direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.C42, C28.29C42, M4(2).2C28, (C2×C8).5C28, C4.3(C4×C28), (C2×C56).15C4, (C2×C28).508D4, C23.6(C7×Q8), (C22×C56).6C2, (C22×C8).3C14, (C7×M4(2)).6C4, C14.9(C8.C4), (C22×C14).18Q8, (C2×M4(2)).7C14, C28.104(C22⋊C4), (C14×M4(2)).19C2, (C22×C28).572C22, C14.25(C2.C42), (C2×C4).41(C2×C28), C2.3(C7×C8.C4), (C2×C4).113(C7×D4), C22.17(C7×C4⋊C4), C4.26(C7×C22⋊C4), (C2×C14).60(C4⋊C4), (C2×C28).262(C2×C4), C2.6(C7×C2.C42), (C22×C4).105(C2×C14), SmallGroup(448,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.C42
G = < a,b,c,d | a7=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c >
Subgroups: 122 in 90 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C22×C8, C2×M4(2), C56, C2×C28, C2×C28, C22×C14, C4.C42, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C22×C56, C14×M4(2), C7×C4.C42
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C8.C4, C2×C28, C7×D4, C7×Q8, C4.C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×C8.C4, C7×C4.C42
►On 224 points
Generators in S224
(1 145 33 137 25 129 17)(2 146 34 138 26 130 18)(3 147 35 139 27 131 19)(4 148 36 140 28 132 20)(5 149 37 141 29 133 21)(6 150 38 142 30 134 22)(7 151 39 143 31 135 23)(8 152 40 144 32 136 24)(9 62 166 54 158 46 122)(10 63 167 55 159 47 123)(11 64 168 56 160 48 124)(12 57 161 49 153 41 125)(13 58 162 50 154 42 126)(14 59 163 51 155 43 127)(15 60 164 52 156 44 128)(16 61 165 53 157 45 121)(65 169 193 81 185 73 177)(66 170 194 82 186 74 178)(67 171 195 83 187 75 179)(68 172 196 84 188 76 180)(69 173 197 85 189 77 181)(70 174 198 86 190 78 182)(71 175 199 87 191 79 183)(72 176 200 88 192 80 184)(89 119 217 105 209 97 201)(90 120 218 106 210 98 202)(91 113 219 107 211 99 203)(92 114 220 108 212 100 204)(93 115 221 109 213 101 205)(94 116 222 110 214 102 206)(95 117 223 111 215 103 207)(96 118 224 112 216 104 208)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 47 45 43)(42 44 46 48)(49 55 53 51)(50 52 54 56)(57 63 61 59)(58 60 62 64)(65 71 69 67)(66 68 70 72)(73 79 77 75)(74 76 78 80)(81 87 85 83)(82 84 86 88)(89 95 93 91)(90 92 94 96)(97 103 101 99)(98 100 102 104)(105 111 109 107)(106 108 110 112)(113 119 117 115)(114 116 118 120)(121 127 125 123)(122 124 126 128)(129 135 133 131)(130 132 134 136)(137 143 141 139)(138 140 142 144)(145 151 149 147)(146 148 150 152)(153 159 157 155)(154 156 158 160)(161 167 165 163)(162 164 166 168)(169 175 173 171)(170 172 174 176)(177 183 181 179)(178 180 182 184)(185 191 189 187)(186 188 190 192)(193 199 197 195)(194 196 198 200)(201 207 205 203)(202 204 206 208)(209 215 213 211)(210 212 214 216)(217 223 221 219)(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 91 171 45 5 95 175 41)(2 90 176 48 6 94 172 44)(3 93 173 47 7 89 169 43)(4 92 170 42 8 96 174 46)(9 36 220 82 13 40 224 86)(10 39 217 81 14 35 221 85)(11 38 222 84 15 34 218 88)(12 33 219 83 16 37 223 87)(17 203 67 157 21 207 71 153)(18 202 72 160 22 206 68 156)(19 205 69 159 23 201 65 155)(20 204 66 154 24 208 70 158)(25 211 75 165 29 215 79 161)(26 210 80 168 30 214 76 164)(27 213 77 167 31 209 73 163)(28 212 74 162 32 216 78 166)(49 129 99 179 53 133 103 183)(50 136 104 182 54 132 100 178)(51 131 101 181 55 135 97 177)(52 130 98 184 56 134 102 180)(57 137 107 187 61 141 111 191)(58 144 112 190 62 140 108 186)(59 139 109 189 63 143 105 185)(60 138 106 192 64 142 110 188)(113 195 121 149 117 199 125 145)(114 194 126 152 118 198 122 148)(115 197 123 151 119 193 127 147)(116 196 128 146 120 200 124 150)
G:=sub<Sym(224)| (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,62,166,54,158,46,122)(10,63,167,55,159,47,123)(11,64,168,56,160,48,124)(12,57,161,49,153,41,125)(13,58,162,50,154,42,126)(14,59,163,51,155,43,127)(15,60,164,52,156,44,128)(16,61,165,53,157,45,121)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,119,217,105,209,97,201)(90,120,218,106,210,98,202)(91,113,219,107,211,99,203)(92,114,220,108,212,100,204)(93,115,221,109,213,101,205)(94,116,222,110,214,102,206)(95,117,223,111,215,103,207)(96,118,224,112,216,104,208), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,47,45,43)(42,44,46,48)(49,55,53,51)(50,52,54,56)(57,63,61,59)(58,60,62,64)(65,71,69,67)(66,68,70,72)(73,79,77,75)(74,76,78,80)(81,87,85,83)(82,84,86,88)(89,95,93,91)(90,92,94,96)(97,103,101,99)(98,100,102,104)(105,111,109,107)(106,108,110,112)(113,119,117,115)(114,116,118,120)(121,127,125,123)(122,124,126,128)(129,135,133,131)(130,132,134,136)(137,143,141,139)(138,140,142,144)(145,151,149,147)(146,148,150,152)(153,159,157,155)(154,156,158,160)(161,167,165,163)(162,164,166,168)(169,175,173,171)(170,172,174,176)(177,183,181,179)(178,180,182,184)(185,191,189,187)(186,188,190,192)(193,199,197,195)(194,196,198,200)(201,207,205,203)(202,204,206,208)(209,215,213,211)(210,212,214,216)(217,223,221,219)(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), (1,91,171,45,5,95,175,41)(2,90,176,48,6,94,172,44)(3,93,173,47,7,89,169,43)(4,92,170,42,8,96,174,46)(9,36,220,82,13,40,224,86)(10,39,217,81,14,35,221,85)(11,38,222,84,15,34,218,88)(12,33,219,83,16,37,223,87)(17,203,67,157,21,207,71,153)(18,202,72,160,22,206,68,156)(19,205,69,159,23,201,65,155)(20,204,66,154,24,208,70,158)(25,211,75,165,29,215,79,161)(26,210,80,168,30,214,76,164)(27,213,77,167,31,209,73,163)(28,212,74,162,32,216,78,166)(49,129,99,179,53,133,103,183)(50,136,104,182,54,132,100,178)(51,131,101,181,55,135,97,177)(52,130,98,184,56,134,102,180)(57,137,107,187,61,141,111,191)(58,144,112,190,62,140,108,186)(59,139,109,189,63,143,105,185)(60,138,106,192,64,142,110,188)(113,195,121,149,117,199,125,145)(114,194,126,152,118,198,122,148)(115,197,123,151,119,193,127,147)(116,196,128,146,120,200,124,150)>;
G:=Group( (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,62,166,54,158,46,122)(10,63,167,55,159,47,123)(11,64,168,56,160,48,124)(12,57,161,49,153,41,125)(13,58,162,50,154,42,126)(14,59,163,51,155,43,127)(15,60,164,52,156,44,128)(16,61,165,53,157,45,121)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,119,217,105,209,97,201)(90,120,218,106,210,98,202)(91,113,219,107,211,99,203)(92,114,220,108,212,100,204)(93,115,221,109,213,101,205)(94,116,222,110,214,102,206)(95,117,223,111,215,103,207)(96,118,224,112,216,104,208), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,47,45,43)(42,44,46,48)(49,55,53,51)(50,52,54,56)(57,63,61,59)(58,60,62,64)(65,71,69,67)(66,68,70,72)(73,79,77,75)(74,76,78,80)(81,87,85,83)(82,84,86,88)(89,95,93,91)(90,92,94,96)(97,103,101,99)(98,100,102,104)(105,111,109,107)(106,108,110,112)(113,119,117,115)(114,116,118,120)(121,127,125,123)(122,124,126,128)(129,135,133,131)(130,132,134,136)(137,143,141,139)(138,140,142,144)(145,151,149,147)(146,148,150,152)(153,159,157,155)(154,156,158,160)(161,167,165,163)(162,164,166,168)(169,175,173,171)(170,172,174,176)(177,183,181,179)(178,180,182,184)(185,191,189,187)(186,188,190,192)(193,199,197,195)(194,196,198,200)(201,207,205,203)(202,204,206,208)(209,215,213,211)(210,212,214,216)(217,223,221,219)(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), (1,91,171,45,5,95,175,41)(2,90,176,48,6,94,172,44)(3,93,173,47,7,89,169,43)(4,92,170,42,8,96,174,46)(9,36,220,82,13,40,224,86)(10,39,217,81,14,35,221,85)(11,38,222,84,15,34,218,88)(12,33,219,83,16,37,223,87)(17,203,67,157,21,207,71,153)(18,202,72,160,22,206,68,156)(19,205,69,159,23,201,65,155)(20,204,66,154,24,208,70,158)(25,211,75,165,29,215,79,161)(26,210,80,168,30,214,76,164)(27,213,77,167,31,209,73,163)(28,212,74,162,32,216,78,166)(49,129,99,179,53,133,103,183)(50,136,104,182,54,132,100,178)(51,131,101,181,55,135,97,177)(52,130,98,184,56,134,102,180)(57,137,107,187,61,141,111,191)(58,144,112,190,62,140,108,186)(59,139,109,189,63,143,105,185)(60,138,106,192,64,142,110,188)(113,195,121,149,117,199,125,145)(114,194,126,152,118,198,122,148)(115,197,123,151,119,193,127,147)(116,196,128,146,120,200,124,150) );
G=PermutationGroup([[(1,145,33,137,25,129,17),(2,146,34,138,26,130,18),(3,147,35,139,27,131,19),(4,148,36,140,28,132,20),(5,149,37,141,29,133,21),(6,150,38,142,30,134,22),(7,151,39,143,31,135,23),(8,152,40,144,32,136,24),(9,62,166,54,158,46,122),(10,63,167,55,159,47,123),(11,64,168,56,160,48,124),(12,57,161,49,153,41,125),(13,58,162,50,154,42,126),(14,59,163,51,155,43,127),(15,60,164,52,156,44,128),(16,61,165,53,157,45,121),(65,169,193,81,185,73,177),(66,170,194,82,186,74,178),(67,171,195,83,187,75,179),(68,172,196,84,188,76,180),(69,173,197,85,189,77,181),(70,174,198,86,190,78,182),(71,175,199,87,191,79,183),(72,176,200,88,192,80,184),(89,119,217,105,209,97,201),(90,120,218,106,210,98,202),(91,113,219,107,211,99,203),(92,114,220,108,212,100,204),(93,115,221,109,213,101,205),(94,116,222,110,214,102,206),(95,117,223,111,215,103,207),(96,118,224,112,216,104,208)], [(1,7,5,3),(2,4,6,8),(9,11,13,15),(10,16,14,12),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32),(33,39,37,35),(34,36,38,40),(41,47,45,43),(42,44,46,48),(49,55,53,51),(50,52,54,56),(57,63,61,59),(58,60,62,64),(65,71,69,67),(66,68,70,72),(73,79,77,75),(74,76,78,80),(81,87,85,83),(82,84,86,88),(89,95,93,91),(90,92,94,96),(97,103,101,99),(98,100,102,104),(105,111,109,107),(106,108,110,112),(113,119,117,115),(114,116,118,120),(121,127,125,123),(122,124,126,128),(129,135,133,131),(130,132,134,136),(137,143,141,139),(138,140,142,144),(145,151,149,147),(146,148,150,152),(153,159,157,155),(154,156,158,160),(161,167,165,163),(162,164,166,168),(169,175,173,171),(170,172,174,176),(177,183,181,179),(178,180,182,184),(185,191,189,187),(186,188,190,192),(193,199,197,195),(194,196,198,200),(201,207,205,203),(202,204,206,208),(209,215,213,211),(210,212,214,216),(217,223,221,219),(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)], [(1,91,171,45,5,95,175,41),(2,90,176,48,6,94,172,44),(3,93,173,47,7,89,169,43),(4,92,170,42,8,96,174,46),(9,36,220,82,13,40,224,86),(10,39,217,81,14,35,221,85),(11,38,222,84,15,34,218,88),(12,33,219,83,16,37,223,87),(17,203,67,157,21,207,71,153),(18,202,72,160,22,206,68,156),(19,205,69,159,23,201,65,155),(20,204,66,154,24,208,70,158),(25,211,75,165,29,215,79,161),(26,210,80,168,30,214,76,164),(27,213,77,167,31,209,73,163),(28,212,74,162,32,216,78,166),(49,129,99,179,53,133,103,183),(50,136,104,182,54,132,100,178),(51,131,101,181,55,135,97,177),(52,130,98,184,56,134,102,180),(57,137,107,187,61,141,111,191),(58,144,112,190,62,140,108,186),(59,139,109,189,63,143,105,185),(60,138,106,192,64,142,110,188),(113,195,121,149,117,199,125,145),(114,194,126,152,118,198,122,148),(115,197,123,151,119,193,127,147),(116,196,128,146,120,200,124,150)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56AV | 56AW | ··· | 56CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | D4 | Q8 | C8.C4 | C7×D4 | C7×Q8 | C7×C8.C4 |
| kernel | C7×C4.C42 | C22×C56 | C14×M4(2) | C2×C56 | C7×M4(2) | C4.C42 | C22×C8 | C2×M4(2) | C2×C8 | M4(2) | C2×C28 | C22×C14 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 4 | 8 | 6 | 6 | 12 | 24 | 48 | 3 | 1 | 8 | 18 | 6 | 48 |
Matrix representation of C7×C4.C42 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 1 | 0 | 0 |
| 0 | 15 | 37 |
| 0 | 0 | 98 |
| 98 | 0 | 0 |
| 0 | 76 | 97 |
| 0 | 30 | 37 |
| 15 | 0 | 0 |
| 0 | 44 | 84 |
| 0 | 0 | 95 |
G:=sub<GL(3,GF(113))| [1,0,0,0,28,0,0,0,28],[1,0,0,0,15,0,0,37,98],[98,0,0,0,76,30,0,97,37],[15,0,0,0,44,0,0,84,95] >;
C7×C4.C42 in GAP, Magma, Sage, TeX
C_7\times C_4.C_4^2
% in TeX
G:=Group("C7xC4.C4^2"); // GroupNames label
G:=SmallGroup(448,145);
// by ID
G=gap.SmallGroup(448,145);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,7059,248,172,14117,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations