direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4⋊C8, D4⋊C56, C28.67D8, C28.54SD16, C28.19M4(2), C4⋊C8⋊1C14, (C4×C56)⋊2C2, (C4×C8)⋊1C14, (C7×D4)⋊3C8, C4⋊C4.3C28, C4.1(C2×C56), C4.16(C7×D8), C14.20C4≀C2, C28.30(C2×C8), (C2×D4).4C28, (C4×D4).1C14, (D4×C14).14C4, (D4×C28).16C2, (C2×C28).528D4, C4.13(C7×SD16), C4.1(C7×M4(2)), C42.62(C2×C14), C14.22(C22⋊C8), (C4×C28).346C22, C14.32(D4⋊C4), (C7×C4⋊C8)⋊3C2, C2.1(C7×C4≀C2), (C7×C4⋊C4).15C4, (C2×C4).93(C7×D4), C2.5(C7×C22⋊C8), (C2×C4).38(C2×C28), C2.1(C7×D4⋊C4), (C2×C28).258(C2×C4), C22.25(C7×C22⋊C4), (C2×C14).120(C22⋊C4), SmallGroup(448,129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4⋊C8
G = < a,b,c,d | a7=b4=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 154 in 82 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C28, C28, C2×C14, C2×C14, C4×C8, C4⋊C8, C4×D4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D4⋊C8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C22×C28, D4×C14, C4×C56, C7×C4⋊C8, D4×C28, C7×D4⋊C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C14, C22⋊C4, C2×C8, M4(2), D8, SD16, C28, C2×C14, C22⋊C8, D4⋊C4, C4≀C2, C56, C2×C28, C7×D4, D4⋊C8, C7×C22⋊C4, C2×C56, C7×M4(2), C7×D8, C7×SD16, C7×C22⋊C8, C7×D4⋊C4, C7×C4≀C2, C7×D4⋊C8
►On 224 points
Generators in S224
(1 128 59 51 43 35 27)(2 121 60 52 44 36 28)(3 122 61 53 45 37 29)(4 123 62 54 46 38 30)(5 124 63 55 47 39 31)(6 125 64 56 48 40 32)(7 126 57 49 41 33 25)(8 127 58 50 42 34 26)(9 162 154 146 138 130 20)(10 163 155 147 139 131 21)(11 164 156 148 140 132 22)(12 165 157 149 141 133 23)(13 166 158 150 142 134 24)(14 167 159 151 143 135 17)(15 168 160 152 144 136 18)(16 161 153 145 137 129 19)(65 119 105 97 89 81 73)(66 120 106 98 90 82 74)(67 113 107 99 91 83 75)(68 114 108 100 92 84 76)(69 115 109 101 93 85 77)(70 116 110 102 94 86 78)(71 117 111 103 95 87 79)(72 118 112 104 96 88 80)(169 217 209 201 193 185 177)(170 218 210 202 194 186 178)(171 219 211 203 195 187 179)(172 220 212 204 196 188 180)(173 221 213 205 197 189 181)(174 222 214 206 198 190 182)(175 223 215 207 199 191 183)(176 224 216 208 200 192 184)
(1 71 175 131)(2 132 176 72)(3 65 169 133)(4 134 170 66)(5 67 171 135)(6 136 172 68)(7 69 173 129)(8 130 174 70)(9 214 110 58)(10 59 111 215)(11 216 112 60)(12 61 105 209)(13 210 106 62)(14 63 107 211)(15 212 108 64)(16 57 109 213)(17 124 113 219)(18 220 114 125)(19 126 115 221)(20 222 116 127)(21 128 117 223)(22 224 118 121)(23 122 119 217)(24 218 120 123)(25 77 181 137)(26 138 182 78)(27 79 183 139)(28 140 184 80)(29 73 177 141)(30 142 178 74)(31 75 179 143)(32 144 180 76)(33 85 189 145)(34 146 190 86)(35 87 191 147)(36 148 192 88)(37 81 185 149)(38 150 186 82)(39 83 187 151)(40 152 188 84)(41 93 197 153)(42 154 198 94)(43 95 199 155)(44 156 200 96)(45 89 193 157)(46 158 194 90)(47 91 195 159)(48 160 196 92)(49 101 205 161)(50 162 206 102)(51 103 207 163)(52 164 208 104)(53 97 201 165)(54 166 202 98)(55 99 203 167)(56 168 204 100)
(1 131)(2 176)(3 133)(4 170)(5 135)(6 172)(7 129)(8 174)(10 59)(12 61)(14 63)(16 57)(17 124)(19 126)(21 128)(23 122)(25 137)(26 182)(27 139)(28 184)(29 141)(30 178)(31 143)(32 180)(33 145)(34 190)(35 147)(36 192)(37 149)(38 186)(39 151)(40 188)(41 153)(42 198)(43 155)(44 200)(45 157)(46 194)(47 159)(48 196)(49 161)(50 206)(51 163)(52 208)(53 165)(54 202)(55 167)(56 204)(58 214)(60 216)(62 210)(64 212)(65 169)(67 171)(69 173)(71 175)(73 177)(75 179)(77 181)(79 183)(81 185)(83 187)(85 189)(87 191)(89 193)(91 195)(93 197)(95 199)(97 201)(99 203)(101 205)(103 207)(105 209)(107 211)(109 213)(111 215)(113 219)(115 221)(117 223)(119 217)(121 224)(123 218)(125 220)(127 222)
(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)| (1,128,59,51,43,35,27)(2,121,60,52,44,36,28)(3,122,61,53,45,37,29)(4,123,62,54,46,38,30)(5,124,63,55,47,39,31)(6,125,64,56,48,40,32)(7,126,57,49,41,33,25)(8,127,58,50,42,34,26)(9,162,154,146,138,130,20)(10,163,155,147,139,131,21)(11,164,156,148,140,132,22)(12,165,157,149,141,133,23)(13,166,158,150,142,134,24)(14,167,159,151,143,135,17)(15,168,160,152,144,136,18)(16,161,153,145,137,129,19)(65,119,105,97,89,81,73)(66,120,106,98,90,82,74)(67,113,107,99,91,83,75)(68,114,108,100,92,84,76)(69,115,109,101,93,85,77)(70,116,110,102,94,86,78)(71,117,111,103,95,87,79)(72,118,112,104,96,88,80)(169,217,209,201,193,185,177)(170,218,210,202,194,186,178)(171,219,211,203,195,187,179)(172,220,212,204,196,188,180)(173,221,213,205,197,189,181)(174,222,214,206,198,190,182)(175,223,215,207,199,191,183)(176,224,216,208,200,192,184), (1,71,175,131)(2,132,176,72)(3,65,169,133)(4,134,170,66)(5,67,171,135)(6,136,172,68)(7,69,173,129)(8,130,174,70)(9,214,110,58)(10,59,111,215)(11,216,112,60)(12,61,105,209)(13,210,106,62)(14,63,107,211)(15,212,108,64)(16,57,109,213)(17,124,113,219)(18,220,114,125)(19,126,115,221)(20,222,116,127)(21,128,117,223)(22,224,118,121)(23,122,119,217)(24,218,120,123)(25,77,181,137)(26,138,182,78)(27,79,183,139)(28,140,184,80)(29,73,177,141)(30,142,178,74)(31,75,179,143)(32,144,180,76)(33,85,189,145)(34,146,190,86)(35,87,191,147)(36,148,192,88)(37,81,185,149)(38,150,186,82)(39,83,187,151)(40,152,188,84)(41,93,197,153)(42,154,198,94)(43,95,199,155)(44,156,200,96)(45,89,193,157)(46,158,194,90)(47,91,195,159)(48,160,196,92)(49,101,205,161)(50,162,206,102)(51,103,207,163)(52,164,208,104)(53,97,201,165)(54,166,202,98)(55,99,203,167)(56,168,204,100), (1,131)(2,176)(3,133)(4,170)(5,135)(6,172)(7,129)(8,174)(10,59)(12,61)(14,63)(16,57)(17,124)(19,126)(21,128)(23,122)(25,137)(26,182)(27,139)(28,184)(29,141)(30,178)(31,143)(32,180)(33,145)(34,190)(35,147)(36,192)(37,149)(38,186)(39,151)(40,188)(41,153)(42,198)(43,155)(44,200)(45,157)(46,194)(47,159)(48,196)(49,161)(50,206)(51,163)(52,208)(53,165)(54,202)(55,167)(56,204)(58,214)(60,216)(62,210)(64,212)(65,169)(67,171)(69,173)(71,175)(73,177)(75,179)(77,181)(79,183)(81,185)(83,187)(85,189)(87,191)(89,193)(91,195)(93,197)(95,199)(97,201)(99,203)(101,205)(103,207)(105,209)(107,211)(109,213)(111,215)(113,219)(115,221)(117,223)(119,217)(121,224)(123,218)(125,220)(127,222), (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( (1,128,59,51,43,35,27)(2,121,60,52,44,36,28)(3,122,61,53,45,37,29)(4,123,62,54,46,38,30)(5,124,63,55,47,39,31)(6,125,64,56,48,40,32)(7,126,57,49,41,33,25)(8,127,58,50,42,34,26)(9,162,154,146,138,130,20)(10,163,155,147,139,131,21)(11,164,156,148,140,132,22)(12,165,157,149,141,133,23)(13,166,158,150,142,134,24)(14,167,159,151,143,135,17)(15,168,160,152,144,136,18)(16,161,153,145,137,129,19)(65,119,105,97,89,81,73)(66,120,106,98,90,82,74)(67,113,107,99,91,83,75)(68,114,108,100,92,84,76)(69,115,109,101,93,85,77)(70,116,110,102,94,86,78)(71,117,111,103,95,87,79)(72,118,112,104,96,88,80)(169,217,209,201,193,185,177)(170,218,210,202,194,186,178)(171,219,211,203,195,187,179)(172,220,212,204,196,188,180)(173,221,213,205,197,189,181)(174,222,214,206,198,190,182)(175,223,215,207,199,191,183)(176,224,216,208,200,192,184), (1,71,175,131)(2,132,176,72)(3,65,169,133)(4,134,170,66)(5,67,171,135)(6,136,172,68)(7,69,173,129)(8,130,174,70)(9,214,110,58)(10,59,111,215)(11,216,112,60)(12,61,105,209)(13,210,106,62)(14,63,107,211)(15,212,108,64)(16,57,109,213)(17,124,113,219)(18,220,114,125)(19,126,115,221)(20,222,116,127)(21,128,117,223)(22,224,118,121)(23,122,119,217)(24,218,120,123)(25,77,181,137)(26,138,182,78)(27,79,183,139)(28,140,184,80)(29,73,177,141)(30,142,178,74)(31,75,179,143)(32,144,180,76)(33,85,189,145)(34,146,190,86)(35,87,191,147)(36,148,192,88)(37,81,185,149)(38,150,186,82)(39,83,187,151)(40,152,188,84)(41,93,197,153)(42,154,198,94)(43,95,199,155)(44,156,200,96)(45,89,193,157)(46,158,194,90)(47,91,195,159)(48,160,196,92)(49,101,205,161)(50,162,206,102)(51,103,207,163)(52,164,208,104)(53,97,201,165)(54,166,202,98)(55,99,203,167)(56,168,204,100), (1,131)(2,176)(3,133)(4,170)(5,135)(6,172)(7,129)(8,174)(10,59)(12,61)(14,63)(16,57)(17,124)(19,126)(21,128)(23,122)(25,137)(26,182)(27,139)(28,184)(29,141)(30,178)(31,143)(32,180)(33,145)(34,190)(35,147)(36,192)(37,149)(38,186)(39,151)(40,188)(41,153)(42,198)(43,155)(44,200)(45,157)(46,194)(47,159)(48,196)(49,161)(50,206)(51,163)(52,208)(53,165)(54,202)(55,167)(56,204)(58,214)(60,216)(62,210)(64,212)(65,169)(67,171)(69,173)(71,175)(73,177)(75,179)(77,181)(79,183)(81,185)(83,187)(85,189)(87,191)(89,193)(91,195)(93,197)(95,199)(97,201)(99,203)(101,205)(103,207)(105,209)(107,211)(109,213)(111,215)(113,219)(115,221)(117,223)(119,217)(121,224)(123,218)(125,220)(127,222), (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([[(1,128,59,51,43,35,27),(2,121,60,52,44,36,28),(3,122,61,53,45,37,29),(4,123,62,54,46,38,30),(5,124,63,55,47,39,31),(6,125,64,56,48,40,32),(7,126,57,49,41,33,25),(8,127,58,50,42,34,26),(9,162,154,146,138,130,20),(10,163,155,147,139,131,21),(11,164,156,148,140,132,22),(12,165,157,149,141,133,23),(13,166,158,150,142,134,24),(14,167,159,151,143,135,17),(15,168,160,152,144,136,18),(16,161,153,145,137,129,19),(65,119,105,97,89,81,73),(66,120,106,98,90,82,74),(67,113,107,99,91,83,75),(68,114,108,100,92,84,76),(69,115,109,101,93,85,77),(70,116,110,102,94,86,78),(71,117,111,103,95,87,79),(72,118,112,104,96,88,80),(169,217,209,201,193,185,177),(170,218,210,202,194,186,178),(171,219,211,203,195,187,179),(172,220,212,204,196,188,180),(173,221,213,205,197,189,181),(174,222,214,206,198,190,182),(175,223,215,207,199,191,183),(176,224,216,208,200,192,184)], [(1,71,175,131),(2,132,176,72),(3,65,169,133),(4,134,170,66),(5,67,171,135),(6,136,172,68),(7,69,173,129),(8,130,174,70),(9,214,110,58),(10,59,111,215),(11,216,112,60),(12,61,105,209),(13,210,106,62),(14,63,107,211),(15,212,108,64),(16,57,109,213),(17,124,113,219),(18,220,114,125),(19,126,115,221),(20,222,116,127),(21,128,117,223),(22,224,118,121),(23,122,119,217),(24,218,120,123),(25,77,181,137),(26,138,182,78),(27,79,183,139),(28,140,184,80),(29,73,177,141),(30,142,178,74),(31,75,179,143),(32,144,180,76),(33,85,189,145),(34,146,190,86),(35,87,191,147),(36,148,192,88),(37,81,185,149),(38,150,186,82),(39,83,187,151),(40,152,188,84),(41,93,197,153),(42,154,198,94),(43,95,199,155),(44,156,200,96),(45,89,193,157),(46,158,194,90),(47,91,195,159),(48,160,196,92),(49,101,205,161),(50,162,206,102),(51,103,207,163),(52,164,208,104),(53,97,201,165),(54,166,202,98),(55,99,203,167),(56,168,204,100)], [(1,131),(2,176),(3,133),(4,170),(5,135),(6,172),(7,129),(8,174),(10,59),(12,61),(14,63),(16,57),(17,124),(19,126),(21,128),(23,122),(25,137),(26,182),(27,139),(28,184),(29,141),(30,178),(31,143),(32,180),(33,145),(34,190),(35,147),(36,192),(37,149),(38,186),(39,151),(40,188),(41,153),(42,198),(43,155),(44,200),(45,157),(46,194),(47,159),(48,196),(49,161),(50,206),(51,163),(52,208),(53,165),(54,202),(55,167),(56,204),(58,214),(60,216),(62,210),(64,212),(65,169),(67,171),(69,173),(71,175),(73,177),(75,179),(77,181),(79,183),(81,185),(83,187),(85,189),(87,191),(89,193),(91,195),(93,197),(95,199),(97,201),(99,203),(101,205),(103,207),(105,209),(107,211),(109,213),(111,215),(113,219),(115,221),(117,223),(119,217),(121,224),(123,218),(125,220),(127,222)], [(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)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28BH | 56A | ··· | 56AV | 56AW | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C14 | C28 | C28 | C56 | D4 | M4(2) | D8 | SD16 | C4≀C2 | C7×D4 | C7×M4(2) | C7×D8 | C7×SD16 | C7×C4≀C2 |
| kernel | C7×D4⋊C8 | C4×C56 | C7×C4⋊C8 | D4×C28 | C7×C4⋊C4 | D4×C14 | D4⋊C8 | C7×D4 | C4×C8 | C4⋊C8 | C4×D4 | C4⋊C4 | C2×D4 | D4 | C2×C28 | C28 | C28 | C28 | C14 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 12 | 12 | 48 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 |
Matrix representation of C7×D4⋊C8 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 1 | 0 | 0 |
| 0 | 112 | 112 |
| 0 | 2 | 1 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 111 | 112 |
| 44 | 0 | 0 |
| 0 | 0 | 106 |
| 0 | 99 | 0 |
G:=sub<GL(3,GF(113))| [1,0,0,0,28,0,0,0,28],[1,0,0,0,112,2,0,112,1],[112,0,0,0,1,111,0,0,112],[44,0,0,0,0,99,0,106,0] >;
C7×D4⋊C8 in GAP, Magma, Sage, TeX
C_7\times D_4\rtimes C_8
% in TeX
G:=Group("C7xD4:C8"); // GroupNames label
G:=SmallGroup(448,129);
// by ID
G=gap.SmallGroup(448,129);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,1970,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations