direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C8.4Q8, C112.5C4, C16.1C28, C28.70D8, C56.20Q8, C8.5(C7×Q8), C4.19(C7×D8), C56.83(C2×C4), (C2×C16).5C14, C8.15(C2×C28), C28.58(C4⋊C4), (C2×C14).6Q16, (C2×C112).11C2, (C2×C28).410D4, C8.C4.3C14, C22.1(C7×Q16), C14.15(C2.D8), (C2×C56).431C22, C4.9(C7×C4⋊C4), C2.5(C7×C2.D8), (C2×C4).64(C7×D4), (C2×C8).89(C2×C14), (C7×C8.C4).6C2, SmallGroup(448,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.4Q8
G = < a,b,c,d | a7=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >
Smallest permutation representation of C7×C8.4Q8
►On 224 points
Generators in S224
(1 207 118 165 17 180 147)(2 208 119 166 18 181 148)(3 193 120 167 19 182 149)(4 194 121 168 20 183 150)(5 195 122 169 21 184 151)(6 196 123 170 22 185 152)(7 197 124 171 23 186 153)(8 198 125 172 24 187 154)(9 199 126 173 25 188 155)(10 200 127 174 26 189 156)(11 201 128 175 27 190 157)(12 202 113 176 28 191 158)(13 203 114 161 29 192 159)(14 204 115 162 30 177 160)(15 205 116 163 31 178 145)(16 206 117 164 32 179 146)(33 53 136 70 213 89 107)(34 54 137 71 214 90 108)(35 55 138 72 215 91 109)(36 56 139 73 216 92 110)(37 57 140 74 217 93 111)(38 58 141 75 218 94 112)(39 59 142 76 219 95 97)(40 60 143 77 220 96 98)(41 61 144 78 221 81 99)(42 62 129 79 222 82 100)(43 63 130 80 223 83 101)(44 64 131 65 224 84 102)(45 49 132 66 209 85 103)(46 50 133 67 210 86 104)(47 51 134 68 211 87 105)(48 52 135 69 212 88 106)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)(33 35 37 39 41 43 45 47)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)(65 67 69 71 73 75 77 79)(66 68 70 72 74 76 78 80)(81 83 85 87 89 91 93 95)(82 84 86 88 90 92 94 96)(97 99 101 103 105 107 109 111)(98 100 102 104 106 108 110 112)(113 123 117 127 121 115 125 119)(114 124 118 128 122 116 126 120)(129 131 133 135 137 139 141 143)(130 132 134 136 138 140 142 144)(145 155 149 159 153 147 157 151)(146 156 150 160 154 148 158 152)(161 171 165 175 169 163 173 167)(162 172 166 176 170 164 174 168)(177 187 181 191 185 179 189 183)(178 188 182 192 186 180 190 184)(193 203 197 207 201 195 205 199)(194 204 198 208 202 196 206 200)(209 211 213 215 217 219 221 223)(210 212 214 216 218 220 222 224)
(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 63 13 51 9 55 5 59)(2 62 14 50 10 54 6 58)(3 61 15 49 11 53 7 57)(4 60 16 64 12 52 8 56)(17 83 29 87 25 91 21 95)(18 82 30 86 26 90 22 94)(19 81 31 85 27 89 23 93)(20 96 32 84 28 88 24 92)(33 153 37 149 41 145 45 157)(34 152 38 148 42 160 46 156)(35 151 39 147 43 159 47 155)(36 150 40 146 44 158 48 154)(65 113 69 125 73 121 77 117)(66 128 70 124 74 120 78 116)(67 127 71 123 75 119 79 115)(68 126 72 122 76 118 80 114)(97 180 101 192 105 188 109 184)(98 179 102 191 106 187 110 183)(99 178 103 190 107 186 111 182)(100 177 104 189 108 185 112 181)(129 204 133 200 137 196 141 208)(130 203 134 199 138 195 142 207)(131 202 135 198 139 194 143 206)(132 201 136 197 140 193 144 205)(161 211 173 215 169 219 165 223)(162 210 174 214 170 218 166 222)(163 209 175 213 171 217 167 221)(164 224 176 212 172 216 168 220)
G:=sub<Sym(224)| (1,207,118,165,17,180,147)(2,208,119,166,18,181,148)(3,193,120,167,19,182,149)(4,194,121,168,20,183,150)(5,195,122,169,21,184,151)(6,196,123,170,22,185,152)(7,197,124,171,23,186,153)(8,198,125,172,24,187,154)(9,199,126,173,25,188,155)(10,200,127,174,26,189,156)(11,201,128,175,27,190,157)(12,202,113,176,28,191,158)(13,203,114,161,29,192,159)(14,204,115,162,30,177,160)(15,205,116,163,31,178,145)(16,206,117,164,32,179,146)(33,53,136,70,213,89,107)(34,54,137,71,214,90,108)(35,55,138,72,215,91,109)(36,56,139,73,216,92,110)(37,57,140,74,217,93,111)(38,58,141,75,218,94,112)(39,59,142,76,219,95,97)(40,60,143,77,220,96,98)(41,61,144,78,221,81,99)(42,62,129,79,222,82,100)(43,63,130,80,223,83,101)(44,64,131,65,224,84,102)(45,49,132,66,209,85,103)(46,50,133,67,210,86,104)(47,51,134,68,211,87,105)(48,52,135,69,212,88,106), (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,68,70,72,74,76,78,80)(81,83,85,87,89,91,93,95)(82,84,86,88,90,92,94,96)(97,99,101,103,105,107,109,111)(98,100,102,104,106,108,110,112)(113,123,117,127,121,115,125,119)(114,124,118,128,122,116,126,120)(129,131,133,135,137,139,141,143)(130,132,134,136,138,140,142,144)(145,155,149,159,153,147,157,151)(146,156,150,160,154,148,158,152)(161,171,165,175,169,163,173,167)(162,172,166,176,170,164,174,168)(177,187,181,191,185,179,189,183)(178,188,182,192,186,180,190,184)(193,203,197,207,201,195,205,199)(194,204,198,208,202,196,206,200)(209,211,213,215,217,219,221,223)(210,212,214,216,218,220,222,224), (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), (1,63,13,51,9,55,5,59)(2,62,14,50,10,54,6,58)(3,61,15,49,11,53,7,57)(4,60,16,64,12,52,8,56)(17,83,29,87,25,91,21,95)(18,82,30,86,26,90,22,94)(19,81,31,85,27,89,23,93)(20,96,32,84,28,88,24,92)(33,153,37,149,41,145,45,157)(34,152,38,148,42,160,46,156)(35,151,39,147,43,159,47,155)(36,150,40,146,44,158,48,154)(65,113,69,125,73,121,77,117)(66,128,70,124,74,120,78,116)(67,127,71,123,75,119,79,115)(68,126,72,122,76,118,80,114)(97,180,101,192,105,188,109,184)(98,179,102,191,106,187,110,183)(99,178,103,190,107,186,111,182)(100,177,104,189,108,185,112,181)(129,204,133,200,137,196,141,208)(130,203,134,199,138,195,142,207)(131,202,135,198,139,194,143,206)(132,201,136,197,140,193,144,205)(161,211,173,215,169,219,165,223)(162,210,174,214,170,218,166,222)(163,209,175,213,171,217,167,221)(164,224,176,212,172,216,168,220)>;
G:=Group( (1,207,118,165,17,180,147)(2,208,119,166,18,181,148)(3,193,120,167,19,182,149)(4,194,121,168,20,183,150)(5,195,122,169,21,184,151)(6,196,123,170,22,185,152)(7,197,124,171,23,186,153)(8,198,125,172,24,187,154)(9,199,126,173,25,188,155)(10,200,127,174,26,189,156)(11,201,128,175,27,190,157)(12,202,113,176,28,191,158)(13,203,114,161,29,192,159)(14,204,115,162,30,177,160)(15,205,116,163,31,178,145)(16,206,117,164,32,179,146)(33,53,136,70,213,89,107)(34,54,137,71,214,90,108)(35,55,138,72,215,91,109)(36,56,139,73,216,92,110)(37,57,140,74,217,93,111)(38,58,141,75,218,94,112)(39,59,142,76,219,95,97)(40,60,143,77,220,96,98)(41,61,144,78,221,81,99)(42,62,129,79,222,82,100)(43,63,130,80,223,83,101)(44,64,131,65,224,84,102)(45,49,132,66,209,85,103)(46,50,133,67,210,86,104)(47,51,134,68,211,87,105)(48,52,135,69,212,88,106), (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,68,70,72,74,76,78,80)(81,83,85,87,89,91,93,95)(82,84,86,88,90,92,94,96)(97,99,101,103,105,107,109,111)(98,100,102,104,106,108,110,112)(113,123,117,127,121,115,125,119)(114,124,118,128,122,116,126,120)(129,131,133,135,137,139,141,143)(130,132,134,136,138,140,142,144)(145,155,149,159,153,147,157,151)(146,156,150,160,154,148,158,152)(161,171,165,175,169,163,173,167)(162,172,166,176,170,164,174,168)(177,187,181,191,185,179,189,183)(178,188,182,192,186,180,190,184)(193,203,197,207,201,195,205,199)(194,204,198,208,202,196,206,200)(209,211,213,215,217,219,221,223)(210,212,214,216,218,220,222,224), (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), (1,63,13,51,9,55,5,59)(2,62,14,50,10,54,6,58)(3,61,15,49,11,53,7,57)(4,60,16,64,12,52,8,56)(17,83,29,87,25,91,21,95)(18,82,30,86,26,90,22,94)(19,81,31,85,27,89,23,93)(20,96,32,84,28,88,24,92)(33,153,37,149,41,145,45,157)(34,152,38,148,42,160,46,156)(35,151,39,147,43,159,47,155)(36,150,40,146,44,158,48,154)(65,113,69,125,73,121,77,117)(66,128,70,124,74,120,78,116)(67,127,71,123,75,119,79,115)(68,126,72,122,76,118,80,114)(97,180,101,192,105,188,109,184)(98,179,102,191,106,187,110,183)(99,178,103,190,107,186,111,182)(100,177,104,189,108,185,112,181)(129,204,133,200,137,196,141,208)(130,203,134,199,138,195,142,207)(131,202,135,198,139,194,143,206)(132,201,136,197,140,193,144,205)(161,211,173,215,169,219,165,223)(162,210,174,214,170,218,166,222)(163,209,175,213,171,217,167,221)(164,224,176,212,172,216,168,220) );
G=PermutationGroup([[(1,207,118,165,17,180,147),(2,208,119,166,18,181,148),(3,193,120,167,19,182,149),(4,194,121,168,20,183,150),(5,195,122,169,21,184,151),(6,196,123,170,22,185,152),(7,197,124,171,23,186,153),(8,198,125,172,24,187,154),(9,199,126,173,25,188,155),(10,200,127,174,26,189,156),(11,201,128,175,27,190,157),(12,202,113,176,28,191,158),(13,203,114,161,29,192,159),(14,204,115,162,30,177,160),(15,205,116,163,31,178,145),(16,206,117,164,32,179,146),(33,53,136,70,213,89,107),(34,54,137,71,214,90,108),(35,55,138,72,215,91,109),(36,56,139,73,216,92,110),(37,57,140,74,217,93,111),(38,58,141,75,218,94,112),(39,59,142,76,219,95,97),(40,60,143,77,220,96,98),(41,61,144,78,221,81,99),(42,62,129,79,222,82,100),(43,63,130,80,223,83,101),(44,64,131,65,224,84,102),(45,49,132,66,209,85,103),(46,50,133,67,210,86,104),(47,51,134,68,211,87,105),(48,52,135,69,212,88,106)], [(1,11,5,15,9,3,13,7),(2,12,6,16,10,4,14,8),(17,27,21,31,25,19,29,23),(18,28,22,32,26,20,30,24),(33,35,37,39,41,43,45,47),(34,36,38,40,42,44,46,48),(49,51,53,55,57,59,61,63),(50,52,54,56,58,60,62,64),(65,67,69,71,73,75,77,79),(66,68,70,72,74,76,78,80),(81,83,85,87,89,91,93,95),(82,84,86,88,90,92,94,96),(97,99,101,103,105,107,109,111),(98,100,102,104,106,108,110,112),(113,123,117,127,121,115,125,119),(114,124,118,128,122,116,126,120),(129,131,133,135,137,139,141,143),(130,132,134,136,138,140,142,144),(145,155,149,159,153,147,157,151),(146,156,150,160,154,148,158,152),(161,171,165,175,169,163,173,167),(162,172,166,176,170,164,174,168),(177,187,181,191,185,179,189,183),(178,188,182,192,186,180,190,184),(193,203,197,207,201,195,205,199),(194,204,198,208,202,196,206,200),(209,211,213,215,217,219,221,223),(210,212,214,216,218,220,222,224)], [(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)], [(1,63,13,51,9,55,5,59),(2,62,14,50,10,54,6,58),(3,61,15,49,11,53,7,57),(4,60,16,64,12,52,8,56),(17,83,29,87,25,91,21,95),(18,82,30,86,26,90,22,94),(19,81,31,85,27,89,23,93),(20,96,32,84,28,88,24,92),(33,153,37,149,41,145,45,157),(34,152,38,148,42,160,46,156),(35,151,39,147,43,159,47,155),(36,150,40,146,44,158,48,154),(65,113,69,125,73,121,77,117),(66,128,70,124,74,120,78,116),(67,127,71,123,75,119,79,115),(68,126,72,122,76,118,80,114),(97,180,101,192,105,188,109,184),(98,179,102,191,106,187,110,183),(99,178,103,190,107,186,111,182),(100,177,104,189,108,185,112,181),(129,204,133,200,137,196,141,208),(130,203,134,199,138,195,142,207),(131,202,135,198,139,194,143,206),(132,201,136,197,140,193,144,205),(161,211,173,215,169,219,165,223),(162,210,174,214,170,218,166,222),(163,209,175,213,171,217,167,221),(164,224,176,212,172,216,168,220)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14F | 14G | ··· | 14L | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X | 56Y | ··· | 56AV | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | Q8 | D4 | D8 | Q16 | C7×Q8 | C7×D4 | C8.4Q8 | C7×D8 | C7×Q16 | C7×C8.4Q8 |
| kernel | C7×C8.4Q8 | C7×C8.C4 | C2×C112 | C112 | C8.4Q8 | C8.C4 | C2×C16 | C16 | C56 | C2×C28 | C28 | C2×C14 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 1 | 1 | 2 | 2 | 6 | 6 | 8 | 12 | 12 | 48 |
Matrix representation of C7×C8.4Q8 ►in GL2(𝔽113) generated by
| 109 | 0 |
| 0 | 109 |
| 18 | 107 |
| 0 | 69 |
| 78 | 54 |
| 0 | 71 |
| 70 | 31 |
| 83 | 43 |
G:=sub<GL(2,GF(113))| [109,0,0,109],[18,0,107,69],[78,0,54,71],[70,83,31,43] >;
C7×C8.4Q8 in GAP, Magma, Sage, TeX
C_7\times C_8._4Q_8
% in TeX
G:=Group("C7xC8.4Q8"); // GroupNames label
G:=SmallGroup(448,172);
// by ID
G=gap.SmallGroup(448,172);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,988,3923,360,172,14117,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^6*c^3>;
// generators/relations
Export