direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4⋊Q8, C28⋊4Q8, C28.40D4, C42.5C14, C4⋊(C7×Q8), C4.5(C7×D4), C4⋊C4.5C14, C2.5(Q8×C14), (C4×C28).11C2, C14.73(C2×D4), C2.10(D4×C14), (C2×Q8).3C14, (Q8×C14).8C2, C14.22(C2×Q8), (C2×C14).83C23, (C2×C28).126C22, C22.18(C22×C14), (C7×C4⋊C4).12C2, (C2×C4).9(C2×C14), SmallGroup(224,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4⋊Q8
G = < a,b,c,d | a7=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 84 in 68 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×Q8, C28, C28, C2×C14, C4⋊Q8, C2×C28, C2×C28, C7×Q8, C4×C28, C7×C4⋊C4, Q8×C14, C7×C4⋊Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C2×C14, C4⋊Q8, C7×D4, C7×Q8, C22×C14, D4×C14, Q8×C14, C7×C4⋊Q8
►Regular action 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)
(1 106 43 119)(2 107 44 113)(3 108 45 114)(4 109 46 115)(5 110 47 116)(6 111 48 117)(7 112 49 118)(8 161 224 149)(9 155 218 150)(10 156 219 151)(11 157 220 152)(12 158 221 153)(13 159 222 154)(14 160 223 148)(15 134 24 147)(16 135 25 141)(17 136 26 142)(18 137 27 143)(19 138 28 144)(20 139 22 145)(21 140 23 146)(29 128 40 123)(30 129 41 124)(31 130 42 125)(32 131 36 126)(33 132 37 120)(34 133 38 121)(35 127 39 122)(50 91 63 78)(51 85 57 79)(52 86 58 80)(53 87 59 81)(54 88 60 82)(55 89 61 83)(56 90 62 84)(64 104 76 92)(65 105 77 93)(66 99 71 94)(67 100 72 95)(68 101 73 96)(69 102 74 97)(70 103 75 98)(162 190 175 203)(163 191 169 197)(164 192 170 198)(165 193 171 199)(166 194 172 200)(167 195 173 201)(168 196 174 202)(176 204 188 216)(177 205 189 217)(178 206 183 211)(179 207 184 212)(180 208 185 213)(181 209 186 214)(182 210 187 215)
(1 78 39 94)(2 79 40 95)(3 80 41 96)(4 81 42 97)(5 82 36 98)(6 83 37 92)(7 84 38 93)(8 189 23 174)(9 183 24 175)(10 184 25 169)(11 185 26 170)(12 186 27 171)(13 187 28 172)(14 188 22 173)(15 162 218 178)(16 163 219 179)(17 164 220 180)(18 165 221 181)(19 166 222 182)(20 167 223 176)(21 168 224 177)(29 100 44 85)(30 101 45 86)(31 102 46 87)(32 103 47 88)(33 104 48 89)(34 105 49 90)(35 99 43 91)(50 122 66 106)(51 123 67 107)(52 124 68 108)(53 125 69 109)(54 126 70 110)(55 120 64 111)(56 121 65 112)(57 128 72 113)(58 129 73 114)(59 130 74 115)(60 131 75 116)(61 132 76 117)(62 133 77 118)(63 127 71 119)(134 190 150 206)(135 191 151 207)(136 192 152 208)(137 193 153 209)(138 194 154 210)(139 195 148 204)(140 196 149 205)(141 197 156 212)(142 198 157 213)(143 199 158 214)(144 200 159 215)(145 201 160 216)(146 202 161 217)(147 203 155 211)
(1 134 39 150)(2 135 40 151)(3 136 41 152)(4 137 42 153)(5 138 36 154)(6 139 37 148)(7 140 38 149)(8 118 23 133)(9 119 24 127)(10 113 25 128)(11 114 26 129)(12 115 27 130)(13 116 28 131)(14 117 22 132)(15 122 218 106)(16 123 219 107)(17 124 220 108)(18 125 221 109)(19 126 222 110)(20 120 223 111)(21 121 224 112)(29 156 44 141)(30 157 45 142)(31 158 46 143)(32 159 47 144)(33 160 48 145)(34 161 49 146)(35 155 43 147)(50 178 66 162)(51 179 67 163)(52 180 68 164)(53 181 69 165)(54 182 70 166)(55 176 64 167)(56 177 65 168)(57 184 72 169)(58 185 73 170)(59 186 74 171)(60 187 75 172)(61 188 76 173)(62 189 77 174)(63 183 71 175)(78 206 94 190)(79 207 95 191)(80 208 96 192)(81 209 97 193)(82 210 98 194)(83 204 92 195)(84 205 93 196)(85 212 100 197)(86 213 101 198)(87 214 102 199)(88 215 103 200)(89 216 104 201)(90 217 105 202)(91 211 99 203)
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), (1,106,43,119)(2,107,44,113)(3,108,45,114)(4,109,46,115)(5,110,47,116)(6,111,48,117)(7,112,49,118)(8,161,224,149)(9,155,218,150)(10,156,219,151)(11,157,220,152)(12,158,221,153)(13,159,222,154)(14,160,223,148)(15,134,24,147)(16,135,25,141)(17,136,26,142)(18,137,27,143)(19,138,28,144)(20,139,22,145)(21,140,23,146)(29,128,40,123)(30,129,41,124)(31,130,42,125)(32,131,36,126)(33,132,37,120)(34,133,38,121)(35,127,39,122)(50,91,63,78)(51,85,57,79)(52,86,58,80)(53,87,59,81)(54,88,60,82)(55,89,61,83)(56,90,62,84)(64,104,76,92)(65,105,77,93)(66,99,71,94)(67,100,72,95)(68,101,73,96)(69,102,74,97)(70,103,75,98)(162,190,175,203)(163,191,169,197)(164,192,170,198)(165,193,171,199)(166,194,172,200)(167,195,173,201)(168,196,174,202)(176,204,188,216)(177,205,189,217)(178,206,183,211)(179,207,184,212)(180,208,185,213)(181,209,186,214)(182,210,187,215), (1,78,39,94)(2,79,40,95)(3,80,41,96)(4,81,42,97)(5,82,36,98)(6,83,37,92)(7,84,38,93)(8,189,23,174)(9,183,24,175)(10,184,25,169)(11,185,26,170)(12,186,27,171)(13,187,28,172)(14,188,22,173)(15,162,218,178)(16,163,219,179)(17,164,220,180)(18,165,221,181)(19,166,222,182)(20,167,223,176)(21,168,224,177)(29,100,44,85)(30,101,45,86)(31,102,46,87)(32,103,47,88)(33,104,48,89)(34,105,49,90)(35,99,43,91)(50,122,66,106)(51,123,67,107)(52,124,68,108)(53,125,69,109)(54,126,70,110)(55,120,64,111)(56,121,65,112)(57,128,72,113)(58,129,73,114)(59,130,74,115)(60,131,75,116)(61,132,76,117)(62,133,77,118)(63,127,71,119)(134,190,150,206)(135,191,151,207)(136,192,152,208)(137,193,153,209)(138,194,154,210)(139,195,148,204)(140,196,149,205)(141,197,156,212)(142,198,157,213)(143,199,158,214)(144,200,159,215)(145,201,160,216)(146,202,161,217)(147,203,155,211), (1,134,39,150)(2,135,40,151)(3,136,41,152)(4,137,42,153)(5,138,36,154)(6,139,37,148)(7,140,38,149)(8,118,23,133)(9,119,24,127)(10,113,25,128)(11,114,26,129)(12,115,27,130)(13,116,28,131)(14,117,22,132)(15,122,218,106)(16,123,219,107)(17,124,220,108)(18,125,221,109)(19,126,222,110)(20,120,223,111)(21,121,224,112)(29,156,44,141)(30,157,45,142)(31,158,46,143)(32,159,47,144)(33,160,48,145)(34,161,49,146)(35,155,43,147)(50,178,66,162)(51,179,67,163)(52,180,68,164)(53,181,69,165)(54,182,70,166)(55,176,64,167)(56,177,65,168)(57,184,72,169)(58,185,73,170)(59,186,74,171)(60,187,75,172)(61,188,76,173)(62,189,77,174)(63,183,71,175)(78,206,94,190)(79,207,95,191)(80,208,96,192)(81,209,97,193)(82,210,98,194)(83,204,92,195)(84,205,93,196)(85,212,100,197)(86,213,101,198)(87,214,102,199)(88,215,103,200)(89,216,104,201)(90,217,105,202)(91,211,99,203)>;
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), (1,106,43,119)(2,107,44,113)(3,108,45,114)(4,109,46,115)(5,110,47,116)(6,111,48,117)(7,112,49,118)(8,161,224,149)(9,155,218,150)(10,156,219,151)(11,157,220,152)(12,158,221,153)(13,159,222,154)(14,160,223,148)(15,134,24,147)(16,135,25,141)(17,136,26,142)(18,137,27,143)(19,138,28,144)(20,139,22,145)(21,140,23,146)(29,128,40,123)(30,129,41,124)(31,130,42,125)(32,131,36,126)(33,132,37,120)(34,133,38,121)(35,127,39,122)(50,91,63,78)(51,85,57,79)(52,86,58,80)(53,87,59,81)(54,88,60,82)(55,89,61,83)(56,90,62,84)(64,104,76,92)(65,105,77,93)(66,99,71,94)(67,100,72,95)(68,101,73,96)(69,102,74,97)(70,103,75,98)(162,190,175,203)(163,191,169,197)(164,192,170,198)(165,193,171,199)(166,194,172,200)(167,195,173,201)(168,196,174,202)(176,204,188,216)(177,205,189,217)(178,206,183,211)(179,207,184,212)(180,208,185,213)(181,209,186,214)(182,210,187,215), (1,78,39,94)(2,79,40,95)(3,80,41,96)(4,81,42,97)(5,82,36,98)(6,83,37,92)(7,84,38,93)(8,189,23,174)(9,183,24,175)(10,184,25,169)(11,185,26,170)(12,186,27,171)(13,187,28,172)(14,188,22,173)(15,162,218,178)(16,163,219,179)(17,164,220,180)(18,165,221,181)(19,166,222,182)(20,167,223,176)(21,168,224,177)(29,100,44,85)(30,101,45,86)(31,102,46,87)(32,103,47,88)(33,104,48,89)(34,105,49,90)(35,99,43,91)(50,122,66,106)(51,123,67,107)(52,124,68,108)(53,125,69,109)(54,126,70,110)(55,120,64,111)(56,121,65,112)(57,128,72,113)(58,129,73,114)(59,130,74,115)(60,131,75,116)(61,132,76,117)(62,133,77,118)(63,127,71,119)(134,190,150,206)(135,191,151,207)(136,192,152,208)(137,193,153,209)(138,194,154,210)(139,195,148,204)(140,196,149,205)(141,197,156,212)(142,198,157,213)(143,199,158,214)(144,200,159,215)(145,201,160,216)(146,202,161,217)(147,203,155,211), (1,134,39,150)(2,135,40,151)(3,136,41,152)(4,137,42,153)(5,138,36,154)(6,139,37,148)(7,140,38,149)(8,118,23,133)(9,119,24,127)(10,113,25,128)(11,114,26,129)(12,115,27,130)(13,116,28,131)(14,117,22,132)(15,122,218,106)(16,123,219,107)(17,124,220,108)(18,125,221,109)(19,126,222,110)(20,120,223,111)(21,121,224,112)(29,156,44,141)(30,157,45,142)(31,158,46,143)(32,159,47,144)(33,160,48,145)(34,161,49,146)(35,155,43,147)(50,178,66,162)(51,179,67,163)(52,180,68,164)(53,181,69,165)(54,182,70,166)(55,176,64,167)(56,177,65,168)(57,184,72,169)(58,185,73,170)(59,186,74,171)(60,187,75,172)(61,188,76,173)(62,189,77,174)(63,183,71,175)(78,206,94,190)(79,207,95,191)(80,208,96,192)(81,209,97,193)(82,210,98,194)(83,204,92,195)(84,205,93,196)(85,212,100,197)(86,213,101,198)(87,214,102,199)(88,215,103,200)(89,216,104,201)(90,217,105,202)(91,211,99,203) );
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)], [(1,106,43,119),(2,107,44,113),(3,108,45,114),(4,109,46,115),(5,110,47,116),(6,111,48,117),(7,112,49,118),(8,161,224,149),(9,155,218,150),(10,156,219,151),(11,157,220,152),(12,158,221,153),(13,159,222,154),(14,160,223,148),(15,134,24,147),(16,135,25,141),(17,136,26,142),(18,137,27,143),(19,138,28,144),(20,139,22,145),(21,140,23,146),(29,128,40,123),(30,129,41,124),(31,130,42,125),(32,131,36,126),(33,132,37,120),(34,133,38,121),(35,127,39,122),(50,91,63,78),(51,85,57,79),(52,86,58,80),(53,87,59,81),(54,88,60,82),(55,89,61,83),(56,90,62,84),(64,104,76,92),(65,105,77,93),(66,99,71,94),(67,100,72,95),(68,101,73,96),(69,102,74,97),(70,103,75,98),(162,190,175,203),(163,191,169,197),(164,192,170,198),(165,193,171,199),(166,194,172,200),(167,195,173,201),(168,196,174,202),(176,204,188,216),(177,205,189,217),(178,206,183,211),(179,207,184,212),(180,208,185,213),(181,209,186,214),(182,210,187,215)], [(1,78,39,94),(2,79,40,95),(3,80,41,96),(4,81,42,97),(5,82,36,98),(6,83,37,92),(7,84,38,93),(8,189,23,174),(9,183,24,175),(10,184,25,169),(11,185,26,170),(12,186,27,171),(13,187,28,172),(14,188,22,173),(15,162,218,178),(16,163,219,179),(17,164,220,180),(18,165,221,181),(19,166,222,182),(20,167,223,176),(21,168,224,177),(29,100,44,85),(30,101,45,86),(31,102,46,87),(32,103,47,88),(33,104,48,89),(34,105,49,90),(35,99,43,91),(50,122,66,106),(51,123,67,107),(52,124,68,108),(53,125,69,109),(54,126,70,110),(55,120,64,111),(56,121,65,112),(57,128,72,113),(58,129,73,114),(59,130,74,115),(60,131,75,116),(61,132,76,117),(62,133,77,118),(63,127,71,119),(134,190,150,206),(135,191,151,207),(136,192,152,208),(137,193,153,209),(138,194,154,210),(139,195,148,204),(140,196,149,205),(141,197,156,212),(142,198,157,213),(143,199,158,214),(144,200,159,215),(145,201,160,216),(146,202,161,217),(147,203,155,211)], [(1,134,39,150),(2,135,40,151),(3,136,41,152),(4,137,42,153),(5,138,36,154),(6,139,37,148),(7,140,38,149),(8,118,23,133),(9,119,24,127),(10,113,25,128),(11,114,26,129),(12,115,27,130),(13,116,28,131),(14,117,22,132),(15,122,218,106),(16,123,219,107),(17,124,220,108),(18,125,221,109),(19,126,222,110),(20,120,223,111),(21,121,224,112),(29,156,44,141),(30,157,45,142),(31,158,46,143),(32,159,47,144),(33,160,48,145),(34,161,49,146),(35,155,43,147),(50,178,66,162),(51,179,67,163),(52,180,68,164),(53,181,69,165),(54,182,70,166),(55,176,64,167),(56,177,65,168),(57,184,72,169),(58,185,73,170),(59,186,74,171),(60,187,75,172),(61,188,76,173),(62,189,77,174),(63,183,71,175),(78,206,94,190),(79,207,95,191),(80,208,96,192),(81,209,97,193),(82,210,98,194),(83,204,92,195),(84,205,93,196),(85,212,100,197),(86,213,101,198),(87,214,102,199),(88,215,103,200),(89,216,104,201),(90,217,105,202),(91,211,99,203)]])
C7×C4⋊Q8 is a maximal subgroup of
C28.5Q16 C28.10D8 C42.3Dic7 C28.17D8 C28.SD16 C42.76D14 C28.Q16 C42.77D14 C28⋊5SD16 D28⋊5Q8 C28⋊6SD16 C42.80D14 D28⋊6Q8 C28.D8 C42.82D14 C28⋊Q16 Dic14⋊5Q8 C28⋊3Q16 C28.11Q16 Dic14⋊6Q8 D28.15D4 Dic14⋊8Q8 Dic14⋊9Q8 C42.171D14 C42.240D14 D28⋊12D4 D28⋊8Q8 C42.241D14 C42.174D14 D28⋊9Q8 C42.176D14 C42.177D14 C42.178D14 C42.179D14 C42.180D14 C7×D4×Q8 C7×Q82
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 14A | ··· | 14R | 28A | ··· | 28AJ | 28AK | ··· | 28BH |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | Q8 | C7×D4 | C7×Q8 |
| kernel | C7×C4⋊Q8 | C4×C28 | C7×C4⋊C4 | Q8×C14 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C28 | C28 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 2 | 4 | 12 | 24 |
Matrix representation of C7×C4⋊Q8 ►in GL4(𝔽29) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 1 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 28 | 1 |
| 5 | 11 | 0 | 0 |
| 11 | 24 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 17 | 12 |
G:=sub<GL(4,GF(29))| [7,0,0,0,0,7,0,0,0,0,1,0,0,0,0,1],[0,28,0,0,1,0,0,0,0,0,1,1,0,0,27,28],[28,0,0,0,0,28,0,0,0,0,28,28,0,0,2,1],[5,11,0,0,11,24,0,0,0,0,17,17,0,0,0,12] >;
C7×C4⋊Q8 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes Q_8
% in TeX
G:=Group("C7xC4:Q8"); // GroupNames label
G:=SmallGroup(224,163);
// by ID
G=gap.SmallGroup(224,163);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,697,343,2090,518]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations