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