metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊7D8, D4⋊1D28, C42.49D14, (C7×D4)⋊8D4, (C4×D4)⋊3D7, C4⋊3(D4⋊D7), (D4×C28)⋊3C2, C7⋊3(C4⋊D8), C28⋊4D4⋊8C2, C28⋊C8⋊23C2, C28.17(C2×D4), C14.52(C2×D8), C4.13(C2×D28), (C2×C28).60D4, C4⋊C4.243D14, C4.9(C4○D28), C14.D8⋊30C2, (C2×D4).190D14, C28.50(C4○D4), (C4×C28).86C22, C2.8(D4⋊D14), C2.12(C28⋊7D4), C14.64(C4⋊D4), (C2×C28).337C23, (C2×D28).93C22, C14.109(C8⋊C22), (D4×C14).232C22, (C2×D4⋊D7)⋊7C2, C2.7(C2×D4⋊D7), (C2×C7⋊C8).93C22, (C2×C14).468(C2×D4), (C2×C4).245(C7⋊D4), (C7×C4⋊C4).274C22, (C2×C4).437(C22×D7), C22.149(C2×C7⋊D4), SmallGroup(448,549)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊7D8
G = < a,b,c | a28=b8=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 884 in 140 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C28, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C7⋊C8, D28, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C4⋊D8, C2×C7⋊C8, D4⋊D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×D28, C2×D28, C22×C28, D4×C14, C28⋊C8, C14.D8, C28⋊4D4, C2×D4⋊D7, D4×C28, C28⋊7D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C4⋊D4, C2×D8, C8⋊C22, D28, C7⋊D4, C22×D7, C4⋊D8, D4⋊D7, C2×D28, C4○D28, C2×C7⋊D4, C28⋊7D4, C2×D4⋊D7, D4⋊D14, C28⋊7D8
►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 33 124 88 195 147 206 83)(2 32 125 87 196 146 207 82)(3 31 126 86 169 145 208 81)(4 30 127 85 170 144 209 80)(5 29 128 112 171 143 210 79)(6 56 129 111 172 142 211 78)(7 55 130 110 173 141 212 77)(8 54 131 109 174 168 213 76)(9 53 132 108 175 167 214 75)(10 52 133 107 176 166 215 74)(11 51 134 106 177 165 216 73)(12 50 135 105 178 164 217 72)(13 49 136 104 179 163 218 71)(14 48 137 103 180 162 219 70)(15 47 138 102 181 161 220 69)(16 46 139 101 182 160 221 68)(17 45 140 100 183 159 222 67)(18 44 113 99 184 158 223 66)(19 43 114 98 185 157 224 65)(20 42 115 97 186 156 197 64)(21 41 116 96 187 155 198 63)(22 40 117 95 188 154 199 62)(23 39 118 94 189 153 200 61)(24 38 119 93 190 152 201 60)(25 37 120 92 191 151 202 59)(26 36 121 91 192 150 203 58)(27 35 122 90 193 149 204 57)(28 34 123 89 194 148 205 84)
(1 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 70)(29 224)(30 223)(31 222)(32 221)(33 220)(34 219)(35 218)(36 217)(37 216)(38 215)(39 214)(40 213)(41 212)(42 211)(43 210)(44 209)(45 208)(46 207)(47 206)(48 205)(49 204)(50 203)(51 202)(52 201)(53 200)(54 199)(55 198)(56 197)(85 184)(86 183)(87 182)(88 181)(89 180)(90 179)(91 178)(92 177)(93 176)(94 175)(95 174)(96 173)(97 172)(98 171)(99 170)(100 169)(101 196)(102 195)(103 194)(104 193)(105 192)(106 191)(107 190)(108 189)(109 188)(110 187)(111 186)(112 185)(113 144)(114 143)(115 142)(116 141)(117 168)(118 167)(119 166)(120 165)(121 164)(122 163)(123 162)(124 161)(125 160)(126 159)(127 158)(128 157)(129 156)(130 155)(131 154)(132 153)(133 152)(134 151)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)
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), (1,33,124,88,195,147,206,83)(2,32,125,87,196,146,207,82)(3,31,126,86,169,145,208,81)(4,30,127,85,170,144,209,80)(5,29,128,112,171,143,210,79)(6,56,129,111,172,142,211,78)(7,55,130,110,173,141,212,77)(8,54,131,109,174,168,213,76)(9,53,132,108,175,167,214,75)(10,52,133,107,176,166,215,74)(11,51,134,106,177,165,216,73)(12,50,135,105,178,164,217,72)(13,49,136,104,179,163,218,71)(14,48,137,103,180,162,219,70)(15,47,138,102,181,161,220,69)(16,46,139,101,182,160,221,68)(17,45,140,100,183,159,222,67)(18,44,113,99,184,158,223,66)(19,43,114,98,185,157,224,65)(20,42,115,97,186,156,197,64)(21,41,116,96,187,155,198,63)(22,40,117,95,188,154,199,62)(23,39,118,94,189,153,200,61)(24,38,119,93,190,152,201,60)(25,37,120,92,191,151,202,59)(26,36,121,91,192,150,203,58)(27,35,122,90,193,149,204,57)(28,34,123,89,194,148,205,84), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,224)(30,223)(31,222)(32,221)(33,220)(34,219)(35,218)(36,217)(37,216)(38,215)(39,214)(40,213)(41,212)(42,211)(43,210)(44,209)(45,208)(46,207)(47,206)(48,205)(49,204)(50,203)(51,202)(52,201)(53,200)(54,199)(55,198)(56,197)(85,184)(86,183)(87,182)(88,181)(89,180)(90,179)(91,178)(92,177)(93,176)(94,175)(95,174)(96,173)(97,172)(98,171)(99,170)(100,169)(101,196)(102,195)(103,194)(104,193)(105,192)(106,191)(107,190)(108,189)(109,188)(110,187)(111,186)(112,185)(113,144)(114,143)(115,142)(116,141)(117,168)(118,167)(119,166)(120,165)(121,164)(122,163)(123,162)(124,161)(125,160)(126,159)(127,158)(128,157)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)>;
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), (1,33,124,88,195,147,206,83)(2,32,125,87,196,146,207,82)(3,31,126,86,169,145,208,81)(4,30,127,85,170,144,209,80)(5,29,128,112,171,143,210,79)(6,56,129,111,172,142,211,78)(7,55,130,110,173,141,212,77)(8,54,131,109,174,168,213,76)(9,53,132,108,175,167,214,75)(10,52,133,107,176,166,215,74)(11,51,134,106,177,165,216,73)(12,50,135,105,178,164,217,72)(13,49,136,104,179,163,218,71)(14,48,137,103,180,162,219,70)(15,47,138,102,181,161,220,69)(16,46,139,101,182,160,221,68)(17,45,140,100,183,159,222,67)(18,44,113,99,184,158,223,66)(19,43,114,98,185,157,224,65)(20,42,115,97,186,156,197,64)(21,41,116,96,187,155,198,63)(22,40,117,95,188,154,199,62)(23,39,118,94,189,153,200,61)(24,38,119,93,190,152,201,60)(25,37,120,92,191,151,202,59)(26,36,121,91,192,150,203,58)(27,35,122,90,193,149,204,57)(28,34,123,89,194,148,205,84), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,224)(30,223)(31,222)(32,221)(33,220)(34,219)(35,218)(36,217)(37,216)(38,215)(39,214)(40,213)(41,212)(42,211)(43,210)(44,209)(45,208)(46,207)(47,206)(48,205)(49,204)(50,203)(51,202)(52,201)(53,200)(54,199)(55,198)(56,197)(85,184)(86,183)(87,182)(88,181)(89,180)(90,179)(91,178)(92,177)(93,176)(94,175)(95,174)(96,173)(97,172)(98,171)(99,170)(100,169)(101,196)(102,195)(103,194)(104,193)(105,192)(106,191)(107,190)(108,189)(109,188)(110,187)(111,186)(112,185)(113,144)(114,143)(115,142)(116,141)(117,168)(118,167)(119,166)(120,165)(121,164)(122,163)(123,162)(124,161)(125,160)(126,159)(127,158)(128,157)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145) );
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)], [(1,33,124,88,195,147,206,83),(2,32,125,87,196,146,207,82),(3,31,126,86,169,145,208,81),(4,30,127,85,170,144,209,80),(5,29,128,112,171,143,210,79),(6,56,129,111,172,142,211,78),(7,55,130,110,173,141,212,77),(8,54,131,109,174,168,213,76),(9,53,132,108,175,167,214,75),(10,52,133,107,176,166,215,74),(11,51,134,106,177,165,216,73),(12,50,135,105,178,164,217,72),(13,49,136,104,179,163,218,71),(14,48,137,103,180,162,219,70),(15,47,138,102,181,161,220,69),(16,46,139,101,182,160,221,68),(17,45,140,100,183,159,222,67),(18,44,113,99,184,158,223,66),(19,43,114,98,185,157,224,65),(20,42,115,97,186,156,197,64),(21,41,116,96,187,155,198,63),(22,40,117,95,188,154,199,62),(23,39,118,94,189,153,200,61),(24,38,119,93,190,152,201,60),(25,37,120,92,191,151,202,59),(26,36,121,91,192,150,203,58),(27,35,122,90,193,149,204,57),(28,34,123,89,194,148,205,84)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,84),(15,83),(16,82),(17,81),(18,80),(19,79),(20,78),(21,77),(22,76),(23,75),(24,74),(25,73),(26,72),(27,71),(28,70),(29,224),(30,223),(31,222),(32,221),(33,220),(34,219),(35,218),(36,217),(37,216),(38,215),(39,214),(40,213),(41,212),(42,211),(43,210),(44,209),(45,208),(46,207),(47,206),(48,205),(49,204),(50,203),(51,202),(52,201),(53,200),(54,199),(55,198),(56,197),(85,184),(86,183),(87,182),(88,181),(89,180),(90,179),(91,178),(92,177),(93,176),(94,175),(95,174),(96,173),(97,172),(98,171),(99,170),(100,169),(101,196),(102,195),(103,194),(104,193),(105,192),(106,191),(107,190),(108,189),(109,188),(110,187),(111,186),(112,185),(113,144),(114,143),(115,142),(116,141),(117,168),(118,167),(119,166),(120,165),(121,164),(122,163),(123,162),(124,161),(125,160),(126,159),(127,158),(128,157),(129,156),(130,155),(131,154),(132,153),(133,152),(134,151),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 56 | 56 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | D4 | D7 | D8 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D28 | C4○D28 | C8⋊C22 | D4⋊D7 | D4⋊D14 |
| kernel | C28⋊7D8 | C28⋊C8 | C14.D8 | C28⋊4D4 | C2×D4⋊D7 | 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 | 1 | 2 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of C28⋊7D8 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 55 | 32 |
| 0 | 0 | 81 | 4 |
| 82 | 31 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 64 | 13 |
| 0 | 0 | 24 | 49 |
| 82 | 31 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 49 | 100 |
| 0 | 0 | 89 | 64 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,55,81,0,0,32,4],[82,82,0,0,31,82,0,0,0,0,64,24,0,0,13,49],[82,31,0,0,31,31,0,0,0,0,49,89,0,0,100,64] >;
C28⋊7D8 in GAP, Magma, Sage, TeX
C_{28}\rtimes_7D_8 % in TeX
G:=Group("C28:7D8"); // GroupNames label
G:=SmallGroup(448,549);
// by ID
G=gap.SmallGroup(448,549);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations