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,91,0,0,0,0,18,13],[59,28,0,0,0,0,29,54,0,0,0,0,0,0,51,71,0,0,0,0,62,11,0,0,0,0,0,0,98,0,0,0,0,0,0,98],[88,54,0,0,0,0,1,25,0,0,0,0,0,0,112,110,0,0,0,0,0,1,0,0,0,0,0,0,100,66,0,0,0,0,18,13] >;
C28.D8 in GAP, Magma, Sage, TeX
C_{28}.D_8 % in TeX
G:=Group("C28.D8"); // GroupNames label
G:=SmallGroup(448,622);
// by ID
G=gap.SmallGroup(448,622);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,219,100,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=a^14*b^-1>;
// generators/relations