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), C8⋊D5.C22, C8.C22⋊5D5, C40⋊C2.C22, C8.7(C22×D5), Q16⋊D5⋊4C2, C4○D4.15D10, D10.58(C2×D4), C8.D10⋊4C2, SD16⋊D5⋊4C2, C20.247(C2×D4), (C2×Q8).92D10, (C8×D5).2C22, C22.17(D4×D5), C4.26(C23×D5), Q8⋊D5.2C22, (C5×SD16).C22, SD16⋊3D5⋊4C2, D20.2C4⋊4C2, D4.8D10⋊6C2, (Q8×D5).3C22, C5⋊2C8.28C23, Dic5.64(C2×D4), D4.19(C22×D5), (C5×D4).19C23, (C4×D5).17C23, 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
Subgroups: 870 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4, D4 [×10], Q8, Q8 [×2], Q8 [×10], D5 [×3], C10, C10 [×2], C2×C8 [×3], M4(2), M4(2) [×2], D8, SD16 [×2], SD16 [×4], Q16 [×2], Q16 [×7], C2×Q8, C2×Q8 [×7], C4○D4, C4○D4 [×12], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10, C2×C10, C2×C10, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22, C8.C22 [×5], 2- (1+4) [×2], C5⋊2C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], Dic10 [×5], C4×D5 [×2], C4×D5 [×7], D20 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, Q8○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C5⋊Q16 [×4], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×Dic10, C2×Dic10, C4○D20 [×2], C4○D20 [×3], D4⋊2D5 [×2], D4⋊2D5 [×2], Q8×D5 [×4], Q8×D5, Q8⋊2D5 [×2], Q8⋊2D5, Q8×C10, C5×C4○D4, D20.2C4, C8.D10, SD16⋊D5 [×2], SD16⋊3D5 [×2], D5×Q16 [×2], Q16⋊D5 [×2], C2×C5⋊Q16, D4.8D10, C5×C8.C22, Q8.10D10, D4.10D10, D20.44D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D20.44D4
Generators and relations
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 >
►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)
(1 132)(2 131)(3 130)(4 129)(5 128)(6 127)(7 126)(8 125)(9 124)(10 123)(11 122)(12 121)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 160)(22 159)(23 158)(24 157)(25 156)(26 155)(27 154)(28 153)(29 152)(30 151)(31 150)(32 149)(33 148)(34 147)(35 146)(36 145)(37 144)(38 143)(39 142)(40 141)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 80)(58 79)(59 78)(60 77)(81 105)(82 104)(83 103)(84 102)(85 101)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)
(1 78 141 118 11 68 151 108)(2 69 142 109 12 79 152 119)(3 80 143 120 13 70 153 110)(4 71 144 111 14 61 154 101)(5 62 145 102 15 72 155 112)(6 73 146 113 16 63 156 103)(7 64 147 104 17 74 157 114)(8 75 148 115 18 65 158 105)(9 66 149 106 19 76 159 116)(10 77 150 117 20 67 160 107)(21 89 123 50 31 99 133 60)(22 100 124 41 32 90 134 51)(23 91 125 52 33 81 135 42)(24 82 126 43 34 92 136 53)(25 93 127 54 35 83 137 44)(26 84 128 45 36 94 138 55)(27 95 129 56 37 85 139 46)(28 86 130 47 38 96 140 57)(29 97 131 58 39 87 121 48)(30 88 132 49 40 98 122 59)
(1 36 11 26)(2 27 12 37)(3 38 13 28)(4 29 14 39)(5 40 15 30)(6 31 16 21)(7 22 17 32)(8 33 18 23)(9 24 19 34)(10 35 20 25)(41 64 51 74)(42 75 52 65)(43 66 53 76)(44 77 54 67)(45 68 55 78)(46 79 56 69)(47 70 57 80)(48 61 58 71)(49 72 59 62)(50 63 60 73)(81 115 91 105)(82 106 92 116)(83 117 93 107)(84 108 94 118)(85 119 95 109)(86 110 96 120)(87 101 97 111)(88 112 98 102)(89 103 99 113)(90 114 100 104)(121 154 131 144)(122 145 132 155)(123 156 133 146)(124 147 134 157)(125 158 135 148)(126 149 136 159)(127 160 137 150)(128 151 138 141)(129 142 139 152)(130 153 140 143)
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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,160)(22,159)(23,158)(24,157)(25,156)(26,155)(27,154)(28,153)(29,152)(30,151)(31,150)(32,149)(33,148)(34,147)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,80)(58,79)(59,78)(60,77)(81,105)(82,104)(83,103)(84,102)(85,101)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106), (1,78,141,118,11,68,151,108)(2,69,142,109,12,79,152,119)(3,80,143,120,13,70,153,110)(4,71,144,111,14,61,154,101)(5,62,145,102,15,72,155,112)(6,73,146,113,16,63,156,103)(7,64,147,104,17,74,157,114)(8,75,148,115,18,65,158,105)(9,66,149,106,19,76,159,116)(10,77,150,117,20,67,160,107)(21,89,123,50,31,99,133,60)(22,100,124,41,32,90,134,51)(23,91,125,52,33,81,135,42)(24,82,126,43,34,92,136,53)(25,93,127,54,35,83,137,44)(26,84,128,45,36,94,138,55)(27,95,129,56,37,85,139,46)(28,86,130,47,38,96,140,57)(29,97,131,58,39,87,121,48)(30,88,132,49,40,98,122,59), (1,36,11,26)(2,27,12,37)(3,38,13,28)(4,29,14,39)(5,40,15,30)(6,31,16,21)(7,22,17,32)(8,33,18,23)(9,24,19,34)(10,35,20,25)(41,64,51,74)(42,75,52,65)(43,66,53,76)(44,77,54,67)(45,68,55,78)(46,79,56,69)(47,70,57,80)(48,61,58,71)(49,72,59,62)(50,63,60,73)(81,115,91,105)(82,106,92,116)(83,117,93,107)(84,108,94,118)(85,119,95,109)(86,110,96,120)(87,101,97,111)(88,112,98,102)(89,103,99,113)(90,114,100,104)(121,154,131,144)(122,145,132,155)(123,156,133,146)(124,147,134,157)(125,158,135,148)(126,149,136,159)(127,160,137,150)(128,151,138,141)(129,142,139,152)(130,153,140,143)>;
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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,160)(22,159)(23,158)(24,157)(25,156)(26,155)(27,154)(28,153)(29,152)(30,151)(31,150)(32,149)(33,148)(34,147)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,80)(58,79)(59,78)(60,77)(81,105)(82,104)(83,103)(84,102)(85,101)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106), (1,78,141,118,11,68,151,108)(2,69,142,109,12,79,152,119)(3,80,143,120,13,70,153,110)(4,71,144,111,14,61,154,101)(5,62,145,102,15,72,155,112)(6,73,146,113,16,63,156,103)(7,64,147,104,17,74,157,114)(8,75,148,115,18,65,158,105)(9,66,149,106,19,76,159,116)(10,77,150,117,20,67,160,107)(21,89,123,50,31,99,133,60)(22,100,124,41,32,90,134,51)(23,91,125,52,33,81,135,42)(24,82,126,43,34,92,136,53)(25,93,127,54,35,83,137,44)(26,84,128,45,36,94,138,55)(27,95,129,56,37,85,139,46)(28,86,130,47,38,96,140,57)(29,97,131,58,39,87,121,48)(30,88,132,49,40,98,122,59), (1,36,11,26)(2,27,12,37)(3,38,13,28)(4,29,14,39)(5,40,15,30)(6,31,16,21)(7,22,17,32)(8,33,18,23)(9,24,19,34)(10,35,20,25)(41,64,51,74)(42,75,52,65)(43,66,53,76)(44,77,54,67)(45,68,55,78)(46,79,56,69)(47,70,57,80)(48,61,58,71)(49,72,59,62)(50,63,60,73)(81,115,91,105)(82,106,92,116)(83,117,93,107)(84,108,94,118)(85,119,95,109)(86,110,96,120)(87,101,97,111)(88,112,98,102)(89,103,99,113)(90,114,100,104)(121,154,131,144)(122,145,132,155)(123,156,133,146)(124,147,134,157)(125,158,135,148)(126,149,136,159)(127,160,137,150)(128,151,138,141)(129,142,139,152)(130,153,140,143) );
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)], [(1,132),(2,131),(3,130),(4,129),(5,128),(6,127),(7,126),(8,125),(9,124),(10,123),(11,122),(12,121),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,160),(22,159),(23,158),(24,157),(25,156),(26,155),(27,154),(28,153),(29,152),(30,151),(31,150),(32,149),(33,148),(34,147),(35,146),(36,145),(37,144),(38,143),(39,142),(40,141),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,80),(58,79),(59,78),(60,77),(81,105),(82,104),(83,103),(84,102),(85,101),(86,120),(87,119),(88,118),(89,117),(90,116),(91,115),(92,114),(93,113),(94,112),(95,111),(96,110),(97,109),(98,108),(99,107),(100,106)], [(1,78,141,118,11,68,151,108),(2,69,142,109,12,79,152,119),(3,80,143,120,13,70,153,110),(4,71,144,111,14,61,154,101),(5,62,145,102,15,72,155,112),(6,73,146,113,16,63,156,103),(7,64,147,104,17,74,157,114),(8,75,148,115,18,65,158,105),(9,66,149,106,19,76,159,116),(10,77,150,117,20,67,160,107),(21,89,123,50,31,99,133,60),(22,100,124,41,32,90,134,51),(23,91,125,52,33,81,135,42),(24,82,126,43,34,92,136,53),(25,93,127,54,35,83,137,44),(26,84,128,45,36,94,138,55),(27,95,129,56,37,85,139,46),(28,86,130,47,38,96,140,57),(29,97,131,58,39,87,121,48),(30,88,132,49,40,98,122,59)], [(1,36,11,26),(2,27,12,37),(3,38,13,28),(4,29,14,39),(5,40,15,30),(6,31,16,21),(7,22,17,32),(8,33,18,23),(9,24,19,34),(10,35,20,25),(41,64,51,74),(42,75,52,65),(43,66,53,76),(44,77,54,67),(45,68,55,78),(46,79,56,69),(47,70,57,80),(48,61,58,71),(49,72,59,62),(50,63,60,73),(81,115,91,105),(82,106,92,116),(83,117,93,107),(84,108,94,118),(85,119,95,109),(86,110,96,120),(87,101,97,111),(88,112,98,102),(89,103,99,113),(90,114,100,104),(121,154,131,144),(122,145,132,155),(123,156,133,146),(124,147,134,157),(125,158,135,148),(126,149,136,159),(127,160,137,150),(128,151,138,141),(129,142,139,152),(130,153,140,143)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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