metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊3D8, (C7×D4)⋊4D4, (C2×D8)⋊2D7, C7⋊5(C4⋊D8), C2.26(D7×D8), (C14×D8)⋊12C2, C28⋊D4⋊3C2, D4⋊1(C7⋊D4), (D4×Dic7)⋊4C2, C14.43(C2×D8), (C2×C8).33D14, Dic7⋊C8⋊27C2, C28.162(C2×D4), C2.D56⋊28C2, (C2×D4).140D14, C28.90(C4○D4), D4⋊Dic7⋊25C2, C4.7(D4⋊2D7), C22.251(D4×D7), C2.26(D8⋊D7), C14.46(C8⋊C22), (C2×C56).247C22, (C2×C28).427C23, (C2×Dic7).179D4, (D4×C14).77C22, C14.107(C4⋊D4), (C2×D28).113C22, C4⋊Dic7.162C22, (C4×Dic7).42C22, C2.22(Dic7⋊D4), (C2×D4⋊D7)⋊16C2, C4.34(C2×C7⋊D4), (C2×C14).340(C2×D4), (C2×C7⋊C8).145C22, (C2×C4).517(C22×D7), SmallGroup(448,684)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊D8
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c-1 >
Subgroups: 836 in 140 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C2×D8, C7⋊C8, C56, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C4⋊D8, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D4⋊D7, C23.D7, C2×C56, C7×D8, C2×D28, C22×Dic7, C2×C7⋊D4, D4×C14, Dic7⋊C8, C2.D56, D4⋊Dic7, C2×D4⋊D7, D4×Dic7, C28⋊D4, C14×D8, Dic7⋊D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C4⋊D4, C2×D8, C8⋊C22, C7⋊D4, C22×D7, C4⋊D8, D4×D7, D4⋊2D7, C2×C7⋊D4, D7×D8, D8⋊D7, Dic7⋊D4, Dic7⋊D8
►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)
(1 115 8 122)(2 114 9 121)(3 113 10 120)(4 126 11 119)(5 125 12 118)(6 124 13 117)(7 123 14 116)(15 66 22 59)(16 65 23 58)(17 64 24 57)(18 63 25 70)(19 62 26 69)(20 61 27 68)(21 60 28 67)(29 175 36 182)(30 174 37 181)(31 173 38 180)(32 172 39 179)(33 171 40 178)(34 170 41 177)(35 169 42 176)(43 128 50 135)(44 127 51 134)(45 140 52 133)(46 139 53 132)(47 138 54 131)(48 137 55 130)(49 136 56 129)(71 146 78 153)(72 145 79 152)(73 144 80 151)(74 143 81 150)(75 142 82 149)(76 141 83 148)(77 154 84 147)(85 104 92 111)(86 103 93 110)(87 102 94 109)(88 101 95 108)(89 100 96 107)(90 99 97 106)(91 112 98 105)(155 186 162 193)(156 185 163 192)(157 184 164 191)(158 183 165 190)(159 196 166 189)(160 195 167 188)(161 194 168 187)(197 214 204 221)(198 213 205 220)(199 212 206 219)(200 211 207 218)(201 224 208 217)(202 223 209 216)(203 222 210 215)
(1 83 65 98 140 38 199 185)(2 84 66 85 127 39 200 186)(3 71 67 86 128 40 201 187)(4 72 68 87 129 41 202 188)(5 73 69 88 130 42 203 189)(6 74 70 89 131 29 204 190)(7 75 57 90 132 30 205 191)(8 76 58 91 133 31 206 192)(9 77 59 92 134 32 207 193)(10 78 60 93 135 33 208 194)(11 79 61 94 136 34 209 195)(12 80 62 95 137 35 210 196)(13 81 63 96 138 36 197 183)(14 82 64 97 139 37 198 184)(15 104 44 179 218 162 121 147)(16 105 45 180 219 163 122 148)(17 106 46 181 220 164 123 149)(18 107 47 182 221 165 124 150)(19 108 48 169 222 166 125 151)(20 109 49 170 223 167 126 152)(21 110 50 171 224 168 113 153)(22 111 51 172 211 155 114 154)(23 112 52 173 212 156 115 141)(24 99 53 174 213 157 116 142)(25 100 54 175 214 158 117 143)(26 101 55 176 215 159 118 144)(27 102 56 177 216 160 119 145)(28 103 43 178 217 161 120 146)
(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(22 211)(23 212)(24 213)(25 214)(26 215)(27 216)(28 217)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 85)(40 86)(41 87)(42 88)(57 205)(58 206)(59 207)(60 208)(61 209)(62 210)(63 197)(64 198)(65 199)(66 200)(67 201)(68 202)(69 203)(70 204)(71 187)(72 188)(73 189)(74 190)(75 191)(76 192)(77 193)(78 194)(79 195)(80 196)(81 183)(82 184)(83 185)(84 186)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 169)(109 170)(110 171)(111 172)(112 173)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(151 166)(152 167)(153 168)(154 155)
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), (1,115,8,122)(2,114,9,121)(3,113,10,120)(4,126,11,119)(5,125,12,118)(6,124,13,117)(7,123,14,116)(15,66,22,59)(16,65,23,58)(17,64,24,57)(18,63,25,70)(19,62,26,69)(20,61,27,68)(21,60,28,67)(29,175,36,182)(30,174,37,181)(31,173,38,180)(32,172,39,179)(33,171,40,178)(34,170,41,177)(35,169,42,176)(43,128,50,135)(44,127,51,134)(45,140,52,133)(46,139,53,132)(47,138,54,131)(48,137,55,130)(49,136,56,129)(71,146,78,153)(72,145,79,152)(73,144,80,151)(74,143,81,150)(75,142,82,149)(76,141,83,148)(77,154,84,147)(85,104,92,111)(86,103,93,110)(87,102,94,109)(88,101,95,108)(89,100,96,107)(90,99,97,106)(91,112,98,105)(155,186,162,193)(156,185,163,192)(157,184,164,191)(158,183,165,190)(159,196,166,189)(160,195,167,188)(161,194,168,187)(197,214,204,221)(198,213,205,220)(199,212,206,219)(200,211,207,218)(201,224,208,217)(202,223,209,216)(203,222,210,215), (1,83,65,98,140,38,199,185)(2,84,66,85,127,39,200,186)(3,71,67,86,128,40,201,187)(4,72,68,87,129,41,202,188)(5,73,69,88,130,42,203,189)(6,74,70,89,131,29,204,190)(7,75,57,90,132,30,205,191)(8,76,58,91,133,31,206,192)(9,77,59,92,134,32,207,193)(10,78,60,93,135,33,208,194)(11,79,61,94,136,34,209,195)(12,80,62,95,137,35,210,196)(13,81,63,96,138,36,197,183)(14,82,64,97,139,37,198,184)(15,104,44,179,218,162,121,147)(16,105,45,180,219,163,122,148)(17,106,46,181,220,164,123,149)(18,107,47,182,221,165,124,150)(19,108,48,169,222,166,125,151)(20,109,49,170,223,167,126,152)(21,110,50,171,224,168,113,153)(22,111,51,172,211,155,114,154)(23,112,52,173,212,156,115,141)(24,99,53,174,213,157,116,142)(25,100,54,175,214,158,117,143)(26,101,55,176,215,159,118,144)(27,102,56,177,216,160,119,145)(28,103,43,178,217,161,120,146), (15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,85)(40,86)(41,87)(42,88)(57,205)(58,206)(59,207)(60,208)(61,209)(62,210)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,195)(80,196)(81,183)(82,184)(83,185)(84,186)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,169)(109,170)(110,171)(111,172)(112,173)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(151,166)(152,167)(153,168)(154,155)>;
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), (1,115,8,122)(2,114,9,121)(3,113,10,120)(4,126,11,119)(5,125,12,118)(6,124,13,117)(7,123,14,116)(15,66,22,59)(16,65,23,58)(17,64,24,57)(18,63,25,70)(19,62,26,69)(20,61,27,68)(21,60,28,67)(29,175,36,182)(30,174,37,181)(31,173,38,180)(32,172,39,179)(33,171,40,178)(34,170,41,177)(35,169,42,176)(43,128,50,135)(44,127,51,134)(45,140,52,133)(46,139,53,132)(47,138,54,131)(48,137,55,130)(49,136,56,129)(71,146,78,153)(72,145,79,152)(73,144,80,151)(74,143,81,150)(75,142,82,149)(76,141,83,148)(77,154,84,147)(85,104,92,111)(86,103,93,110)(87,102,94,109)(88,101,95,108)(89,100,96,107)(90,99,97,106)(91,112,98,105)(155,186,162,193)(156,185,163,192)(157,184,164,191)(158,183,165,190)(159,196,166,189)(160,195,167,188)(161,194,168,187)(197,214,204,221)(198,213,205,220)(199,212,206,219)(200,211,207,218)(201,224,208,217)(202,223,209,216)(203,222,210,215), (1,83,65,98,140,38,199,185)(2,84,66,85,127,39,200,186)(3,71,67,86,128,40,201,187)(4,72,68,87,129,41,202,188)(5,73,69,88,130,42,203,189)(6,74,70,89,131,29,204,190)(7,75,57,90,132,30,205,191)(8,76,58,91,133,31,206,192)(9,77,59,92,134,32,207,193)(10,78,60,93,135,33,208,194)(11,79,61,94,136,34,209,195)(12,80,62,95,137,35,210,196)(13,81,63,96,138,36,197,183)(14,82,64,97,139,37,198,184)(15,104,44,179,218,162,121,147)(16,105,45,180,219,163,122,148)(17,106,46,181,220,164,123,149)(18,107,47,182,221,165,124,150)(19,108,48,169,222,166,125,151)(20,109,49,170,223,167,126,152)(21,110,50,171,224,168,113,153)(22,111,51,172,211,155,114,154)(23,112,52,173,212,156,115,141)(24,99,53,174,213,157,116,142)(25,100,54,175,214,158,117,143)(26,101,55,176,215,159,118,144)(27,102,56,177,216,160,119,145)(28,103,43,178,217,161,120,146), (15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,85)(40,86)(41,87)(42,88)(57,205)(58,206)(59,207)(60,208)(61,209)(62,210)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,195)(80,196)(81,183)(82,184)(83,185)(84,186)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,169)(109,170)(110,171)(111,172)(112,173)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(151,166)(152,167)(153,168)(154,155) );
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)], [(1,115,8,122),(2,114,9,121),(3,113,10,120),(4,126,11,119),(5,125,12,118),(6,124,13,117),(7,123,14,116),(15,66,22,59),(16,65,23,58),(17,64,24,57),(18,63,25,70),(19,62,26,69),(20,61,27,68),(21,60,28,67),(29,175,36,182),(30,174,37,181),(31,173,38,180),(32,172,39,179),(33,171,40,178),(34,170,41,177),(35,169,42,176),(43,128,50,135),(44,127,51,134),(45,140,52,133),(46,139,53,132),(47,138,54,131),(48,137,55,130),(49,136,56,129),(71,146,78,153),(72,145,79,152),(73,144,80,151),(74,143,81,150),(75,142,82,149),(76,141,83,148),(77,154,84,147),(85,104,92,111),(86,103,93,110),(87,102,94,109),(88,101,95,108),(89,100,96,107),(90,99,97,106),(91,112,98,105),(155,186,162,193),(156,185,163,192),(157,184,164,191),(158,183,165,190),(159,196,166,189),(160,195,167,188),(161,194,168,187),(197,214,204,221),(198,213,205,220),(199,212,206,219),(200,211,207,218),(201,224,208,217),(202,223,209,216),(203,222,210,215)], [(1,83,65,98,140,38,199,185),(2,84,66,85,127,39,200,186),(3,71,67,86,128,40,201,187),(4,72,68,87,129,41,202,188),(5,73,69,88,130,42,203,189),(6,74,70,89,131,29,204,190),(7,75,57,90,132,30,205,191),(8,76,58,91,133,31,206,192),(9,77,59,92,134,32,207,193),(10,78,60,93,135,33,208,194),(11,79,61,94,136,34,209,195),(12,80,62,95,137,35,210,196),(13,81,63,96,138,36,197,183),(14,82,64,97,139,37,198,184),(15,104,44,179,218,162,121,147),(16,105,45,180,219,163,122,148),(17,106,46,181,220,164,123,149),(18,107,47,182,221,165,124,150),(19,108,48,169,222,166,125,151),(20,109,49,170,223,167,126,152),(21,110,50,171,224,168,113,153),(22,111,51,172,211,155,114,154),(23,112,52,173,212,156,115,141),(24,99,53,174,213,157,116,142),(25,100,54,175,214,158,117,143),(26,101,55,176,215,159,118,144),(27,102,56,177,216,160,119,145),(28,103,43,178,217,161,120,146)], [(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(22,211),(23,212),(24,213),(25,214),(26,215),(27,216),(28,217),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,85),(40,86),(41,87),(42,88),(57,205),(58,206),(59,207),(60,208),(61,209),(62,210),(63,197),(64,198),(65,199),(66,200),(67,201),(68,202),(69,203),(70,204),(71,187),(72,188),(73,189),(74,190),(75,191),(76,192),(77,193),(78,194),(79,195),(80,196),(81,183),(82,184),(83,185),(84,186),(99,174),(100,175),(101,176),(102,177),(103,178),(104,179),(105,180),(106,181),(107,182),(108,169),(109,170),(110,171),(111,172),(112,173),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,162),(148,163),(149,164),(150,165),(151,166),(152,167),(153,168),(154,155)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 56 | 2 | 2 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | C4○D4 | D14 | D14 | C7⋊D4 | C8⋊C22 | D4⋊2D7 | D4×D7 | D7×D8 | D8⋊D7 |
| kernel | Dic7⋊D8 | Dic7⋊C8 | C2.D56 | D4⋊Dic7 | C2×D4⋊D7 | D4×Dic7 | C28⋊D4 | C14×D8 | C2×Dic7 | C7×D4 | C2×D8 | Dic7 | C28 | C2×C8 | C2×D4 | D4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Dic7⋊D8 ►in GL4(𝔽113) generated by
| 1 | 112 | 0 | 0 |
| 11 | 103 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 28 | 0 | 0 |
| 18 | 88 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 84 | 108 | 0 | 0 |
| 55 | 29 | 0 | 0 |
| 0 | 0 | 62 | 14 |
| 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 44 | 112 |
G:=sub<GL(4,GF(113))| [1,11,0,0,112,103,0,0,0,0,1,0,0,0,0,1],[25,18,0,0,28,88,0,0,0,0,112,0,0,0,0,112],[84,55,0,0,108,29,0,0,0,0,62,8,0,0,14,0],[1,0,0,0,0,1,0,0,0,0,1,44,0,0,0,112] >;
Dic7⋊D8 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes D_8 % in TeX
G:=Group("Dic7:D8"); // GroupNames label
G:=SmallGroup(448,684);
// by ID
G=gap.SmallGroup(448,684);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,422,135,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations