metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊12D4, C42.172D10, C10.352- (1+4), C4⋊Q8⋊10D5, C4.73(D4×D5), (C4×D20)⋊51C2, C20⋊7(C4○D4), C5⋊7(D4⋊6D4), C20.71(C2×D4), C4⋊2D20⋊40C2, C4⋊C4.123D10, C4⋊2(Q8⋊2D5), D10.48(C2×D4), D10⋊3Q8⋊35C2, (C2×Q8).145D10, (C2×C20).103C23, (C2×C10).270C24, (C4×C20).211C22, C10.100(C22×D4), D10.13D4⋊46C2, (C2×D20).279C22, C4⋊Dic5.384C22, (Q8×C10).137C22, C22.291(C23×D5), (C2×Dic5).141C23, C10.D4.60C22, (C22×D5).241C23, D10⋊C4.151C22, C2.36(Q8.10D10), C2.73(C2×D4×D5), (D5×C4⋊C4)⋊44C2, (C5×C4⋊Q8)⋊12C2, (C2×Q8⋊2D5)⋊13C2, C10.121(C2×C4○D4), C2.28(C2×Q8⋊2D5), (C2×C4×D5).153C22, (C2×C4).93(C22×D5), (C5×C4⋊C4).213C22, SmallGroup(320,1398)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1110 in 292 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×14], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×20], D4 [×14], Q8 [×4], C23 [×4], D5 [×6], C10 [×3], C42, C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×8], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], C20 [×4], C20 [×5], D10 [×4], D10 [×10], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C4×D5 [×16], D20 [×4], D20 [×10], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5 [×4], D4⋊6D4, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5 [×8], C2×D20 [×2], C2×D20 [×4], Q8⋊2D5 [×8], Q8×C10 [×2], C4×D20 [×2], D5×C4⋊C4 [×2], D10.13D4 [×4], C4⋊2D20 [×2], D10⋊3Q8 [×2], C5×C4⋊Q8, C2×Q8⋊2D5 [×2], D20⋊12D4
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, D4×D5 [×2], Q8⋊2D5 [×2], C23×D5, C2×D4×D5, C2×Q8⋊2D5, Q8.10D10, D20⋊12D4
Generators and relations
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, 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)
(1 121)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 124)(19 123)(20 122)(21 112)(22 111)(23 110)(24 109)(25 108)(26 107)(27 106)(28 105)(29 104)(30 103)(31 102)(32 101)(33 120)(34 119)(35 118)(36 117)(37 116)(38 115)(39 114)(40 113)(41 61)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(81 146)(82 145)(83 144)(84 143)(85 142)(86 141)(87 160)(88 159)(89 158)(90 157)(91 156)(92 155)(93 154)(94 153)(95 152)(96 151)(97 150)(98 149)(99 148)(100 147)
(1 69 117 156)(2 70 118 157)(3 71 119 158)(4 72 120 159)(5 73 101 160)(6 74 102 141)(7 75 103 142)(8 76 104 143)(9 77 105 144)(10 78 106 145)(11 79 107 146)(12 80 108 147)(13 61 109 148)(14 62 110 149)(15 63 111 150)(16 64 112 151)(17 65 113 152)(18 66 114 153)(19 67 115 154)(20 68 116 155)(21 96 126 58)(22 97 127 59)(23 98 128 60)(24 99 129 41)(25 100 130 42)(26 81 131 43)(27 82 132 44)(28 83 133 45)(29 84 134 46)(30 85 135 47)(31 86 136 48)(32 87 137 49)(33 88 138 50)(34 89 139 51)(35 90 140 52)(36 91 121 53)(37 92 122 54)(38 93 123 55)(39 94 124 56)(40 95 125 57)
(1 117)(2 106)(3 115)(4 104)(5 113)(6 102)(7 111)(8 120)(9 109)(10 118)(11 107)(12 116)(13 105)(14 114)(15 103)(16 112)(17 101)(18 110)(19 119)(20 108)(21 128)(22 137)(23 126)(24 135)(25 124)(26 133)(27 122)(28 131)(29 140)(30 129)(31 138)(32 127)(33 136)(34 125)(35 134)(36 123)(37 132)(38 121)(39 130)(40 139)(41 47)(42 56)(43 45)(44 54)(46 52)(48 50)(49 59)(51 57)(53 55)(58 60)(61 77)(62 66)(63 75)(65 73)(67 71)(68 80)(70 78)(72 76)(81 83)(82 92)(84 90)(85 99)(86 88)(87 97)(89 95)(91 93)(94 100)(96 98)(142 150)(143 159)(144 148)(145 157)(147 155)(149 153)(152 160)(154 158)
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), (1,121)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,61)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(81,146)(82,145)(83,144)(84,143)(85,142)(86,141)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147), (1,69,117,156)(2,70,118,157)(3,71,119,158)(4,72,120,159)(5,73,101,160)(6,74,102,141)(7,75,103,142)(8,76,104,143)(9,77,105,144)(10,78,106,145)(11,79,107,146)(12,80,108,147)(13,61,109,148)(14,62,110,149)(15,63,111,150)(16,64,112,151)(17,65,113,152)(18,66,114,153)(19,67,115,154)(20,68,116,155)(21,96,126,58)(22,97,127,59)(23,98,128,60)(24,99,129,41)(25,100,130,42)(26,81,131,43)(27,82,132,44)(28,83,133,45)(29,84,134,46)(30,85,135,47)(31,86,136,48)(32,87,137,49)(33,88,138,50)(34,89,139,51)(35,90,140,52)(36,91,121,53)(37,92,122,54)(38,93,123,55)(39,94,124,56)(40,95,125,57), (1,117)(2,106)(3,115)(4,104)(5,113)(6,102)(7,111)(8,120)(9,109)(10,118)(11,107)(12,116)(13,105)(14,114)(15,103)(16,112)(17,101)(18,110)(19,119)(20,108)(21,128)(22,137)(23,126)(24,135)(25,124)(26,133)(27,122)(28,131)(29,140)(30,129)(31,138)(32,127)(33,136)(34,125)(35,134)(36,123)(37,132)(38,121)(39,130)(40,139)(41,47)(42,56)(43,45)(44,54)(46,52)(48,50)(49,59)(51,57)(53,55)(58,60)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(81,83)(82,92)(84,90)(85,99)(86,88)(87,97)(89,95)(91,93)(94,100)(96,98)(142,150)(143,159)(144,148)(145,157)(147,155)(149,153)(152,160)(154,158)>;
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), (1,121)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,61)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(81,146)(82,145)(83,144)(84,143)(85,142)(86,141)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147), (1,69,117,156)(2,70,118,157)(3,71,119,158)(4,72,120,159)(5,73,101,160)(6,74,102,141)(7,75,103,142)(8,76,104,143)(9,77,105,144)(10,78,106,145)(11,79,107,146)(12,80,108,147)(13,61,109,148)(14,62,110,149)(15,63,111,150)(16,64,112,151)(17,65,113,152)(18,66,114,153)(19,67,115,154)(20,68,116,155)(21,96,126,58)(22,97,127,59)(23,98,128,60)(24,99,129,41)(25,100,130,42)(26,81,131,43)(27,82,132,44)(28,83,133,45)(29,84,134,46)(30,85,135,47)(31,86,136,48)(32,87,137,49)(33,88,138,50)(34,89,139,51)(35,90,140,52)(36,91,121,53)(37,92,122,54)(38,93,123,55)(39,94,124,56)(40,95,125,57), (1,117)(2,106)(3,115)(4,104)(5,113)(6,102)(7,111)(8,120)(9,109)(10,118)(11,107)(12,116)(13,105)(14,114)(15,103)(16,112)(17,101)(18,110)(19,119)(20,108)(21,128)(22,137)(23,126)(24,135)(25,124)(26,133)(27,122)(28,131)(29,140)(30,129)(31,138)(32,127)(33,136)(34,125)(35,134)(36,123)(37,132)(38,121)(39,130)(40,139)(41,47)(42,56)(43,45)(44,54)(46,52)(48,50)(49,59)(51,57)(53,55)(58,60)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(81,83)(82,92)(84,90)(85,99)(86,88)(87,97)(89,95)(91,93)(94,100)(96,98)(142,150)(143,159)(144,148)(145,157)(147,155)(149,153)(152,160)(154,158) );
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)], [(1,121),(2,140),(3,139),(4,138),(5,137),(6,136),(7,135),(8,134),(9,133),(10,132),(11,131),(12,130),(13,129),(14,128),(15,127),(16,126),(17,125),(18,124),(19,123),(20,122),(21,112),(22,111),(23,110),(24,109),(25,108),(26,107),(27,106),(28,105),(29,104),(30,103),(31,102),(32,101),(33,120),(34,119),(35,118),(36,117),(37,116),(38,115),(39,114),(40,113),(41,61),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(81,146),(82,145),(83,144),(84,143),(85,142),(86,141),(87,160),(88,159),(89,158),(90,157),(91,156),(92,155),(93,154),(94,153),(95,152),(96,151),(97,150),(98,149),(99,148),(100,147)], [(1,69,117,156),(2,70,118,157),(3,71,119,158),(4,72,120,159),(5,73,101,160),(6,74,102,141),(7,75,103,142),(8,76,104,143),(9,77,105,144),(10,78,106,145),(11,79,107,146),(12,80,108,147),(13,61,109,148),(14,62,110,149),(15,63,111,150),(16,64,112,151),(17,65,113,152),(18,66,114,153),(19,67,115,154),(20,68,116,155),(21,96,126,58),(22,97,127,59),(23,98,128,60),(24,99,129,41),(25,100,130,42),(26,81,131,43),(27,82,132,44),(28,83,133,45),(29,84,134,46),(30,85,135,47),(31,86,136,48),(32,87,137,49),(33,88,138,50),(34,89,139,51),(35,90,140,52),(36,91,121,53),(37,92,122,54),(38,93,123,55),(39,94,124,56),(40,95,125,57)], [(1,117),(2,106),(3,115),(4,104),(5,113),(6,102),(7,111),(8,120),(9,109),(10,118),(11,107),(12,116),(13,105),(14,114),(15,103),(16,112),(17,101),(18,110),(19,119),(20,108),(21,128),(22,137),(23,126),(24,135),(25,124),(26,133),(27,122),(28,131),(29,140),(30,129),(31,138),(32,127),(33,136),(34,125),(35,134),(36,123),(37,132),(38,121),(39,130),(40,139),(41,47),(42,56),(43,45),(44,54),(46,52),(48,50),(49,59),(51,57),(53,55),(58,60),(61,77),(62,66),(63,75),(65,73),(67,71),(68,80),(70,78),(72,76),(81,83),(82,92),(84,90),(85,99),(86,88),(87,97),(89,95),(91,93),(94,100),(96,98),(142,150),(143,159),(144,148),(145,157),(147,155),(149,153),(152,160),(154,158)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 1 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 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 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 18 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,32,2,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,0,40,0,0,0,0,0,0,32,2,0,0,0,0,1,9],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,18,0,0,0,0,0,40] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | Q8⋊2D5 | Q8.10D10 |
| kernel | D20⋊12D4 | C4×D20 | D5×C4⋊C4 | D10.13D4 | C4⋊2D20 | D10⋊3Q8 | C5×C4⋊Q8 | C2×Q8⋊2D5 | D20 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{12}D_4 % in TeX
G:=Group("D20:12D4"); // GroupNames label
G:=SmallGroup(320,1398);
// by ID
G=gap.SmallGroup(320,1398);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,297,136,12550]);
// 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,d*a*d=a^9,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀