direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8xDic7, C56:3C4, C14.3C42, C7:C8:5C4, C7:2(C4xC8), C2.2(C8xD7), C14.3(C2xC8), (C2xC8).10D7, C4.20(C4xD7), C28.40(C2xC4), (C2xC56).11C2, (C2xC4).90D14, C2.2(C4xDic7), C22.8(C4xD7), (C2xDic7).7C4, C4.12(C2xDic7), (C4xDic7).10C2, (C2xC28).104C22, (C2xC7:C8).12C2, (C2xC14).9(C2xC4), SmallGroup(224,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C8xDic7 |
Generators and relations for C8xDic7
G = < a,b,c | a8=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Quotients: C1, C2, C4, C22, C8, C2xC4, D7, C42, C2xC8, Dic7, D14, C4xC8, C4xD7, C2xDic7, C8xD7, C4xDic7, C8xDic7
Smallest permutation representation of C8xDic7
►Regular action on 224 points
Generators in S224
(1 153 19 137 45 40 73 123)(2 154 20 138 46 41 74 124)(3 141 21 139 47 42 75 125)(4 142 22 140 48 29 76 126)(5 143 23 127 49 30 77 113)(6 144 24 128 50 31 78 114)(7 145 25 129 51 32 79 115)(8 146 26 130 52 33 80 116)(9 147 27 131 53 34 81 117)(10 148 28 132 54 35 82 118)(11 149 15 133 55 36 83 119)(12 150 16 134 56 37 84 120)(13 151 17 135 43 38 71 121)(14 152 18 136 44 39 72 122)(57 87 169 103 204 185 163 221)(58 88 170 104 205 186 164 222)(59 89 171 105 206 187 165 223)(60 90 172 106 207 188 166 224)(61 91 173 107 208 189 167 211)(62 92 174 108 209 190 168 212)(63 93 175 109 210 191 155 213)(64 94 176 110 197 192 156 214)(65 95 177 111 198 193 157 215)(66 96 178 112 199 194 158 216)(67 97 179 99 200 195 159 217)(68 98 180 100 201 196 160 218)(69 85 181 101 202 183 161 219)(70 86 182 102 203 184 162 220)
(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)
(1 175 8 182)(2 174 9 181)(3 173 10 180)(4 172 11 179)(5 171 12 178)(6 170 13 177)(7 169 14 176)(15 200 22 207)(16 199 23 206)(17 198 24 205)(18 197 25 204)(19 210 26 203)(20 209 27 202)(21 208 28 201)(29 224 36 217)(30 223 37 216)(31 222 38 215)(32 221 39 214)(33 220 40 213)(34 219 41 212)(35 218 42 211)(43 157 50 164)(44 156 51 163)(45 155 52 162)(46 168 53 161)(47 167 54 160)(48 166 55 159)(49 165 56 158)(57 72 64 79)(58 71 65 78)(59 84 66 77)(60 83 67 76)(61 82 68 75)(62 81 69 74)(63 80 70 73)(85 124 92 117)(86 123 93 116)(87 122 94 115)(88 121 95 114)(89 120 96 113)(90 119 97 126)(91 118 98 125)(99 142 106 149)(100 141 107 148)(101 154 108 147)(102 153 109 146)(103 152 110 145)(104 151 111 144)(105 150 112 143)(127 187 134 194)(128 186 135 193)(129 185 136 192)(130 184 137 191)(131 183 138 190)(132 196 139 189)(133 195 140 188)
G:=sub<Sym(224)| (1,153,19,137,45,40,73,123)(2,154,20,138,46,41,74,124)(3,141,21,139,47,42,75,125)(4,142,22,140,48,29,76,126)(5,143,23,127,49,30,77,113)(6,144,24,128,50,31,78,114)(7,145,25,129,51,32,79,115)(8,146,26,130,52,33,80,116)(9,147,27,131,53,34,81,117)(10,148,28,132,54,35,82,118)(11,149,15,133,55,36,83,119)(12,150,16,134,56,37,84,120)(13,151,17,135,43,38,71,121)(14,152,18,136,44,39,72,122)(57,87,169,103,204,185,163,221)(58,88,170,104,205,186,164,222)(59,89,171,105,206,187,165,223)(60,90,172,106,207,188,166,224)(61,91,173,107,208,189,167,211)(62,92,174,108,209,190,168,212)(63,93,175,109,210,191,155,213)(64,94,176,110,197,192,156,214)(65,95,177,111,198,193,157,215)(66,96,178,112,199,194,158,216)(67,97,179,99,200,195,159,217)(68,98,180,100,201,196,160,218)(69,85,181,101,202,183,161,219)(70,86,182,102,203,184,162,220), (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), (1,175,8,182)(2,174,9,181)(3,173,10,180)(4,172,11,179)(5,171,12,178)(6,170,13,177)(7,169,14,176)(15,200,22,207)(16,199,23,206)(17,198,24,205)(18,197,25,204)(19,210,26,203)(20,209,27,202)(21,208,28,201)(29,224,36,217)(30,223,37,216)(31,222,38,215)(32,221,39,214)(33,220,40,213)(34,219,41,212)(35,218,42,211)(43,157,50,164)(44,156,51,163)(45,155,52,162)(46,168,53,161)(47,167,54,160)(48,166,55,159)(49,165,56,158)(57,72,64,79)(58,71,65,78)(59,84,66,77)(60,83,67,76)(61,82,68,75)(62,81,69,74)(63,80,70,73)(85,124,92,117)(86,123,93,116)(87,122,94,115)(88,121,95,114)(89,120,96,113)(90,119,97,126)(91,118,98,125)(99,142,106,149)(100,141,107,148)(101,154,108,147)(102,153,109,146)(103,152,110,145)(104,151,111,144)(105,150,112,143)(127,187,134,194)(128,186,135,193)(129,185,136,192)(130,184,137,191)(131,183,138,190)(132,196,139,189)(133,195,140,188)>;
G:=Group( (1,153,19,137,45,40,73,123)(2,154,20,138,46,41,74,124)(3,141,21,139,47,42,75,125)(4,142,22,140,48,29,76,126)(5,143,23,127,49,30,77,113)(6,144,24,128,50,31,78,114)(7,145,25,129,51,32,79,115)(8,146,26,130,52,33,80,116)(9,147,27,131,53,34,81,117)(10,148,28,132,54,35,82,118)(11,149,15,133,55,36,83,119)(12,150,16,134,56,37,84,120)(13,151,17,135,43,38,71,121)(14,152,18,136,44,39,72,122)(57,87,169,103,204,185,163,221)(58,88,170,104,205,186,164,222)(59,89,171,105,206,187,165,223)(60,90,172,106,207,188,166,224)(61,91,173,107,208,189,167,211)(62,92,174,108,209,190,168,212)(63,93,175,109,210,191,155,213)(64,94,176,110,197,192,156,214)(65,95,177,111,198,193,157,215)(66,96,178,112,199,194,158,216)(67,97,179,99,200,195,159,217)(68,98,180,100,201,196,160,218)(69,85,181,101,202,183,161,219)(70,86,182,102,203,184,162,220), (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), (1,175,8,182)(2,174,9,181)(3,173,10,180)(4,172,11,179)(5,171,12,178)(6,170,13,177)(7,169,14,176)(15,200,22,207)(16,199,23,206)(17,198,24,205)(18,197,25,204)(19,210,26,203)(20,209,27,202)(21,208,28,201)(29,224,36,217)(30,223,37,216)(31,222,38,215)(32,221,39,214)(33,220,40,213)(34,219,41,212)(35,218,42,211)(43,157,50,164)(44,156,51,163)(45,155,52,162)(46,168,53,161)(47,167,54,160)(48,166,55,159)(49,165,56,158)(57,72,64,79)(58,71,65,78)(59,84,66,77)(60,83,67,76)(61,82,68,75)(62,81,69,74)(63,80,70,73)(85,124,92,117)(86,123,93,116)(87,122,94,115)(88,121,95,114)(89,120,96,113)(90,119,97,126)(91,118,98,125)(99,142,106,149)(100,141,107,148)(101,154,108,147)(102,153,109,146)(103,152,110,145)(104,151,111,144)(105,150,112,143)(127,187,134,194)(128,186,135,193)(129,185,136,192)(130,184,137,191)(131,183,138,190)(132,196,139,189)(133,195,140,188) );
G=PermutationGroup([[(1,153,19,137,45,40,73,123),(2,154,20,138,46,41,74,124),(3,141,21,139,47,42,75,125),(4,142,22,140,48,29,76,126),(5,143,23,127,49,30,77,113),(6,144,24,128,50,31,78,114),(7,145,25,129,51,32,79,115),(8,146,26,130,52,33,80,116),(9,147,27,131,53,34,81,117),(10,148,28,132,54,35,82,118),(11,149,15,133,55,36,83,119),(12,150,16,134,56,37,84,120),(13,151,17,135,43,38,71,121),(14,152,18,136,44,39,72,122),(57,87,169,103,204,185,163,221),(58,88,170,104,205,186,164,222),(59,89,171,105,206,187,165,223),(60,90,172,106,207,188,166,224),(61,91,173,107,208,189,167,211),(62,92,174,108,209,190,168,212),(63,93,175,109,210,191,155,213),(64,94,176,110,197,192,156,214),(65,95,177,111,198,193,157,215),(66,96,178,112,199,194,158,216),(67,97,179,99,200,195,159,217),(68,98,180,100,201,196,160,218),(69,85,181,101,202,183,161,219),(70,86,182,102,203,184,162,220)], [(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)], [(1,175,8,182),(2,174,9,181),(3,173,10,180),(4,172,11,179),(5,171,12,178),(6,170,13,177),(7,169,14,176),(15,200,22,207),(16,199,23,206),(17,198,24,205),(18,197,25,204),(19,210,26,203),(20,209,27,202),(21,208,28,201),(29,224,36,217),(30,223,37,216),(31,222,38,215),(32,221,39,214),(33,220,40,213),(34,219,41,212),(35,218,42,211),(43,157,50,164),(44,156,51,163),(45,155,52,162),(46,168,53,161),(47,167,54,160),(48,166,55,159),(49,165,56,158),(57,72,64,79),(58,71,65,78),(59,84,66,77),(60,83,67,76),(61,82,68,75),(62,81,69,74),(63,80,70,73),(85,124,92,117),(86,123,93,116),(87,122,94,115),(88,121,95,114),(89,120,96,113),(90,119,97,126),(91,118,98,125),(99,142,106,149),(100,141,107,148),(101,154,108,147),(102,153,109,146),(103,152,110,145),(104,151,111,144),(105,150,112,143),(127,187,134,194),(128,186,135,193),(129,185,136,192),(130,184,137,191),(131,183,138,190),(132,196,139,189),(133,195,140,188)]])
C8xDic7 is a maximal subgroup of
Dic7:C16 C112:9C4 C56.9Q8 D7xC4xC8 D14.C42 C56:Q8 Dic7.C42 D14.4C42 Dic7.5M4(2) C56:C4:C2 C7:D4:C8 Dic7:M4(2) Dic7:4D8 Dic7:6SD16 Dic7.SD16 (C8xDic7):C2 Dic7:7SD16 Dic7:4Q16 Dic7.1Q16 Q8:Dic7:C2 C42.27D14 Dic14:C8 C42.200D14 C42.31D14 Dic7:8SD16 C56:5Q8 C56.8Q8 Dic7:5D8 Dic28:6C4 C56:2Q8 C56.4Q8 D56:7C4 C28.12C42 C28.7C42 C56:18D4 C56.93D4 C56:5D4 C56.22D4 C56.43D4 C56:15D4 C56.26D4 C56.28D4 D8:5Dic7
C8xDic7 is a maximal quotient of
C42.279D14 C112:9C4 (C2xC56):5C4
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D7 | Dic7 | D14 | C4xD7 | C4xD7 | C8xD7 |
| kernel | C8xDic7 | C2xC7:C8 | C4xDic7 | C2xC56 | C7:C8 | C56 | C2xDic7 | Dic7 | C2xC8 | C8 | C2xC4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 3 | 6 | 3 | 6 | 6 | 24 |
Matrix representation of C8xDic7 ►in GL3(F113) generated by
| 15 | 0 | 0 |
| 0 | 95 | 0 |
| 0 | 0 | 95 |
| 112 | 0 | 0 |
| 0 | 0 | 112 |
| 0 | 1 | 34 |
| 15 | 0 | 0 |
| 0 | 104 | 66 |
| 0 | 33 | 9 |
G:=sub<GL(3,GF(113))| [15,0,0,0,95,0,0,0,95],[112,0,0,0,0,1,0,112,34],[15,0,0,0,104,33,0,66,9] >;
C8xDic7 in GAP, Magma, Sage, TeX
C_8\times {\rm Dic}_7 % in TeX
G:=Group("C8xDic7"); // GroupNames label
G:=SmallGroup(224,19);
// by ID
G=gap.SmallGroup(224,19);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,55,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^8=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export