metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊24D4, C42.109D10, C10.592- (1+4), (C4×D4)⋊13D5, (D4×C20)⋊15C2, (C4×D20)⋊29C2, C20⋊2(C4○D4), C4⋊3(C4○D20), C20⋊2D4⋊8C2, C5⋊2(D4⋊6D4), C4.140(D4×D5), C4⋊C4.316D10, C20⋊2Q8⋊24C2, D10.38(C2×D4), C20.346(C2×D4), (C2×D4).214D10, (C2×C10).95C24, C10.50(C22×D4), D10.12D4⋊6C2, C20.48D4⋊20C2, (C4×C20).152C22, (C2×C20).783C23, C22⋊C4.110D10, (C22×C4).208D10, C23.95(C22×D5), (C2×D20).295C22, (D4×C10).257C22, C4⋊Dic5.199C22, (C2×Dic5).41C23, C22.120(C23×D5), C23.D5.12C22, D10⋊C4.98C22, (C22×C20).107C22, (C22×C10).165C23, C10.D4.65C22, (C22×D5).183C23, C2.16(D4.10D10), (C2×Dic10).247C22, C2.23(C2×D4×D5), (D5×C4⋊C4)⋊15C2, (C2×C4○D20)⋊8C2, C2.46(C2×C4○D20), C10.42(C2×C4○D4), (C2×C4×D5).73C22, (C5×C4⋊C4).326C22, (C2×C4).579(C22×D5), (C2×C5⋊D4).121C22, (C5×C22⋊C4).122C22, SmallGroup(320,1223)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1030 in 292 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×14], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×14], Q8 [×4], C23 [×2], C23 [×2], D5 [×4], C10 [×3], C10 [×2], 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 [×4], D10 [×4], C2×C10, C2×C10 [×6], 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 [×12], D20 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], D4⋊6D4, C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20⋊2Q8, C4×D20, D10.12D4 [×4], D5×C4⋊C4 [×2], C20.48D4 [×2], C20⋊2D4 [×2], D4×C20, C2×C4○D20 [×2], D20⋊24D4
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), C22×D5 [×7], D4⋊6D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4.10D10, D20⋊24D4
Generators and relations
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)
(1 113 48 77)(2 114 49 78)(3 115 50 79)(4 116 51 80)(5 117 52 61)(6 118 53 62)(7 119 54 63)(8 120 55 64)(9 101 56 65)(10 102 57 66)(11 103 58 67)(12 104 59 68)(13 105 60 69)(14 106 41 70)(15 107 42 71)(16 108 43 72)(17 109 44 73)(18 110 45 74)(19 111 46 75)(20 112 47 76)(21 125 155 94)(22 126 156 95)(23 127 157 96)(24 128 158 97)(25 129 159 98)(26 130 160 99)(27 131 141 100)(28 132 142 81)(29 133 143 82)(30 134 144 83)(31 135 145 84)(32 136 146 85)(33 137 147 86)(34 138 148 87)(35 139 149 88)(36 140 150 89)(37 121 151 90)(38 122 152 91)(39 123 153 92)(40 124 154 93)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 136)(62 137)(63 138)(64 139)(65 140)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155), (1,113,48,77)(2,114,49,78)(3,115,50,79)(4,116,51,80)(5,117,52,61)(6,118,53,62)(7,119,54,63)(8,120,55,64)(9,101,56,65)(10,102,57,66)(11,103,58,67)(12,104,59,68)(13,105,60,69)(14,106,41,70)(15,107,42,71)(16,108,43,72)(17,109,44,73)(18,110,45,74)(19,111,46,75)(20,112,47,76)(21,125,155,94)(22,126,156,95)(23,127,157,96)(24,128,158,97)(25,129,159,98)(26,130,160,99)(27,131,141,100)(28,132,142,81)(29,133,143,82)(30,134,144,83)(31,135,145,84)(32,136,146,85)(33,137,147,86)(34,138,148,87)(35,139,149,88)(36,140,150,89)(37,121,151,90)(38,122,152,91)(39,123,153,92)(40,124,154,93), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,136)(62,137)(63,138)(64,139)(65,140)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155), (1,113,48,77)(2,114,49,78)(3,115,50,79)(4,116,51,80)(5,117,52,61)(6,118,53,62)(7,119,54,63)(8,120,55,64)(9,101,56,65)(10,102,57,66)(11,103,58,67)(12,104,59,68)(13,105,60,69)(14,106,41,70)(15,107,42,71)(16,108,43,72)(17,109,44,73)(18,110,45,74)(19,111,46,75)(20,112,47,76)(21,125,155,94)(22,126,156,95)(23,127,157,96)(24,128,158,97)(25,129,159,98)(26,130,160,99)(27,131,141,100)(28,132,142,81)(29,133,143,82)(30,134,144,83)(31,135,145,84)(32,136,146,85)(33,137,147,86)(34,138,148,87)(35,139,149,88)(36,140,150,89)(37,121,151,90)(38,122,152,91)(39,123,153,92)(40,124,154,93), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,136)(62,137)(63,138)(64,139)(65,140)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,85),(82,84),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(121,127),(122,126),(123,125),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(141,147),(142,146),(143,145),(148,160),(149,159),(150,158),(151,157),(152,156),(153,155)], [(1,113,48,77),(2,114,49,78),(3,115,50,79),(4,116,51,80),(5,117,52,61),(6,118,53,62),(7,119,54,63),(8,120,55,64),(9,101,56,65),(10,102,57,66),(11,103,58,67),(12,104,59,68),(13,105,60,69),(14,106,41,70),(15,107,42,71),(16,108,43,72),(17,109,44,73),(18,110,45,74),(19,111,46,75),(20,112,47,76),(21,125,155,94),(22,126,156,95),(23,127,157,96),(24,128,158,97),(25,129,159,98),(26,130,160,99),(27,131,141,100),(28,132,142,81),(29,133,143,82),(30,134,144,83),(31,135,145,84),(32,136,146,85),(33,137,147,86),(34,138,148,87),(35,139,149,88),(36,140,150,89),(37,121,151,90),(38,122,152,91),(39,123,153,92),(40,124,154,93)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,21),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(41,155),(42,156),(43,157),(44,158),(45,159),(46,160),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,153),(60,154),(61,136),(62,137),(63,138),(64,139),(65,140),(66,121),(67,122),(68,123),(69,124),(70,125),(71,126),(72,127),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,111),(100,112)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 37 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 25 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 5 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [32,37,0,0,0,0,0,9,0,0,0,0,0,0,34,8,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,25,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | ··· | 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 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 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 | C4○D20 | 2- (1+4) | D4×D5 | D4.10D10 |
| kernel | D20⋊24D4 | C20⋊2Q8 | C4×D20 | D10.12D4 | D5×C4⋊C4 | C20.48D4 | C20⋊2D4 | D4×C20 | C2×C4○D20 | D20 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | 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 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{24}D_4 % in TeX
G:=Group("D20:24D4"); // GroupNames label
G:=SmallGroup(320,1223);
// by ID
G=gap.SmallGroup(320,1223);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀