metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.37D4, Dic10.36D4, C22⋊Q8⋊2D5, C4⋊C4.66D10, C4.102(D4×D5), (C2×C20).266D4, C20.153(C2×D4), C5⋊4(D4.7D4), (C2×Q8).28D10, D20⋊6C4⋊38C2, C10.48C22≀C2, C10.Q16⋊37C2, (C22×C10).93D4, C10.100(C4○D8), C20.55D4⋊14C2, (C2×C20).366C23, (C22×C4).128D10, C23.27(C5⋊D4), (Q8×C10).46C22, C2.16(C23⋊D10), (C2×D20).251C22, C10.90(C8.C22), C2.19(D4.8D10), C2.11(C20.C23), (C22×C20).170C22, (C2×Dic10).278C22, (C2×Q8⋊D5)⋊9C2, (C2×C5⋊Q16)⋊8C2, (C5×C22⋊Q8)⋊2C2, (C2×C4○D20).10C2, (C2×C10).497(C2×D4), (C2×C4).173(C5⋊D4), (C5×C4⋊C4).113C22, (C2×C4).466(C22×D5), C22.172(C2×C5⋊D4), (C2×C5⋊2C8).115C22, SmallGroup(320,674)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.37D4
G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >
Subgroups: 622 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×7], Q8 [×5], C23, C23, D5 [×2], C10 [×3], C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×3], D10 [×4], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C5⋊2C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×C10, D4.7D4, C2×C5⋊2C8 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C2×C5⋊D4, C22×C20, Q8×C10, D20⋊6C4, C10.Q16, C20.55D4, C2×Q8⋊D5, C2×C5⋊Q16, C5×C22⋊Q8, C2×C4○D20, D20.37D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8.C22, C5⋊D4 [×2], C22×D5, D4.7D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C20.C23, D4.8D10, D20.37D4
►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 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)
(1 99 72 145)(2 90 73 156)(3 81 74 147)(4 92 75 158)(5 83 76 149)(6 94 77 160)(7 85 78 151)(8 96 79 142)(9 87 80 153)(10 98 61 144)(11 89 62 155)(12 100 63 146)(13 91 64 157)(14 82 65 148)(15 93 66 159)(16 84 67 150)(17 95 68 141)(18 86 69 152)(19 97 70 143)(20 88 71 154)(21 118 127 54)(22 109 128 45)(23 120 129 56)(24 111 130 47)(25 102 131 58)(26 113 132 49)(27 104 133 60)(28 115 134 51)(29 106 135 42)(30 117 136 53)(31 108 137 44)(32 119 138 55)(33 110 139 46)(34 101 140 57)(35 112 121 48)(36 103 122 59)(37 114 123 50)(38 105 124 41)(39 116 125 52)(40 107 126 43)
(1 28 11 38)(2 29 12 39)(3 30 13 40)(4 31 14 21)(5 32 15 22)(6 33 16 23)(7 34 17 24)(8 35 18 25)(9 36 19 26)(10 37 20 27)(41 99 51 89)(42 100 52 90)(43 81 53 91)(44 82 54 92)(45 83 55 93)(46 84 56 94)(47 85 57 95)(48 86 58 96)(49 87 59 97)(50 88 60 98)(61 123 71 133)(62 124 72 134)(63 125 73 135)(64 126 74 136)(65 127 75 137)(66 128 76 138)(67 129 77 139)(68 130 78 140)(69 131 79 121)(70 132 80 122)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)(106 146 116 156)(107 147 117 157)(108 148 118 158)(109 149 119 159)(110 150 120 160)
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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158), (1,99,72,145)(2,90,73,156)(3,81,74,147)(4,92,75,158)(5,83,76,149)(6,94,77,160)(7,85,78,151)(8,96,79,142)(9,87,80,153)(10,98,61,144)(11,89,62,155)(12,100,63,146)(13,91,64,157)(14,82,65,148)(15,93,66,159)(16,84,67,150)(17,95,68,141)(18,86,69,152)(19,97,70,143)(20,88,71,154)(21,118,127,54)(22,109,128,45)(23,120,129,56)(24,111,130,47)(25,102,131,58)(26,113,132,49)(27,104,133,60)(28,115,134,51)(29,106,135,42)(30,117,136,53)(31,108,137,44)(32,119,138,55)(33,110,139,46)(34,101,140,57)(35,112,121,48)(36,103,122,59)(37,114,123,50)(38,105,124,41)(39,116,125,52)(40,107,126,43), (1,28,11,38)(2,29,12,39)(3,30,13,40)(4,31,14,21)(5,32,15,22)(6,33,16,23)(7,34,17,24)(8,35,18,25)(9,36,19,26)(10,37,20,27)(41,99,51,89)(42,100,52,90)(43,81,53,91)(44,82,54,92)(45,83,55,93)(46,84,56,94)(47,85,57,95)(48,86,58,96)(49,87,59,97)(50,88,60,98)(61,123,71,133)(62,124,72,134)(63,125,73,135)(64,126,74,136)(65,127,75,137)(66,128,76,138)(67,129,77,139)(68,130,78,140)(69,131,79,121)(70,132,80,122)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155)(106,146,116,156)(107,147,117,157)(108,148,118,158)(109,149,119,159)(110,150,120,160)>;
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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158), (1,99,72,145)(2,90,73,156)(3,81,74,147)(4,92,75,158)(5,83,76,149)(6,94,77,160)(7,85,78,151)(8,96,79,142)(9,87,80,153)(10,98,61,144)(11,89,62,155)(12,100,63,146)(13,91,64,157)(14,82,65,148)(15,93,66,159)(16,84,67,150)(17,95,68,141)(18,86,69,152)(19,97,70,143)(20,88,71,154)(21,118,127,54)(22,109,128,45)(23,120,129,56)(24,111,130,47)(25,102,131,58)(26,113,132,49)(27,104,133,60)(28,115,134,51)(29,106,135,42)(30,117,136,53)(31,108,137,44)(32,119,138,55)(33,110,139,46)(34,101,140,57)(35,112,121,48)(36,103,122,59)(37,114,123,50)(38,105,124,41)(39,116,125,52)(40,107,126,43), (1,28,11,38)(2,29,12,39)(3,30,13,40)(4,31,14,21)(5,32,15,22)(6,33,16,23)(7,34,17,24)(8,35,18,25)(9,36,19,26)(10,37,20,27)(41,99,51,89)(42,100,52,90)(43,81,53,91)(44,82,54,92)(45,83,55,93)(46,84,56,94)(47,85,57,95)(48,86,58,96)(49,87,59,97)(50,88,60,98)(61,123,71,133)(62,124,72,134)(63,125,73,135)(64,126,74,136)(65,127,75,137)(66,128,76,138)(67,129,77,139)(68,130,78,140)(69,131,79,121)(70,132,80,122)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155)(106,146,116,156)(107,147,117,157)(108,148,118,158)(109,149,119,159)(110,150,120,160) );
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),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108),(114,120),(115,119),(116,118),(121,126),(122,125),(123,124),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(141,153),(142,152),(143,151),(144,150),(145,149),(146,148),(154,160),(155,159),(156,158)], [(1,99,72,145),(2,90,73,156),(3,81,74,147),(4,92,75,158),(5,83,76,149),(6,94,77,160),(7,85,78,151),(8,96,79,142),(9,87,80,153),(10,98,61,144),(11,89,62,155),(12,100,63,146),(13,91,64,157),(14,82,65,148),(15,93,66,159),(16,84,67,150),(17,95,68,141),(18,86,69,152),(19,97,70,143),(20,88,71,154),(21,118,127,54),(22,109,128,45),(23,120,129,56),(24,111,130,47),(25,102,131,58),(26,113,132,49),(27,104,133,60),(28,115,134,51),(29,106,135,42),(30,117,136,53),(31,108,137,44),(32,119,138,55),(33,110,139,46),(34,101,140,57),(35,112,121,48),(36,103,122,59),(37,114,123,50),(38,105,124,41),(39,116,125,52),(40,107,126,43)], [(1,28,11,38),(2,29,12,39),(3,30,13,40),(4,31,14,21),(5,32,15,22),(6,33,16,23),(7,34,17,24),(8,35,18,25),(9,36,19,26),(10,37,20,27),(41,99,51,89),(42,100,52,90),(43,81,53,91),(44,82,54,92),(45,83,55,93),(46,84,56,94),(47,85,57,95),(48,86,58,96),(49,87,59,97),(50,88,60,98),(61,123,71,133),(62,124,72,134),(63,125,73,135),(64,126,74,136),(65,127,75,137),(66,128,76,138),(67,129,77,139),(68,130,78,140),(69,131,79,121),(70,132,80,122),(101,141,111,151),(102,142,112,152),(103,143,113,153),(104,144,114,154),(105,145,115,155),(106,146,116,156),(107,147,117,157),(108,148,118,158),(109,149,119,159),(110,150,120,160)])
47 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 | ··· | 20P |
| 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 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 8 | 8 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
47 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | C8.C22 | D4×D5 | C20.C23 | D4.8D10 |
| kernel | D20.37D4 | D20⋊6C4 | C10.Q16 | C20.55D4 | C2×Q8⋊D5 | C2×C5⋊Q16 | C5×C22⋊Q8 | C2×C4○D20 | Dic10 | D20 | C2×C20 | C22×C10 | C22⋊Q8 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C2×C4 | C23 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of D20.37D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 36 | 1 |
| 29 | 29 | 0 | 0 | 0 | 0 |
| 29 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 36 |
| 0 | 0 | 0 | 0 | 13 | 8 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 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 | 5 | 40 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,40,36,0,0,0,0,0,1],[29,29,0,0,0,0,29,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,13,0,0,0,0,36,8],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,0,40] >;
D20.37D4 in GAP, Magma, Sage, TeX
D_{20}._{37}D_4 % in TeX
G:=Group("D20.37D4"); // GroupNames label
G:=SmallGroup(320,674);
// by ID
G=gap.SmallGroup(320,674);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations