metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.32D4, C23.11D20, Dic10.31D4, C22⋊C8⋊5D5, C4.121(D4×D5), D20⋊5C4⋊5C2, C10.9C22≀C2, (C2×Dic20)⋊2C2, C10.8(C4○D8), C20.333(C2×D4), (C2×C4).118D20, (C2×C20).240D4, (C2×C8).108D10, C5⋊1(D4.7D4), (C2×C40).3C22, (C22×C4).83D10, (C22×C10).53D4, C20.44D4⋊10C2, C20.48D4⋊16C2, (C2×C20).743C23, C22.106(C2×D20), C10.9(C8.C22), C4⋊Dic5.12C22, C2.12(C22⋊D20), C2.10(D40⋊7C2), C2.12(C8.D10), (C2×D20).198C22, (C22×C20).96C22, (C2×Dic10).216C22, (C5×C22⋊C8)⋊7C2, (C2×C40⋊C2)⋊10C2, (C2×C4○D20).2C2, (C2×C10).126(C2×D4), (C2×C4).688(C22×D5), SmallGroup(320,360)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.32D4
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a5c3 >
Subgroups: 686 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, D4.7D4, C40⋊C2, Dic20, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C20.44D4, D20⋊5C4, C5×C22⋊C8, C2×C40⋊C2, C2×Dic20, C20.48D4, C2×C4○D20, D20.32D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8.C22, D20, C22×D5, D4.7D4, C2×D20, D4×D5, C22⋊D20, D40⋊7C2, C8.D10, D20.32D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(116 120)(117 119)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
(1 125 99 50 11 135 89 60)(2 126 100 51 12 136 90 41)(3 127 81 52 13 137 91 42)(4 128 82 53 14 138 92 43)(5 129 83 54 15 139 93 44)(6 130 84 55 16 140 94 45)(7 131 85 56 17 121 95 46)(8 132 86 57 18 122 96 47)(9 133 87 58 19 123 97 48)(10 134 88 59 20 124 98 49)(21 142 65 107 31 152 75 117)(22 143 66 108 32 153 76 118)(23 144 67 109 33 154 77 119)(24 145 68 110 34 155 78 120)(25 146 69 111 35 156 79 101)(26 147 70 112 36 157 80 102)(27 148 71 113 37 158 61 103)(28 149 72 114 38 159 62 104)(29 150 73 115 39 160 63 105)(30 151 74 116 40 141 64 106)
(1 74 11 64)(2 75 12 65)(3 76 13 66)(4 77 14 67)(5 78 15 68)(6 79 16 69)(7 80 17 70)(8 61 18 71)(9 62 19 72)(10 63 20 73)(21 90 31 100)(22 91 32 81)(23 92 33 82)(24 93 34 83)(25 94 35 84)(26 95 36 85)(27 96 37 86)(28 97 38 87)(29 98 39 88)(30 99 40 89)(41 102 51 112)(42 103 52 113)(43 104 53 114)(44 105 54 115)(45 106 55 116)(46 107 56 117)(47 108 57 118)(48 109 58 119)(49 110 59 120)(50 111 60 101)(121 142 131 152)(122 143 132 153)(123 144 133 154)(124 145 134 155)(125 146 135 156)(126 147 136 157)(127 148 137 158)(128 149 138 159)(129 150 139 160)(130 151 140 141)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(116,120)(117,119)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,125,99,50,11,135,89,60)(2,126,100,51,12,136,90,41)(3,127,81,52,13,137,91,42)(4,128,82,53,14,138,92,43)(5,129,83,54,15,139,93,44)(6,130,84,55,16,140,94,45)(7,131,85,56,17,121,95,46)(8,132,86,57,18,122,96,47)(9,133,87,58,19,123,97,48)(10,134,88,59,20,124,98,49)(21,142,65,107,31,152,75,117)(22,143,66,108,32,153,76,118)(23,144,67,109,33,154,77,119)(24,145,68,110,34,155,78,120)(25,146,69,111,35,156,79,101)(26,147,70,112,36,157,80,102)(27,148,71,113,37,158,61,103)(28,149,72,114,38,159,62,104)(29,150,73,115,39,160,63,105)(30,151,74,116,40,141,64,106), (1,74,11,64)(2,75,12,65)(3,76,13,66)(4,77,14,67)(5,78,15,68)(6,79,16,69)(7,80,17,70)(8,61,18,71)(9,62,19,72)(10,63,20,73)(21,90,31,100)(22,91,32,81)(23,92,33,82)(24,93,34,83)(25,94,35,84)(26,95,36,85)(27,96,37,86)(28,97,38,87)(29,98,39,88)(30,99,40,89)(41,102,51,112)(42,103,52,113)(43,104,53,114)(44,105,54,115)(45,106,55,116)(46,107,56,117)(47,108,57,118)(48,109,58,119)(49,110,59,120)(50,111,60,101)(121,142,131,152)(122,143,132,153)(123,144,133,154)(124,145,134,155)(125,146,135,156)(126,147,136,157)(127,148,137,158)(128,149,138,159)(129,150,139,160)(130,151,140,141)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(116,120)(117,119)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,125,99,50,11,135,89,60)(2,126,100,51,12,136,90,41)(3,127,81,52,13,137,91,42)(4,128,82,53,14,138,92,43)(5,129,83,54,15,139,93,44)(6,130,84,55,16,140,94,45)(7,131,85,56,17,121,95,46)(8,132,86,57,18,122,96,47)(9,133,87,58,19,123,97,48)(10,134,88,59,20,124,98,49)(21,142,65,107,31,152,75,117)(22,143,66,108,32,153,76,118)(23,144,67,109,33,154,77,119)(24,145,68,110,34,155,78,120)(25,146,69,111,35,156,79,101)(26,147,70,112,36,157,80,102)(27,148,71,113,37,158,61,103)(28,149,72,114,38,159,62,104)(29,150,73,115,39,160,63,105)(30,151,74,116,40,141,64,106), (1,74,11,64)(2,75,12,65)(3,76,13,66)(4,77,14,67)(5,78,15,68)(6,79,16,69)(7,80,17,70)(8,61,18,71)(9,62,19,72)(10,63,20,73)(21,90,31,100)(22,91,32,81)(23,92,33,82)(24,93,34,83)(25,94,35,84)(26,95,36,85)(27,96,37,86)(28,97,38,87)(29,98,39,88)(30,99,40,89)(41,102,51,112)(42,103,52,113)(43,104,53,114)(44,105,54,115)(45,106,55,116)(46,107,56,117)(47,108,57,118)(48,109,58,119)(49,110,59,120)(50,111,60,101)(121,142,131,152)(122,143,132,153)(123,144,133,154)(124,145,134,155)(125,146,135,156)(126,147,136,157)(127,148,137,158)(128,149,138,159)(129,150,139,160)(130,151,140,141) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109),(116,120),(117,119),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(134,140),(135,139),(136,138),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159)], [(1,125,99,50,11,135,89,60),(2,126,100,51,12,136,90,41),(3,127,81,52,13,137,91,42),(4,128,82,53,14,138,92,43),(5,129,83,54,15,139,93,44),(6,130,84,55,16,140,94,45),(7,131,85,56,17,121,95,46),(8,132,86,57,18,122,96,47),(9,133,87,58,19,123,97,48),(10,134,88,59,20,124,98,49),(21,142,65,107,31,152,75,117),(22,143,66,108,32,153,76,118),(23,144,67,109,33,154,77,119),(24,145,68,110,34,155,78,120),(25,146,69,111,35,156,79,101),(26,147,70,112,36,157,80,102),(27,148,71,113,37,158,61,103),(28,149,72,114,38,159,62,104),(29,150,73,115,39,160,63,105),(30,151,74,116,40,141,64,106)], [(1,74,11,64),(2,75,12,65),(3,76,13,66),(4,77,14,67),(5,78,15,68),(6,79,16,69),(7,80,17,70),(8,61,18,71),(9,62,19,72),(10,63,20,73),(21,90,31,100),(22,91,32,81),(23,92,33,82),(24,93,34,83),(25,94,35,84),(26,95,36,85),(27,96,37,86),(28,97,38,87),(29,98,39,88),(30,99,40,89),(41,102,51,112),(42,103,52,113),(43,104,53,114),(44,105,54,115),(45,106,55,116),(46,107,56,117),(47,108,57,118),(48,109,58,119),(49,110,59,120),(50,111,60,101),(121,142,131,152),(122,143,132,153),(123,144,133,154),(124,145,134,155),(125,146,135,156),(126,147,136,157),(127,148,137,158),(128,149,138,159),(129,150,139,160),(130,151,140,141)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 | C8.C22 | D4×D5 | C8.D10 |
| kernel | D20.32D4 | C20.44D4 | D20⋊5C4 | C5×C22⋊C8 | C2×C40⋊C2 | C2×Dic20 | C20.48D4 | C2×C4○D20 | Dic10 | D20 | C2×C20 | C22×C10 | C22⋊C8 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of D20.32D4 ►in GL6(𝔽41)
| 15 | 8 | 0 | 0 | 0 | 0 |
| 23 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 15 | 8 | 0 | 0 | 0 | 0 |
| 13 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 4 | 14 | 0 | 0 | 0 | 0 |
| 30 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 39 |
| 0 | 0 | 0 | 0 | 10 | 12 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 29 | 40 |
G:=sub<GL(6,GF(41))| [15,23,0,0,0,0,8,26,0,0,0,0,0,0,0,1,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[15,13,0,0,0,0,8,26,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,12,0,0,0,0,0,1],[4,30,0,0,0,0,14,13,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,29,10,0,0,0,0,39,12],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,29,0,0,0,0,0,40] >;
D20.32D4 in GAP, Magma, Sage, TeX
D_{20}._{32}D_4 % in TeX
G:=Group("D20.32D4"); // GroupNames label
G:=SmallGroup(320,360);
// by ID
G=gap.SmallGroup(320,360);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^5*c^3>;
// generators/relations