direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C56.C4, C23.14Dic14, (C2×C56).19C4, C56.74(C2×C4), C4.84(C2×D28), C28.48(C4⋊C4), (C2×C28).62Q8, (C2×C4).170D28, C28.304(C2×D4), (C2×C8).314D14, (C2×C28).402D4, (C2×C8).11Dic7, C8.19(C2×Dic7), C14⋊1(C8.C4), (C22×C8).12D7, (C22×C56).18C2, C4.24(C4⋊Dic7), (C2×C4).51Dic14, (C22×C14).23Q8, C28.172(C22×C4), (C2×C56).386C22, (C2×C28).795C23, (C22×C4).426D14, C4.26(C22×Dic7), C22.8(C2×Dic14), C22.14(C4⋊Dic7), C4.Dic7.35C22, (C22×C28).540C22, C7⋊2(C2×C8.C4), C14.47(C2×C4⋊C4), C2.13(C2×C4⋊Dic7), (C2×C14).40(C2×Q8), (C2×C14).52(C4⋊C4), (C2×C28).307(C2×C4), (C2×C4).84(C2×Dic7), (C2×C4.Dic7).5C2, (C2×C4).713(C22×D7), SmallGroup(448,641)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C56.C4
G = < a,b,c,d | a2=b8=1, c14=b4, d2=b4c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c13 >
Subgroups: 292 in 106 conjugacy classes, 71 normal (41 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, C23, C14, C14, C14, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C28, C2×C14, C2×C14, C8.C4, C22×C8, C2×M4(2), C7⋊C8, C56, C2×C28, C22×C14, C2×C8.C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C2×C56, C22×C28, C56.C4, C2×C4.Dic7, C22×C56, C2×C56.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic7, D14, C8.C4, C2×C4⋊C4, Dic14, D28, C2×Dic7, C22×D7, C2×C8.C4, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C56.C4, C2×C4⋊Dic7, C2×C56.C4
►On 224 points
Generators in S224
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 29)(28 30)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 127)(64 128)(65 129)(66 130)(67 131)(68 132)(69 133)(70 134)(71 135)(72 136)(73 137)(74 138)(75 139)(76 140)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 174)(86 175)(87 176)(88 177)(89 178)(90 179)(91 180)(92 181)(93 182)(94 183)(95 184)(96 185)(97 186)(98 187)(99 188)(100 189)(101 190)(102 191)(103 192)(104 193)(105 194)(106 195)(107 196)(108 169)(109 170)(110 171)(111 172)(112 173)(141 214)(142 215)(143 216)(144 217)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 224)(152 197)(153 198)(154 199)(155 200)(156 201)(157 202)(158 203)(159 204)(160 205)(161 206)(162 207)(163 208)(164 209)(165 210)(166 211)(167 212)(168 213)
(1 125 38 68 15 139 52 82)(2 126 39 69 16 140 53 83)(3 127 40 70 17 113 54 84)(4 128 41 71 18 114 55 57)(5 129 42 72 19 115 56 58)(6 130 43 73 20 116 29 59)(7 131 44 74 21 117 30 60)(8 132 45 75 22 118 31 61)(9 133 46 76 23 119 32 62)(10 134 47 77 24 120 33 63)(11 135 48 78 25 121 34 64)(12 136 49 79 26 122 35 65)(13 137 50 80 27 123 36 66)(14 138 51 81 28 124 37 67)(85 147 195 213 99 161 181 199)(86 148 196 214 100 162 182 200)(87 149 169 215 101 163 183 201)(88 150 170 216 102 164 184 202)(89 151 171 217 103 165 185 203)(90 152 172 218 104 166 186 204)(91 153 173 219 105 167 187 205)(92 154 174 220 106 168 188 206)(93 155 175 221 107 141 189 207)(94 156 176 222 108 142 190 208)(95 157 177 223 109 143 191 209)(96 158 178 224 110 144 192 210)(97 159 179 197 111 145 193 211)(98 160 180 198 112 146 194 212)
(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)
(1 149 22 142 15 163 8 156)(2 162 23 155 16 148 9 141)(3 147 24 168 17 161 10 154)(4 160 25 153 18 146 11 167)(5 145 26 166 19 159 12 152)(6 158 27 151 20 144 13 165)(7 143 28 164 21 157 14 150)(29 224 50 217 43 210 36 203)(30 209 51 202 44 223 37 216)(31 222 52 215 45 208 38 201)(32 207 53 200 46 221 39 214)(33 220 54 213 47 206 40 199)(34 205 55 198 48 219 41 212)(35 218 56 211 49 204 42 197)(57 180 78 173 71 194 64 187)(58 193 79 186 72 179 65 172)(59 178 80 171 73 192 66 185)(60 191 81 184 74 177 67 170)(61 176 82 169 75 190 68 183)(62 189 83 182 76 175 69 196)(63 174 84 195 77 188 70 181)(85 120 106 113 99 134 92 127)(86 133 107 126 100 119 93 140)(87 118 108 139 101 132 94 125)(88 131 109 124 102 117 95 138)(89 116 110 137 103 130 96 123)(90 129 111 122 104 115 97 136)(91 114 112 135 105 128 98 121)
G:=sub<Sym(224)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,29)(28,30)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,169)(109,170)(110,171)(111,172)(112,173)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,197)(153,198)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,211)(167,212)(168,213), (1,125,38,68,15,139,52,82)(2,126,39,69,16,140,53,83)(3,127,40,70,17,113,54,84)(4,128,41,71,18,114,55,57)(5,129,42,72,19,115,56,58)(6,130,43,73,20,116,29,59)(7,131,44,74,21,117,30,60)(8,132,45,75,22,118,31,61)(9,133,46,76,23,119,32,62)(10,134,47,77,24,120,33,63)(11,135,48,78,25,121,34,64)(12,136,49,79,26,122,35,65)(13,137,50,80,27,123,36,66)(14,138,51,81,28,124,37,67)(85,147,195,213,99,161,181,199)(86,148,196,214,100,162,182,200)(87,149,169,215,101,163,183,201)(88,150,170,216,102,164,184,202)(89,151,171,217,103,165,185,203)(90,152,172,218,104,166,186,204)(91,153,173,219,105,167,187,205)(92,154,174,220,106,168,188,206)(93,155,175,221,107,141,189,207)(94,156,176,222,108,142,190,208)(95,157,177,223,109,143,191,209)(96,158,178,224,110,144,192,210)(97,159,179,197,111,145,193,211)(98,160,180,198,112,146,194,212), (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), (1,149,22,142,15,163,8,156)(2,162,23,155,16,148,9,141)(3,147,24,168,17,161,10,154)(4,160,25,153,18,146,11,167)(5,145,26,166,19,159,12,152)(6,158,27,151,20,144,13,165)(7,143,28,164,21,157,14,150)(29,224,50,217,43,210,36,203)(30,209,51,202,44,223,37,216)(31,222,52,215,45,208,38,201)(32,207,53,200,46,221,39,214)(33,220,54,213,47,206,40,199)(34,205,55,198,48,219,41,212)(35,218,56,211,49,204,42,197)(57,180,78,173,71,194,64,187)(58,193,79,186,72,179,65,172)(59,178,80,171,73,192,66,185)(60,191,81,184,74,177,67,170)(61,176,82,169,75,190,68,183)(62,189,83,182,76,175,69,196)(63,174,84,195,77,188,70,181)(85,120,106,113,99,134,92,127)(86,133,107,126,100,119,93,140)(87,118,108,139,101,132,94,125)(88,131,109,124,102,117,95,138)(89,116,110,137,103,130,96,123)(90,129,111,122,104,115,97,136)(91,114,112,135,105,128,98,121)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,29)(28,30)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,169)(109,170)(110,171)(111,172)(112,173)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,197)(153,198)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,211)(167,212)(168,213), (1,125,38,68,15,139,52,82)(2,126,39,69,16,140,53,83)(3,127,40,70,17,113,54,84)(4,128,41,71,18,114,55,57)(5,129,42,72,19,115,56,58)(6,130,43,73,20,116,29,59)(7,131,44,74,21,117,30,60)(8,132,45,75,22,118,31,61)(9,133,46,76,23,119,32,62)(10,134,47,77,24,120,33,63)(11,135,48,78,25,121,34,64)(12,136,49,79,26,122,35,65)(13,137,50,80,27,123,36,66)(14,138,51,81,28,124,37,67)(85,147,195,213,99,161,181,199)(86,148,196,214,100,162,182,200)(87,149,169,215,101,163,183,201)(88,150,170,216,102,164,184,202)(89,151,171,217,103,165,185,203)(90,152,172,218,104,166,186,204)(91,153,173,219,105,167,187,205)(92,154,174,220,106,168,188,206)(93,155,175,221,107,141,189,207)(94,156,176,222,108,142,190,208)(95,157,177,223,109,143,191,209)(96,158,178,224,110,144,192,210)(97,159,179,197,111,145,193,211)(98,160,180,198,112,146,194,212), (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), (1,149,22,142,15,163,8,156)(2,162,23,155,16,148,9,141)(3,147,24,168,17,161,10,154)(4,160,25,153,18,146,11,167)(5,145,26,166,19,159,12,152)(6,158,27,151,20,144,13,165)(7,143,28,164,21,157,14,150)(29,224,50,217,43,210,36,203)(30,209,51,202,44,223,37,216)(31,222,52,215,45,208,38,201)(32,207,53,200,46,221,39,214)(33,220,54,213,47,206,40,199)(34,205,55,198,48,219,41,212)(35,218,56,211,49,204,42,197)(57,180,78,173,71,194,64,187)(58,193,79,186,72,179,65,172)(59,178,80,171,73,192,66,185)(60,191,81,184,74,177,67,170)(61,176,82,169,75,190,68,183)(62,189,83,182,76,175,69,196)(63,174,84,195,77,188,70,181)(85,120,106,113,99,134,92,127)(86,133,107,126,100,119,93,140)(87,118,108,139,101,132,94,125)(88,131,109,124,102,117,95,138)(89,116,110,137,103,130,96,123)(90,129,111,122,104,115,97,136)(91,114,112,135,105,128,98,121) );
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,29),(28,30),(57,121),(58,122),(59,123),(60,124),(61,125),(62,126),(63,127),(64,128),(65,129),(66,130),(67,131),(68,132),(69,133),(70,134),(71,135),(72,136),(73,137),(74,138),(75,139),(76,140),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,174),(86,175),(87,176),(88,177),(89,178),(90,179),(91,180),(92,181),(93,182),(94,183),(95,184),(96,185),(97,186),(98,187),(99,188),(100,189),(101,190),(102,191),(103,192),(104,193),(105,194),(106,195),(107,196),(108,169),(109,170),(110,171),(111,172),(112,173),(141,214),(142,215),(143,216),(144,217),(145,218),(146,219),(147,220),(148,221),(149,222),(150,223),(151,224),(152,197),(153,198),(154,199),(155,200),(156,201),(157,202),(158,203),(159,204),(160,205),(161,206),(162,207),(163,208),(164,209),(165,210),(166,211),(167,212),(168,213)], [(1,125,38,68,15,139,52,82),(2,126,39,69,16,140,53,83),(3,127,40,70,17,113,54,84),(4,128,41,71,18,114,55,57),(5,129,42,72,19,115,56,58),(6,130,43,73,20,116,29,59),(7,131,44,74,21,117,30,60),(8,132,45,75,22,118,31,61),(9,133,46,76,23,119,32,62),(10,134,47,77,24,120,33,63),(11,135,48,78,25,121,34,64),(12,136,49,79,26,122,35,65),(13,137,50,80,27,123,36,66),(14,138,51,81,28,124,37,67),(85,147,195,213,99,161,181,199),(86,148,196,214,100,162,182,200),(87,149,169,215,101,163,183,201),(88,150,170,216,102,164,184,202),(89,151,171,217,103,165,185,203),(90,152,172,218,104,166,186,204),(91,153,173,219,105,167,187,205),(92,154,174,220,106,168,188,206),(93,155,175,221,107,141,189,207),(94,156,176,222,108,142,190,208),(95,157,177,223,109,143,191,209),(96,158,178,224,110,144,192,210),(97,159,179,197,111,145,193,211),(98,160,180,198,112,146,194,212)], [(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)], [(1,149,22,142,15,163,8,156),(2,162,23,155,16,148,9,141),(3,147,24,168,17,161,10,154),(4,160,25,153,18,146,11,167),(5,145,26,166,19,159,12,152),(6,158,27,151,20,144,13,165),(7,143,28,164,21,157,14,150),(29,224,50,217,43,210,36,203),(30,209,51,202,44,223,37,216),(31,222,52,215,45,208,38,201),(32,207,53,200,46,221,39,214),(33,220,54,213,47,206,40,199),(34,205,55,198,48,219,41,212),(35,218,56,211,49,204,42,197),(57,180,78,173,71,194,64,187),(58,193,79,186,72,179,65,172),(59,178,80,171,73,192,66,185),(60,191,81,184,74,177,67,170),(61,176,82,169,75,190,68,183),(62,189,83,182,76,175,69,196),(63,174,84,195,77,188,70,181),(85,120,106,113,99,134,92,127),(86,133,107,126,100,119,93,140),(87,118,108,139,101,132,94,125),(88,131,109,124,102,117,95,138),(89,116,110,137,103,130,96,123),(90,129,111,122,104,115,97,136),(91,114,112,135,105,128,98,121)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 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 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D7 | Dic7 | D14 | D14 | C8.C4 | Dic14 | D28 | Dic14 | C56.C4 |
| kernel | C2×C56.C4 | C56.C4 | C2×C4.Dic7 | C22×C56 | C2×C56 | C2×C28 | C2×C28 | C22×C14 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 1 | 1 | 3 | 12 | 6 | 3 | 8 | 6 | 12 | 6 | 48 |
Matrix representation of C2×C56.C4 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 95 | 0 |
| 0 | 0 | 69 |
| 1 | 0 | 0 |
| 0 | 53 | 0 |
| 0 | 0 | 81 |
| 1 | 0 | 0 |
| 0 | 0 | 44 |
| 0 | 44 | 0 |
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[1,0,0,0,95,0,0,0,69],[1,0,0,0,53,0,0,0,81],[1,0,0,0,0,44,0,44,0] >;
C2×C56.C4 in GAP, Magma, Sage, TeX
C_2\times C_{56}.C_4 % in TeX
G:=Group("C2xC56.C4"); // GroupNames label
G:=SmallGroup(448,641);
// by ID
G=gap.SmallGroup(448,641);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,100,136,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^14=b^4,d^2=b^4*c^7,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^13>;
// generators/relations