direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C8.17D4, C56.97D4, C28.63D8, Q16.2C28, M5(2).3C14, C8.3(C2×C28), C8.17(C7×D4), C4.12(C7×D8), C56.44(C2×C4), (C7×Q16).4C4, (C2×C28).281D4, (C2×Q16).5C14, C8.C4.1C14, (C14×Q16).12C2, (C2×C14).25SD16, (C7×M5(2)).7C2, C22.4(C7×SD16), C28.74(C22⋊C4), (C2×C56).267C22, C14.42(D4⋊C4), (C2×C4).12(C7×D4), C4.6(C7×C22⋊C4), (C2×C8).14(C2×C14), (C7×C8.C4).4C2, C2.11(C7×D4⋊C4), SmallGroup(448,166)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.17D4
G = < a,b,c,d | a7=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C7×C8.17D4
►On 224 points
Generators in S224
(1 132 31 64 204 96 66)(2 133 32 49 205 81 67)(3 134 17 50 206 82 68)(4 135 18 51 207 83 69)(5 136 19 52 208 84 70)(6 137 20 53 193 85 71)(7 138 21 54 194 86 72)(8 139 22 55 195 87 73)(9 140 23 56 196 88 74)(10 141 24 57 197 89 75)(11 142 25 58 198 90 76)(12 143 26 59 199 91 77)(13 144 27 60 200 92 78)(14 129 28 61 201 93 79)(15 130 29 62 202 94 80)(16 131 30 63 203 95 65)(33 101 173 187 210 160 124)(34 102 174 188 211 145 125)(35 103 175 189 212 146 126)(36 104 176 190 213 147 127)(37 105 161 191 214 148 128)(38 106 162 192 215 149 113)(39 107 163 177 216 150 114)(40 108 164 178 217 151 115)(41 109 165 179 218 152 116)(42 110 166 180 219 153 117)(43 111 167 181 220 154 118)(44 112 168 182 221 155 119)(45 97 169 183 222 156 120)(46 98 170 184 223 157 121)(47 99 171 185 224 158 122)(48 100 172 186 209 159 123)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)(65 79 77 75 73 71 69 67)(66 80 78 76 74 72 70 68)(81 95 93 91 89 87 85 83)(82 96 94 92 90 88 86 84)(97 111 109 107 105 103 101 99)(98 112 110 108 106 104 102 100)(113 127 125 123 121 119 117 115)(114 128 126 124 122 120 118 116)(129 143 141 139 137 135 133 131)(130 144 142 140 138 136 134 132)(145 159 157 155 153 151 149 147)(146 160 158 156 154 152 150 148)(161 175 173 171 169 167 165 163)(162 176 174 172 170 168 166 164)(177 191 189 187 185 183 181 179)(178 192 190 188 186 184 182 180)(193 207 205 203 201 199 197 195)(194 208 206 204 202 200 198 196)(209 223 221 219 217 215 213 211)(210 224 222 220 218 216 214 212)
(1 99 13 103 9 107 5 111)(2 102 6 98 10 110 14 106)(3 97 15 101 11 105 7 109)(4 100 8 112 12 108 16 104)(17 183 29 187 25 191 21 179)(18 186 22 182 26 178 30 190)(19 181 31 185 27 189 23 177)(20 184 24 180 28 192 32 188)(33 76 37 72 41 68 45 80)(34 71 46 75 42 79 38 67)(35 74 39 70 43 66 47 78)(36 69 48 73 44 77 40 65)(49 211 53 223 57 219 61 215)(50 222 62 210 58 214 54 218)(51 209 55 221 59 217 63 213)(52 220 64 224 60 212 56 216)(81 125 85 121 89 117 93 113)(82 120 94 124 90 128 86 116)(83 123 87 119 91 115 95 127)(84 118 96 122 92 126 88 114)(129 162 133 174 137 170 141 166)(130 173 142 161 138 165 134 169)(131 176 135 172 139 168 143 164)(132 171 144 175 140 163 136 167)(145 193 157 197 153 201 149 205)(146 196 150 208 154 204 158 200)(147 207 159 195 155 199 151 203)(148 194 152 206 156 202 160 198)
(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)
G:=sub<Sym(224)| (1,132,31,64,204,96,66)(2,133,32,49,205,81,67)(3,134,17,50,206,82,68)(4,135,18,51,207,83,69)(5,136,19,52,208,84,70)(6,137,20,53,193,85,71)(7,138,21,54,194,86,72)(8,139,22,55,195,87,73)(9,140,23,56,196,88,74)(10,141,24,57,197,89,75)(11,142,25,58,198,90,76)(12,143,26,59,199,91,77)(13,144,27,60,200,92,78)(14,129,28,61,201,93,79)(15,130,29,62,202,94,80)(16,131,30,63,203,95,65)(33,101,173,187,210,160,124)(34,102,174,188,211,145,125)(35,103,175,189,212,146,126)(36,104,176,190,213,147,127)(37,105,161,191,214,148,128)(38,106,162,192,215,149,113)(39,107,163,177,216,150,114)(40,108,164,178,217,151,115)(41,109,165,179,218,152,116)(42,110,166,180,219,153,117)(43,111,167,181,220,154,118)(44,112,168,182,221,155,119)(45,97,169,183,222,156,120)(46,98,170,184,223,157,121)(47,99,171,185,224,158,122)(48,100,172,186,209,159,123), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84)(97,111,109,107,105,103,101,99)(98,112,110,108,106,104,102,100)(113,127,125,123,121,119,117,115)(114,128,126,124,122,120,118,116)(129,143,141,139,137,135,133,131)(130,144,142,140,138,136,134,132)(145,159,157,155,153,151,149,147)(146,160,158,156,154,152,150,148)(161,175,173,171,169,167,165,163)(162,176,174,172,170,168,166,164)(177,191,189,187,185,183,181,179)(178,192,190,188,186,184,182,180)(193,207,205,203,201,199,197,195)(194,208,206,204,202,200,198,196)(209,223,221,219,217,215,213,211)(210,224,222,220,218,216,214,212), (1,99,13,103,9,107,5,111)(2,102,6,98,10,110,14,106)(3,97,15,101,11,105,7,109)(4,100,8,112,12,108,16,104)(17,183,29,187,25,191,21,179)(18,186,22,182,26,178,30,190)(19,181,31,185,27,189,23,177)(20,184,24,180,28,192,32,188)(33,76,37,72,41,68,45,80)(34,71,46,75,42,79,38,67)(35,74,39,70,43,66,47,78)(36,69,48,73,44,77,40,65)(49,211,53,223,57,219,61,215)(50,222,62,210,58,214,54,218)(51,209,55,221,59,217,63,213)(52,220,64,224,60,212,56,216)(81,125,85,121,89,117,93,113)(82,120,94,124,90,128,86,116)(83,123,87,119,91,115,95,127)(84,118,96,122,92,126,88,114)(129,162,133,174,137,170,141,166)(130,173,142,161,138,165,134,169)(131,176,135,172,139,168,143,164)(132,171,144,175,140,163,136,167)(145,193,157,197,153,201,149,205)(146,196,150,208,154,204,158,200)(147,207,159,195,155,199,151,203)(148,194,152,206,156,202,160,198), (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)>;
G:=Group( (1,132,31,64,204,96,66)(2,133,32,49,205,81,67)(3,134,17,50,206,82,68)(4,135,18,51,207,83,69)(5,136,19,52,208,84,70)(6,137,20,53,193,85,71)(7,138,21,54,194,86,72)(8,139,22,55,195,87,73)(9,140,23,56,196,88,74)(10,141,24,57,197,89,75)(11,142,25,58,198,90,76)(12,143,26,59,199,91,77)(13,144,27,60,200,92,78)(14,129,28,61,201,93,79)(15,130,29,62,202,94,80)(16,131,30,63,203,95,65)(33,101,173,187,210,160,124)(34,102,174,188,211,145,125)(35,103,175,189,212,146,126)(36,104,176,190,213,147,127)(37,105,161,191,214,148,128)(38,106,162,192,215,149,113)(39,107,163,177,216,150,114)(40,108,164,178,217,151,115)(41,109,165,179,218,152,116)(42,110,166,180,219,153,117)(43,111,167,181,220,154,118)(44,112,168,182,221,155,119)(45,97,169,183,222,156,120)(46,98,170,184,223,157,121)(47,99,171,185,224,158,122)(48,100,172,186,209,159,123), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84)(97,111,109,107,105,103,101,99)(98,112,110,108,106,104,102,100)(113,127,125,123,121,119,117,115)(114,128,126,124,122,120,118,116)(129,143,141,139,137,135,133,131)(130,144,142,140,138,136,134,132)(145,159,157,155,153,151,149,147)(146,160,158,156,154,152,150,148)(161,175,173,171,169,167,165,163)(162,176,174,172,170,168,166,164)(177,191,189,187,185,183,181,179)(178,192,190,188,186,184,182,180)(193,207,205,203,201,199,197,195)(194,208,206,204,202,200,198,196)(209,223,221,219,217,215,213,211)(210,224,222,220,218,216,214,212), (1,99,13,103,9,107,5,111)(2,102,6,98,10,110,14,106)(3,97,15,101,11,105,7,109)(4,100,8,112,12,108,16,104)(17,183,29,187,25,191,21,179)(18,186,22,182,26,178,30,190)(19,181,31,185,27,189,23,177)(20,184,24,180,28,192,32,188)(33,76,37,72,41,68,45,80)(34,71,46,75,42,79,38,67)(35,74,39,70,43,66,47,78)(36,69,48,73,44,77,40,65)(49,211,53,223,57,219,61,215)(50,222,62,210,58,214,54,218)(51,209,55,221,59,217,63,213)(52,220,64,224,60,212,56,216)(81,125,85,121,89,117,93,113)(82,120,94,124,90,128,86,116)(83,123,87,119,91,115,95,127)(84,118,96,122,92,126,88,114)(129,162,133,174,137,170,141,166)(130,173,142,161,138,165,134,169)(131,176,135,172,139,168,143,164)(132,171,144,175,140,163,136,167)(145,193,157,197,153,201,149,205)(146,196,150,208,154,204,158,200)(147,207,159,195,155,199,151,203)(148,194,152,206,156,202,160,198), (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) );
G=PermutationGroup([[(1,132,31,64,204,96,66),(2,133,32,49,205,81,67),(3,134,17,50,206,82,68),(4,135,18,51,207,83,69),(5,136,19,52,208,84,70),(6,137,20,53,193,85,71),(7,138,21,54,194,86,72),(8,139,22,55,195,87,73),(9,140,23,56,196,88,74),(10,141,24,57,197,89,75),(11,142,25,58,198,90,76),(12,143,26,59,199,91,77),(13,144,27,60,200,92,78),(14,129,28,61,201,93,79),(15,130,29,62,202,94,80),(16,131,30,63,203,95,65),(33,101,173,187,210,160,124),(34,102,174,188,211,145,125),(35,103,175,189,212,146,126),(36,104,176,190,213,147,127),(37,105,161,191,214,148,128),(38,106,162,192,215,149,113),(39,107,163,177,216,150,114),(40,108,164,178,217,151,115),(41,109,165,179,218,152,116),(42,110,166,180,219,153,117),(43,111,167,181,220,154,118),(44,112,168,182,221,155,119),(45,97,169,183,222,156,120),(46,98,170,184,223,157,121),(47,99,171,185,224,158,122),(48,100,172,186,209,159,123)], [(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20),(33,47,45,43,41,39,37,35),(34,48,46,44,42,40,38,36),(49,63,61,59,57,55,53,51),(50,64,62,60,58,56,54,52),(65,79,77,75,73,71,69,67),(66,80,78,76,74,72,70,68),(81,95,93,91,89,87,85,83),(82,96,94,92,90,88,86,84),(97,111,109,107,105,103,101,99),(98,112,110,108,106,104,102,100),(113,127,125,123,121,119,117,115),(114,128,126,124,122,120,118,116),(129,143,141,139,137,135,133,131),(130,144,142,140,138,136,134,132),(145,159,157,155,153,151,149,147),(146,160,158,156,154,152,150,148),(161,175,173,171,169,167,165,163),(162,176,174,172,170,168,166,164),(177,191,189,187,185,183,181,179),(178,192,190,188,186,184,182,180),(193,207,205,203,201,199,197,195),(194,208,206,204,202,200,198,196),(209,223,221,219,217,215,213,211),(210,224,222,220,218,216,214,212)], [(1,99,13,103,9,107,5,111),(2,102,6,98,10,110,14,106),(3,97,15,101,11,105,7,109),(4,100,8,112,12,108,16,104),(17,183,29,187,25,191,21,179),(18,186,22,182,26,178,30,190),(19,181,31,185,27,189,23,177),(20,184,24,180,28,192,32,188),(33,76,37,72,41,68,45,80),(34,71,46,75,42,79,38,67),(35,74,39,70,43,66,47,78),(36,69,48,73,44,77,40,65),(49,211,53,223,57,219,61,215),(50,222,62,210,58,214,54,218),(51,209,55,221,59,217,63,213),(52,220,64,224,60,212,56,216),(81,125,85,121,89,117,93,113),(82,120,94,124,90,128,86,116),(83,123,87,119,91,115,95,127),(84,118,96,122,92,126,88,114),(129,162,133,174,137,170,141,166),(130,173,142,161,138,165,134,169),(131,176,135,172,139,168,143,164),(132,171,144,175,140,163,136,167),(145,193,157,197,153,201,149,205),(146,196,150,208,154,204,158,200),(147,207,159,195,155,199,151,203),(148,194,152,206,156,202,160,198)], [(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)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14F | 14G | ··· | 14L | 16A | 16B | 16C | 16D | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56L | 56M | ··· | 56R | 56S | ··· | 56AD | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D4 | D4 | D8 | SD16 | C7×D4 | C7×D4 | C7×D8 | C7×SD16 | C8.17D4 | C7×C8.17D4 |
| kernel | C7×C8.17D4 | C7×C8.C4 | C7×M5(2) | C14×Q16 | C7×Q16 | C8.17D4 | C8.C4 | M5(2) | C2×Q16 | Q16 | C56 | C2×C28 | C28 | C2×C14 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C7×C8.17D4 ►in GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 82 | 31 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 9 | 90 | 82 | 82 |
| 23 | 9 | 31 | 82 |
| 53 | 77 | 111 | 0 |
| 63 | 33 | 0 | 111 |
| 101 | 91 | 60 | 36 |
| 53 | 32 | 50 | 80 |
| 89 | 57 | 94 | 58 |
| 68 | 109 | 58 | 19 |
| 1 | 105 | 91 | 65 |
| 75 | 38 | 76 | 50 |
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[82,82,9,23,31,82,90,9,0,0,82,31,0,0,82,82],[53,63,101,53,77,33,91,32,111,0,60,50,0,111,36,80],[89,68,1,75,57,109,105,38,94,58,91,76,58,19,65,50] >;
C7×C8.17D4 in GAP, Magma, Sage, TeX
C_7\times C_8._{17}D_4 % in TeX
G:=Group("C7xC8.17D4"); // GroupNames label
G:=SmallGroup(448,166);
// by ID
G=gap.SmallGroup(448,166);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,1576,3923,5106,136,4911,172,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=1,c^4=b^4,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
Export