metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.44D4, C40.7C23, Q16.6D10, C20.26C24, SD16.1D10, Dic10.44D4, D20.19C23, M4(2).18D10, Dic20.1C22, Dic10.19C23, C5⋊4(Q8○D8), (D5×Q16)⋊2C2, C5⋊D4.7D4, D4⋊D5.C22, C4.118(D4×D5), C40⋊C2.C22, C8⋊D5.C22, C8.C22⋊5D5, C8.7(C22×D5), Q16⋊D5⋊4C2, C4○D4.15D10, D10.58(C2×D4), SD16⋊D5⋊4C2, C20.247(C2×D4), C8.D10⋊4C2, (C2×Q8).92D10, (C8×D5).2C22, C4.26(C23×D5), C22.17(D4×D5), Q8⋊D5.2C22, (C5×SD16).C22, SD16⋊3D5⋊4C2, D4.8D10⋊6C2, D20.2C4⋊4C2, (Q8×D5).3C22, C5⋊2C8.28C23, Dic5.64(C2×D4), (C4×D5).17C23, D4.19(C22×D5), (C5×D4).19C23, D4.D5.2C22, D4.10D10⋊8C2, (C5×Q8).19C23, (C5×Q16).1C22, Q8.19(C22×D5), C5⋊Q16.3C22, (C2×C20).117C23, Q8.10D10⋊6C2, C4○D20.32C22, D4⋊2D5.3C22, C10.127(C22×D4), Q8⋊2D5.3C22, (Q8×C10).153C22, (C5×M4(2)).1C22, (C2×Dic10).207C22, C2.100(C2×D4×D5), (C2×C5⋊Q16)⋊29C2, (C2×C10).72(C2×D4), (C5×C8.C22)⋊4C2, (C5×C4○D4).28C22, (C2×C4).101(C22×D5), (C2×C5⋊2C8).182C22, SmallGroup(320,1451)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.44D4
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, cac-1=dad-1=a11, cbc-1=dbd-1=a10b, dcd-1=a10c3 >
Subgroups: 870 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×Q16, C4○D8, C8.C22, C8.C22, 2- 1+4, C5⋊2C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, Q8○D8, C8×D5, C8⋊D5, C40⋊C2, Dic20, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×Dic10, C2×Dic10, C4○D20, C4○D20, D4⋊2D5, D4⋊2D5, Q8×D5, Q8×D5, Q8⋊2D5, Q8⋊2D5, Q8×C10, C5×C4○D4, D20.2C4, C8.D10, SD16⋊D5, SD16⋊3D5, D5×Q16, Q16⋊D5, C2×C5⋊Q16, D4.8D10, C5×C8.C22, Q8.10D10, D4.10D10, D20.44D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, Q8○D8, D4×D5, C23×D5, C2×D4×D5, D20.44D4
►On 160 points
Generators in S160
(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)
(1 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 160)(29 159)(30 158)(31 157)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(81 123)(82 122)(83 121)(84 140)(85 139)(86 138)(87 137)(88 136)(89 135)(90 134)(91 133)(92 132)(93 131)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)
(1 39 108 83 11 29 118 93)(2 30 109 94 12 40 119 84)(3 21 110 85 13 31 120 95)(4 32 111 96 14 22 101 86)(5 23 112 87 15 33 102 97)(6 34 113 98 16 24 103 88)(7 25 114 89 17 35 104 99)(8 36 115 100 18 26 105 90)(9 27 116 91 19 37 106 81)(10 38 117 82 20 28 107 92)(41 125 75 143 51 135 65 153)(42 136 76 154 52 126 66 144)(43 127 77 145 53 137 67 155)(44 138 78 156 54 128 68 146)(45 129 79 147 55 139 69 157)(46 140 80 158 56 130 70 148)(47 131 61 149 57 121 71 159)(48 122 62 160 58 132 72 150)(49 133 63 151 59 123 73 141)(50 124 64 142 60 134 74 152)
(1 43 11 53)(2 54 12 44)(3 45 13 55)(4 56 14 46)(5 47 15 57)(6 58 16 48)(7 49 17 59)(8 60 18 50)(9 51 19 41)(10 42 20 52)(21 157 31 147)(22 148 32 158)(23 159 33 149)(24 150 34 160)(25 141 35 151)(26 152 36 142)(27 143 37 153)(28 154 38 144)(29 145 39 155)(30 156 40 146)(61 112 71 102)(62 103 72 113)(63 114 73 104)(64 105 74 115)(65 116 75 106)(66 107 76 117)(67 118 77 108)(68 109 78 119)(69 120 79 110)(70 111 80 101)(81 135 91 125)(82 126 92 136)(83 137 93 127)(84 128 94 138)(85 139 95 129)(86 130 96 140)(87 121 97 131)(88 132 98 122)(89 123 99 133)(90 134 100 124)
G:=sub<Sym(160)| (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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,160)(29,159)(30,158)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(81,123)(82,122)(83,121)(84,140)(85,139)(86,138)(87,137)(88,136)(89,135)(90,134)(91,133)(92,132)(93,131)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124), (1,39,108,83,11,29,118,93)(2,30,109,94,12,40,119,84)(3,21,110,85,13,31,120,95)(4,32,111,96,14,22,101,86)(5,23,112,87,15,33,102,97)(6,34,113,98,16,24,103,88)(7,25,114,89,17,35,104,99)(8,36,115,100,18,26,105,90)(9,27,116,91,19,37,106,81)(10,38,117,82,20,28,107,92)(41,125,75,143,51,135,65,153)(42,136,76,154,52,126,66,144)(43,127,77,145,53,137,67,155)(44,138,78,156,54,128,68,146)(45,129,79,147,55,139,69,157)(46,140,80,158,56,130,70,148)(47,131,61,149,57,121,71,159)(48,122,62,160,58,132,72,150)(49,133,63,151,59,123,73,141)(50,124,64,142,60,134,74,152), (1,43,11,53)(2,54,12,44)(3,45,13,55)(4,56,14,46)(5,47,15,57)(6,58,16,48)(7,49,17,59)(8,60,18,50)(9,51,19,41)(10,42,20,52)(21,157,31,147)(22,148,32,158)(23,159,33,149)(24,150,34,160)(25,141,35,151)(26,152,36,142)(27,143,37,153)(28,154,38,144)(29,145,39,155)(30,156,40,146)(61,112,71,102)(62,103,72,113)(63,114,73,104)(64,105,74,115)(65,116,75,106)(66,107,76,117)(67,118,77,108)(68,109,78,119)(69,120,79,110)(70,111,80,101)(81,135,91,125)(82,126,92,136)(83,137,93,127)(84,128,94,138)(85,139,95,129)(86,130,96,140)(87,121,97,131)(88,132,98,122)(89,123,99,133)(90,134,100,124)>;
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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,160)(29,159)(30,158)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(81,123)(82,122)(83,121)(84,140)(85,139)(86,138)(87,137)(88,136)(89,135)(90,134)(91,133)(92,132)(93,131)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124), (1,39,108,83,11,29,118,93)(2,30,109,94,12,40,119,84)(3,21,110,85,13,31,120,95)(4,32,111,96,14,22,101,86)(5,23,112,87,15,33,102,97)(6,34,113,98,16,24,103,88)(7,25,114,89,17,35,104,99)(8,36,115,100,18,26,105,90)(9,27,116,91,19,37,106,81)(10,38,117,82,20,28,107,92)(41,125,75,143,51,135,65,153)(42,136,76,154,52,126,66,144)(43,127,77,145,53,137,67,155)(44,138,78,156,54,128,68,146)(45,129,79,147,55,139,69,157)(46,140,80,158,56,130,70,148)(47,131,61,149,57,121,71,159)(48,122,62,160,58,132,72,150)(49,133,63,151,59,123,73,141)(50,124,64,142,60,134,74,152), (1,43,11,53)(2,54,12,44)(3,45,13,55)(4,56,14,46)(5,47,15,57)(6,58,16,48)(7,49,17,59)(8,60,18,50)(9,51,19,41)(10,42,20,52)(21,157,31,147)(22,148,32,158)(23,159,33,149)(24,150,34,160)(25,141,35,151)(26,152,36,142)(27,143,37,153)(28,154,38,144)(29,145,39,155)(30,156,40,146)(61,112,71,102)(62,103,72,113)(63,114,73,104)(64,105,74,115)(65,116,75,106)(66,107,76,117)(67,118,77,108)(68,109,78,119)(69,120,79,110)(70,111,80,101)(81,135,91,125)(82,126,92,136)(83,137,93,127)(84,128,94,138)(85,139,95,129)(86,130,96,140)(87,121,97,131)(88,132,98,122)(89,123,99,133)(90,134,100,124) );
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)], [(1,71),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,147),(22,146),(23,145),(24,144),(25,143),(26,142),(27,141),(28,160),(29,159),(30,158),(31,157),(32,156),(33,155),(34,154),(35,153),(36,152),(37,151),(38,150),(39,149),(40,148),(41,114),(42,113),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,120),(56,119),(57,118),(58,117),(59,116),(60,115),(81,123),(82,122),(83,121),(84,140),(85,139),(86,138),(87,137),(88,136),(89,135),(90,134),(91,133),(92,132),(93,131),(94,130),(95,129),(96,128),(97,127),(98,126),(99,125),(100,124)], [(1,39,108,83,11,29,118,93),(2,30,109,94,12,40,119,84),(3,21,110,85,13,31,120,95),(4,32,111,96,14,22,101,86),(5,23,112,87,15,33,102,97),(6,34,113,98,16,24,103,88),(7,25,114,89,17,35,104,99),(8,36,115,100,18,26,105,90),(9,27,116,91,19,37,106,81),(10,38,117,82,20,28,107,92),(41,125,75,143,51,135,65,153),(42,136,76,154,52,126,66,144),(43,127,77,145,53,137,67,155),(44,138,78,156,54,128,68,146),(45,129,79,147,55,139,69,157),(46,140,80,158,56,130,70,148),(47,131,61,149,57,121,71,159),(48,122,62,160,58,132,72,150),(49,133,63,151,59,123,73,141),(50,124,64,142,60,134,74,152)], [(1,43,11,53),(2,54,12,44),(3,45,13,55),(4,56,14,46),(5,47,15,57),(6,58,16,48),(7,49,17,59),(8,60,18,50),(9,51,19,41),(10,42,20,52),(21,157,31,147),(22,148,32,158),(23,159,33,149),(24,150,34,160),(25,141,35,151),(26,152,36,142),(27,143,37,153),(28,154,38,144),(29,145,39,155),(30,156,40,146),(61,112,71,102),(62,103,72,113),(63,114,73,104),(64,105,74,115),(65,116,75,106),(66,107,76,117),(67,118,77,108),(68,109,78,119),(69,120,79,110),(70,111,80,101),(81,135,91,125),(82,126,92,136),(83,137,93,127),(84,128,94,138),(85,139,95,129),(86,130,96,140),(87,121,97,131),(88,132,98,122),(89,123,99,133),(90,134,100,124)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | Q8○D8 | D4×D5 | D4×D5 | D20.44D4 |
| kernel | D20.44D4 | D20.2C4 | C8.D10 | SD16⋊D5 | SD16⋊3D5 | D5×Q16 | Q16⋊D5 | C2×C5⋊Q16 | D4.8D10 | C5×C8.C22 | Q8.10D10 | D4.10D10 | Dic10 | D20 | C5⋊D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of D20.44D4 ►in GL8(𝔽41)
| 6 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 29 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 12 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 29 | 0 | 0 |
| 0 | 40 | 35 | 7 | 0 | 0 | 0 | 0 |
| 40 | 0 | 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 32 | 32 |
| 0 | 0 | 0 | 0 | 40 | 0 | 32 | 9 |
| 0 | 0 | 0 | 0 | 9 | 9 | 0 | 1 |
| 0 | 0 | 0 | 0 | 9 | 32 | 40 | 0 |
| 35 | 40 | 6 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 35 | 40 | 6 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 | 0 | 1 |
| 0 | 0 | 0 | 0 | 32 | 9 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 32 | 9 |
| 0 | 0 | 0 | 0 | 0 | 1 | 32 | 32 |
| 6 | 1 | 35 | 39 | 0 | 0 | 0 | 0 |
| 40 | 0 | 2 | 6 | 0 | 0 | 0 | 0 |
| 6 | 1 | 35 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 32 | 9 |
| 0 | 0 | 0 | 0 | 0 | 1 | 32 | 32 |
| 0 | 0 | 0 | 0 | 32 | 32 | 0 | 40 |
| 0 | 0 | 0 | 0 | 9 | 32 | 40 | 0 |
G:=sub<GL(8,GF(41))| [6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,29,0,0,0,0,0,0,29,29,0,0,0,0,29,12,0,0,0,0,0,0,12,12,0,0],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,2,35,1,0,0,0,0,7,6,6,6,0,0,0,0,0,0,0,0,0,40,9,9,0,0,0,0,1,0,9,32,0,0,0,0,32,32,0,40,0,0,0,0,32,9,1,0],[35,1,35,1,0,0,0,0,40,0,40,0,0,0,0,0,6,0,6,40,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,9,32,40,0,0,0,0,0,9,9,0,1,0,0,0,0,0,1,32,32,0,0,0,0,1,0,9,32],[6,40,6,40,0,0,0,0,1,0,1,0,0,0,0,0,35,2,35,1,0,0,0,0,39,6,40,0,0,0,0,0,0,0,0,0,40,0,32,9,0,0,0,0,0,1,32,32,0,0,0,0,32,32,0,40,0,0,0,0,9,32,40,0] >;
D20.44D4 in GAP, Magma, Sage, TeX
D_{20}._{44}D_4 % in TeX
G:=Group("D20.44D4"); // GroupNames label
G:=SmallGroup(320,1451);
// by ID
G=gap.SmallGroup(320,1451);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^3>;
// generators/relations