metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D20, C42.112D10, C10.612- 1+4, (C4×D4)⋊17D5, (C5×D4)⋊11D4, (D4×C20)⋊19C2, (C4×D20)⋊31C2, C5⋊3(D4⋊6D4), C4.23(C2×D20), C20.55(C2×D4), C20⋊14(C4○D4), C20⋊7D4⋊10C2, C4⋊5(D4⋊2D5), C4⋊C4.284D10, C20⋊2Q8⋊25C2, C22.2(C2×D20), D10⋊2Q8⋊15C2, (C2×D4).249D10, (C2×C10).99C24, C2.19(C22×D20), C10.17(C22×D4), (C4×C20).155C22, (C2×C20).160C23, C22⋊C4.113D10, (C22×C4).211D10, C22.D20⋊6C2, C4⋊Dic5.39C22, (C2×D20).220C22, D10⋊C4.5C22, (D4×C10).260C22, (C22×C20).81C22, (C22×D5).34C23, C22.124(C23×D5), C23.173(C22×D5), (C22×C10).169C23, (C2×Dic5).216C23, C2.18(D4.10D10), (C2×Dic10).150C22, (C22×Dic5).97C22, (C2×C10).2(C2×D4), (C2×D4⋊2D5)⋊4C2, (C2×C4⋊Dic5)⋊25C2, C10.74(C2×C4○D4), (C2×C4×D5).74C22, C2.22(C2×D4⋊2D5), (C5×C4⋊C4).329C22, (C2×C4).732(C22×D5), (C2×C5⋊D4).15C22, (C5×C22⋊C4).106C22, SmallGroup(320,1227)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊6D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1030 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic5 [×6], C20 [×4], C20 [×3], D10 [×6], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×8], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×D5 [×2], C22×C10 [×2], D4⋊6D4, C4⋊Dic5, C4⋊Dic5 [×8], D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, D4⋊2D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20⋊2Q8, C4×D20, C22.D20 [×4], D10⋊2Q8 [×2], C2×C4⋊Dic5 [×2], C20⋊7D4 [×2], D4×C20, C2×D4⋊2D5 [×2], D4⋊6D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, D20 [×4], C22×D5 [×7], D4⋊6D4, C2×D20 [×6], D4⋊2D5 [×2], C23×D5, C22×D20, C2×D4⋊2D5, D4.10D10, D4⋊6D20
►On 160 points
Generators in S160
(1 159 69 24)(2 160 70 25)(3 141 71 26)(4 142 72 27)(5 143 73 28)(6 144 74 29)(7 145 75 30)(8 146 76 31)(9 147 77 32)(10 148 78 33)(11 149 79 34)(12 150 80 35)(13 151 61 36)(14 152 62 37)(15 153 63 38)(16 154 64 39)(17 155 65 40)(18 156 66 21)(19 157 67 22)(20 158 68 23)(41 109 136 99)(42 110 137 100)(43 111 138 81)(44 112 139 82)(45 113 140 83)(46 114 121 84)(47 115 122 85)(48 116 123 86)(49 117 124 87)(50 118 125 88)(51 119 126 89)(52 120 127 90)(53 101 128 91)(54 102 129 92)(55 103 130 93)(56 104 131 94)(57 105 132 95)(58 106 133 96)(59 107 134 97)(60 108 135 98)
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 121)(19 122)(20 123)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 81)(39 82)(40 83)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(101 143)(102 144)(103 145)(104 146)(105 147)(106 148)(107 149)(108 150)(109 151)(110 152)(111 153)(112 154)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 141)(120 142)
(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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 123)(50 122)(51 121)(52 140)(53 139)(54 138)(55 137)(56 136)(57 135)(58 134)(59 133)(60 132)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 102)(82 101)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 160)(158 159)
G:=sub<Sym(160)| (1,159,69,24)(2,160,70,25)(3,141,71,26)(4,142,72,27)(5,143,73,28)(6,144,74,29)(7,145,75,30)(8,146,76,31)(9,147,77,32)(10,148,78,33)(11,149,79,34)(12,150,80,35)(13,151,61,36)(14,152,62,37)(15,153,63,38)(16,154,64,39)(17,155,65,40)(18,156,66,21)(19,157,67,22)(20,158,68,23)(41,109,136,99)(42,110,137,100)(43,111,138,81)(44,112,139,82)(45,113,140,83)(46,114,121,84)(47,115,122,85)(48,116,123,86)(49,117,124,87)(50,118,125,88)(51,119,126,89)(52,120,127,90)(53,101,128,91)(54,102,129,92)(55,103,130,93)(56,104,131,94)(57,105,132,95)(58,106,133,96)(59,107,134,97)(60,108,135,98), (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,121)(19,122)(20,123)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,81)(39,82)(40,83)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(101,143)(102,144)(103,145)(104,146)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,141)(120,142), (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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,131)(42,130)(43,129)(44,128)(45,127)(46,126)(47,125)(48,124)(49,123)(50,122)(51,121)(52,140)(53,139)(54,138)(55,137)(56,136)(57,135)(58,134)(59,133)(60,132)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)>;
G:=Group( (1,159,69,24)(2,160,70,25)(3,141,71,26)(4,142,72,27)(5,143,73,28)(6,144,74,29)(7,145,75,30)(8,146,76,31)(9,147,77,32)(10,148,78,33)(11,149,79,34)(12,150,80,35)(13,151,61,36)(14,152,62,37)(15,153,63,38)(16,154,64,39)(17,155,65,40)(18,156,66,21)(19,157,67,22)(20,158,68,23)(41,109,136,99)(42,110,137,100)(43,111,138,81)(44,112,139,82)(45,113,140,83)(46,114,121,84)(47,115,122,85)(48,116,123,86)(49,117,124,87)(50,118,125,88)(51,119,126,89)(52,120,127,90)(53,101,128,91)(54,102,129,92)(55,103,130,93)(56,104,131,94)(57,105,132,95)(58,106,133,96)(59,107,134,97)(60,108,135,98), (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,121)(19,122)(20,123)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,81)(39,82)(40,83)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(101,143)(102,144)(103,145)(104,146)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,141)(120,142), (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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,131)(42,130)(43,129)(44,128)(45,127)(46,126)(47,125)(48,124)(49,123)(50,122)(51,121)(52,140)(53,139)(54,138)(55,137)(56,136)(57,135)(58,134)(59,133)(60,132)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159) );
G=PermutationGroup([(1,159,69,24),(2,160,70,25),(3,141,71,26),(4,142,72,27),(5,143,73,28),(6,144,74,29),(7,145,75,30),(8,146,76,31),(9,147,77,32),(10,148,78,33),(11,149,79,34),(12,150,80,35),(13,151,61,36),(14,152,62,37),(15,153,63,38),(16,154,64,39),(17,155,65,40),(18,156,66,21),(19,157,67,22),(20,158,68,23),(41,109,136,99),(42,110,137,100),(43,111,138,81),(44,112,139,82),(45,113,140,83),(46,114,121,84),(47,115,122,85),(48,116,123,86),(49,117,124,87),(50,118,125,88),(51,119,126,89),(52,120,127,90),(53,101,128,91),(54,102,129,92),(55,103,130,93),(56,104,131,94),(57,105,132,95),(58,106,133,96),(59,107,134,97),(60,108,135,98)], [(1,124),(2,125),(3,126),(4,127),(5,128),(6,129),(7,130),(8,131),(9,132),(10,133),(11,134),(12,135),(13,136),(14,137),(15,138),(16,139),(17,140),(18,121),(19,122),(20,123),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,81),(39,82),(40,83),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(101,143),(102,144),(103,145),(104,146),(105,147),(106,148),(107,149),(108,150),(109,151),(110,152),(111,153),(112,154),(113,155),(114,156),(115,157),(116,158),(117,159),(118,160),(119,141),(120,142)], [(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),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,131),(42,130),(43,129),(44,128),(45,127),(46,126),(47,125),(48,124),(49,123),(50,122),(51,121),(52,140),(53,139),(54,138),(55,137),(56,136),(57,135),(58,134),(59,133),(60,132),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79),(81,102),(82,101),(83,120),(84,119),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,160),(158,159)])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | D20 | 2- 1+4 | D4⋊2D5 | D4.10D10 |
| kernel | D4⋊6D20 | C20⋊2Q8 | C4×D20 | C22.D20 | D10⋊2Q8 | C2×C4⋊Dic5 | C20⋊7D4 | D4×C20 | C2×D4⋊2D5 | C5×D4 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D4⋊6D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 32 | 0 |
| 27 | 2 | 0 | 0 |
| 25 | 11 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 30 | 2 | 0 | 0 |
| 22 | 11 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,9,0],[27,25,0,0,2,11,0,0,0,0,40,0,0,0,0,40],[30,22,0,0,2,11,0,0,0,0,40,0,0,0,0,1] >;
D4⋊6D20 in GAP, Magma, Sage, TeX
D_4\rtimes_6D_{20} % in TeX
G:=Group("D4:6D20"); // GroupNames label
G:=SmallGroup(320,1227);
// by ID
G=gap.SmallGroup(320,1227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations