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, (C2×C20).783C23, (C4×C20).152C22, 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, C10.42(C2×C4○D4), C2.46(C2×C4○D20), (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
Generators and relations for D20⋊24D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 1030 in 292 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, D4⋊6D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D4×C10, C20⋊2Q8, C4×D20, D10.12D4, D5×C4⋊C4, C20.48D4, C20⋊2D4, D4×C20, C2×C4○D20, D20⋊24D4
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, C4○D20, D4×D5, C23×D5, C2×C4○D20, C2×D4×D5, D4.10D10, D20⋊24D4
►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 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(121 129)(122 128)(123 127)(124 126)(130 140)(131 139)(132 138)(133 137)(134 136)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
(1 21 151 65)(2 22 152 66)(3 23 153 67)(4 24 154 68)(5 25 155 69)(6 26 156 70)(7 27 157 71)(8 28 158 72)(9 29 159 73)(10 30 160 74)(11 31 141 75)(12 32 142 76)(13 33 143 77)(14 34 144 78)(15 35 145 79)(16 36 146 80)(17 37 147 61)(18 38 148 62)(19 39 149 63)(20 40 150 64)(41 119 133 82)(42 120 134 83)(43 101 135 84)(44 102 136 85)(45 103 137 86)(46 104 138 87)(47 105 139 88)(48 106 140 89)(49 107 121 90)(50 108 122 91)(51 109 123 92)(52 110 124 93)(53 111 125 94)(54 112 126 95)(55 113 127 96)(56 114 128 97)(57 115 129 98)(58 116 130 99)(59 117 131 100)(60 118 132 81)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 81)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(121 159)(122 160)(123 141)(124 142)(125 143)(126 144)(127 145)(128 146)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,21,151,65)(2,22,152,66)(3,23,153,67)(4,24,154,68)(5,25,155,69)(6,26,156,70)(7,27,157,71)(8,28,158,72)(9,29,159,73)(10,30,160,74)(11,31,141,75)(12,32,142,76)(13,33,143,77)(14,34,144,78)(15,35,145,79)(16,36,146,80)(17,37,147,61)(18,38,148,62)(19,39,149,63)(20,40,150,64)(41,119,133,82)(42,120,134,83)(43,101,135,84)(44,102,136,85)(45,103,137,86)(46,104,138,87)(47,105,139,88)(48,106,140,89)(49,107,121,90)(50,108,122,91)(51,109,123,92)(52,110,124,93)(53,111,125,94)(54,112,126,95)(55,113,127,96)(56,114,128,97)(57,115,129,98)(58,116,130,99)(59,117,131,100)(60,118,132,81), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,81)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(121,159)(122,160)(123,141)(124,142)(125,143)(126,144)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,21,151,65)(2,22,152,66)(3,23,153,67)(4,24,154,68)(5,25,155,69)(6,26,156,70)(7,27,157,71)(8,28,158,72)(9,29,159,73)(10,30,160,74)(11,31,141,75)(12,32,142,76)(13,33,143,77)(14,34,144,78)(15,35,145,79)(16,36,146,80)(17,37,147,61)(18,38,148,62)(19,39,149,63)(20,40,150,64)(41,119,133,82)(42,120,134,83)(43,101,135,84)(44,102,136,85)(45,103,137,86)(46,104,138,87)(47,105,139,88)(48,106,140,89)(49,107,121,90)(50,108,122,91)(51,109,123,92)(52,110,124,93)(53,111,125,94)(54,112,126,95)(55,113,127,96)(56,114,128,97)(57,115,129,98)(58,116,130,99)(59,117,131,100)(60,118,132,81), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,81)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(121,159)(122,160)(123,141)(124,142)(125,143)(126,144)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(36,40),(37,39),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,87),(82,86),(83,85),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(121,129),(122,128),(123,127),(124,126),(130,140),(131,139),(132,138),(133,137),(134,136),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159)], [(1,21,151,65),(2,22,152,66),(3,23,153,67),(4,24,154,68),(5,25,155,69),(6,26,156,70),(7,27,157,71),(8,28,158,72),(9,29,159,73),(10,30,160,74),(11,31,141,75),(12,32,142,76),(13,33,143,77),(14,34,144,78),(15,35,145,79),(16,36,146,80),(17,37,147,61),(18,38,148,62),(19,39,149,63),(20,40,150,64),(41,119,133,82),(42,120,134,83),(43,101,135,84),(44,102,136,85),(45,103,137,86),(46,104,138,87),(47,105,139,88),(48,106,140,89),(49,107,121,90),(50,108,122,91),(51,109,123,92),(52,110,124,93),(53,111,125,94),(54,112,126,95),(55,113,127,96),(56,114,128,97),(57,115,129,98),(58,116,130,99),(59,117,131,100),(60,118,132,81)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,81),(61,115),(62,116),(63,117),(64,118),(65,119),(66,120),(67,101),(68,102),(69,103),(70,104),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,113),(80,114),(121,159),(122,160),(123,141),(124,142),(125,143),(126,144),(127,145),(128,146),(129,147),(130,148),(131,149),(132,150),(133,151),(134,152),(135,153),(136,154),(137,155),(138,156),(139,157),(140,158)]])
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 |
Matrix representation of D20⋊24D4 ►in 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] >;
D20⋊24D4 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