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⋊D20⋊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
Generators and relations for D20⋊12D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, dcd=c-1 >
Subgroups: 1110 in 292 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4⋊6D4, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, Q8×C10, C4×D20, D5×C4⋊C4, D10.13D4, C4⋊D20, D10⋊3Q8, C5×C4⋊Q8, C2×Q8⋊2D5, D20⋊12D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C22×D5, D4⋊6D4, D4×D5, Q8⋊2D5, C23×D5, C2×D4×D5, C2×Q8⋊2D5, Q8.10D10, D20⋊12D4
►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)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 140)(17 139)(18 138)(19 137)(20 136)(21 45)(22 44)(23 43)(24 42)(25 41)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(61 154)(62 153)(63 152)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 144)(72 143)(73 142)(74 141)(75 160)(76 159)(77 158)(78 157)(79 156)(80 155)(81 105)(82 104)(83 103)(84 102)(85 101)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)
(1 22 99 73)(2 23 100 74)(3 24 81 75)(4 25 82 76)(5 26 83 77)(6 27 84 78)(7 28 85 79)(8 29 86 80)(9 30 87 61)(10 31 88 62)(11 32 89 63)(12 33 90 64)(13 34 91 65)(14 35 92 66)(15 36 93 67)(16 37 94 68)(17 38 95 69)(18 39 96 70)(19 40 97 71)(20 21 98 72)(41 104 159 132)(42 105 160 133)(43 106 141 134)(44 107 142 135)(45 108 143 136)(46 109 144 137)(47 110 145 138)(48 111 146 139)(49 112 147 140)(50 113 148 121)(51 114 149 122)(52 115 150 123)(53 116 151 124)(54 117 152 125)(55 118 153 126)(56 119 154 127)(57 120 155 128)(58 101 156 129)(59 102 157 130)(60 103 158 131)
(1 99)(2 88)(3 97)(4 86)(5 95)(6 84)(7 93)(8 82)(9 91)(10 100)(11 89)(12 98)(13 87)(14 96)(15 85)(16 94)(17 83)(18 92)(19 81)(20 90)(21 33)(23 31)(24 40)(25 29)(26 38)(28 36)(30 34)(35 39)(41 59)(42 48)(43 57)(44 46)(45 55)(47 53)(49 51)(50 60)(52 58)(54 56)(61 65)(62 74)(64 72)(66 70)(67 79)(69 77)(71 75)(76 80)(101 123)(102 132)(103 121)(104 130)(105 139)(106 128)(107 137)(108 126)(109 135)(110 124)(111 133)(112 122)(113 131)(114 140)(115 129)(116 138)(117 127)(118 136)(119 125)(120 134)(141 155)(142 144)(143 153)(145 151)(146 160)(147 149)(148 158)(150 156)(152 154)(157 159)
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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,140)(17,139)(18,138)(19,137)(20,136)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,105)(82,104)(83,103)(84,102)(85,101)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106), (1,22,99,73)(2,23,100,74)(3,24,81,75)(4,25,82,76)(5,26,83,77)(6,27,84,78)(7,28,85,79)(8,29,86,80)(9,30,87,61)(10,31,88,62)(11,32,89,63)(12,33,90,64)(13,34,91,65)(14,35,92,66)(15,36,93,67)(16,37,94,68)(17,38,95,69)(18,39,96,70)(19,40,97,71)(20,21,98,72)(41,104,159,132)(42,105,160,133)(43,106,141,134)(44,107,142,135)(45,108,143,136)(46,109,144,137)(47,110,145,138)(48,111,146,139)(49,112,147,140)(50,113,148,121)(51,114,149,122)(52,115,150,123)(53,116,151,124)(54,117,152,125)(55,118,153,126)(56,119,154,127)(57,120,155,128)(58,101,156,129)(59,102,157,130)(60,103,158,131), (1,99)(2,88)(3,97)(4,86)(5,95)(6,84)(7,93)(8,82)(9,91)(10,100)(11,89)(12,98)(13,87)(14,96)(15,85)(16,94)(17,83)(18,92)(19,81)(20,90)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39)(41,59)(42,48)(43,57)(44,46)(45,55)(47,53)(49,51)(50,60)(52,58)(54,56)(61,65)(62,74)(64,72)(66,70)(67,79)(69,77)(71,75)(76,80)(101,123)(102,132)(103,121)(104,130)(105,139)(106,128)(107,137)(108,126)(109,135)(110,124)(111,133)(112,122)(113,131)(114,140)(115,129)(116,138)(117,127)(118,136)(119,125)(120,134)(141,155)(142,144)(143,153)(145,151)(146,160)(147,149)(148,158)(150,156)(152,154)(157,159)>;
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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,140)(17,139)(18,138)(19,137)(20,136)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,105)(82,104)(83,103)(84,102)(85,101)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106), (1,22,99,73)(2,23,100,74)(3,24,81,75)(4,25,82,76)(5,26,83,77)(6,27,84,78)(7,28,85,79)(8,29,86,80)(9,30,87,61)(10,31,88,62)(11,32,89,63)(12,33,90,64)(13,34,91,65)(14,35,92,66)(15,36,93,67)(16,37,94,68)(17,38,95,69)(18,39,96,70)(19,40,97,71)(20,21,98,72)(41,104,159,132)(42,105,160,133)(43,106,141,134)(44,107,142,135)(45,108,143,136)(46,109,144,137)(47,110,145,138)(48,111,146,139)(49,112,147,140)(50,113,148,121)(51,114,149,122)(52,115,150,123)(53,116,151,124)(54,117,152,125)(55,118,153,126)(56,119,154,127)(57,120,155,128)(58,101,156,129)(59,102,157,130)(60,103,158,131), (1,99)(2,88)(3,97)(4,86)(5,95)(6,84)(7,93)(8,82)(9,91)(10,100)(11,89)(12,98)(13,87)(14,96)(15,85)(16,94)(17,83)(18,92)(19,81)(20,90)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39)(41,59)(42,48)(43,57)(44,46)(45,55)(47,53)(49,51)(50,60)(52,58)(54,56)(61,65)(62,74)(64,72)(66,70)(67,79)(69,77)(71,75)(76,80)(101,123)(102,132)(103,121)(104,130)(105,139)(106,128)(107,137)(108,126)(109,135)(110,124)(111,133)(112,122)(113,131)(114,140)(115,129)(116,138)(117,127)(118,136)(119,125)(120,134)(141,155)(142,144)(143,153)(145,151)(146,160)(147,149)(148,158)(150,156)(152,154)(157,159) );
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)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,126),(11,125),(12,124),(13,123),(14,122),(15,121),(16,140),(17,139),(18,138),(19,137),(20,136),(21,45),(22,44),(23,43),(24,42),(25,41),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(61,154),(62,153),(63,152),(64,151),(65,150),(66,149),(67,148),(68,147),(69,146),(70,145),(71,144),(72,143),(73,142),(74,141),(75,160),(76,159),(77,158),(78,157),(79,156),(80,155),(81,105),(82,104),(83,103),(84,102),(85,101),(86,120),(87,119),(88,118),(89,117),(90,116),(91,115),(92,114),(93,113),(94,112),(95,111),(96,110),(97,109),(98,108),(99,107),(100,106)], [(1,22,99,73),(2,23,100,74),(3,24,81,75),(4,25,82,76),(5,26,83,77),(6,27,84,78),(7,28,85,79),(8,29,86,80),(9,30,87,61),(10,31,88,62),(11,32,89,63),(12,33,90,64),(13,34,91,65),(14,35,92,66),(15,36,93,67),(16,37,94,68),(17,38,95,69),(18,39,96,70),(19,40,97,71),(20,21,98,72),(41,104,159,132),(42,105,160,133),(43,106,141,134),(44,107,142,135),(45,108,143,136),(46,109,144,137),(47,110,145,138),(48,111,146,139),(49,112,147,140),(50,113,148,121),(51,114,149,122),(52,115,150,123),(53,116,151,124),(54,117,152,125),(55,118,153,126),(56,119,154,127),(57,120,155,128),(58,101,156,129),(59,102,157,130),(60,103,158,131)], [(1,99),(2,88),(3,97),(4,86),(5,95),(6,84),(7,93),(8,82),(9,91),(10,100),(11,89),(12,98),(13,87),(14,96),(15,85),(16,94),(17,83),(18,92),(19,81),(20,90),(21,33),(23,31),(24,40),(25,29),(26,38),(28,36),(30,34),(35,39),(41,59),(42,48),(43,57),(44,46),(45,55),(47,53),(49,51),(50,60),(52,58),(54,56),(61,65),(62,74),(64,72),(66,70),(67,79),(69,77),(71,75),(76,80),(101,123),(102,132),(103,121),(104,130),(105,139),(106,128),(107,137),(108,126),(109,135),(110,124),(111,133),(112,122),(113,131),(114,140),(115,129),(116,138),(117,127),(118,136),(119,125),(120,134),(141,155),(142,144),(143,153),(145,151),(146,160),(147,149),(148,158),(150,156),(152,154),(157,159)]])
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⋊D20 | 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 |
Matrix representation of D20⋊12D4 ►in 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] >;
D20⋊12D4 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