metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.3Dic10, C42.46D10, (C4×D4).4D5, C5⋊6(D4.Q8), (C5×D4).3Q8, C20⋊3C8⋊19C2, (D4×C20).4C2, (C2×C20).59D4, C20.27(C2×Q8), C4⋊C4.240D10, (C2×D4).187D10, C20.47(C4○D4), C4.61(C4○D20), C10.87(C4○D8), C10.D8⋊31C2, (C4×C20).80C22, C20.Q8⋊31C2, C20.6Q8⋊11C2, C4.11(C2×Dic10), D4⋊Dic5.9C2, C10.86(C8⋊C22), (C2×C20).334C23, C10.63(C22⋊Q8), C2.8(D4.D10), (D4×C10).229C22, C4⋊Dic5.138C22, C2.10(D4.8D10), C2.14(C20.48D4), (C2×C10).465(C2×D4), (C2×C4).216(C5⋊D4), (C5×C4⋊C4).271C22, (C2×C5⋊2C8).91C22, (C2×C4).434(C22×D5), C22.148(C2×C5⋊D4), SmallGroup(320,636)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.3Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=a2c10, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 310 in 102 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C2×D4, Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×4], D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C5⋊2C8 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4.Q8, C2×C5⋊2C8 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C20⋊3C8, C10.D8, C20.Q8, D4⋊Dic5 [×2], C20.6Q8, D4×C20, D4.3Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8⋊C22, Dic10 [×2], C5⋊D4 [×2], C22×D5, D4.Q8, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, D4.D10, D4.8D10, D4.3Dic10
►On 160 points
Generators in S160
(1 58 38 148)(2 59 39 149)(3 60 40 150)(4 41 21 151)(5 42 22 152)(6 43 23 153)(7 44 24 154)(8 45 25 155)(9 46 26 156)(10 47 27 157)(11 48 28 158)(12 49 29 159)(13 50 30 160)(14 51 31 141)(15 52 32 142)(16 53 33 143)(17 54 34 144)(18 55 35 145)(19 56 36 146)(20 57 37 147)(61 118 134 88)(62 119 135 89)(63 120 136 90)(64 101 137 91)(65 102 138 92)(66 103 139 93)(67 104 140 94)(68 105 121 95)(69 106 122 96)(70 107 123 97)(71 108 124 98)(72 109 125 99)(73 110 126 100)(74 111 127 81)(75 112 128 82)(76 113 129 83)(77 114 130 84)(78 115 131 85)(79 116 132 86)(80 117 133 87)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(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)
(1 132 28 69)(2 78 29 121)(3 130 30 67)(4 76 31 139)(5 128 32 65)(6 74 33 137)(7 126 34 63)(8 72 35 135)(9 124 36 61)(10 70 37 133)(11 122 38 79)(12 68 39 131)(13 140 40 77)(14 66 21 129)(15 138 22 75)(16 64 23 127)(17 136 24 73)(18 62 25 125)(19 134 26 71)(20 80 27 123)(41 83 141 103)(42 112 142 92)(43 81 143 101)(44 110 144 90)(45 99 145 119)(46 108 146 88)(47 97 147 117)(48 106 148 86)(49 95 149 115)(50 104 150 84)(51 93 151 113)(52 102 152 82)(53 91 153 111)(54 120 154 100)(55 89 155 109)(56 118 156 98)(57 87 157 107)(58 116 158 96)(59 85 159 105)(60 114 160 94)
G:=sub<Sym(160)| (1,58,38,148)(2,59,39,149)(3,60,40,150)(4,41,21,151)(5,42,22,152)(6,43,23,153)(7,44,24,154)(8,45,25,155)(9,46,26,156)(10,47,27,157)(11,48,28,158)(12,49,29,159)(13,50,30,160)(14,51,31,141)(15,52,32,142)(16,53,33,143)(17,54,34,144)(18,55,35,145)(19,56,36,146)(20,57,37,147)(61,118,134,88)(62,119,135,89)(63,120,136,90)(64,101,137,91)(65,102,138,92)(66,103,139,93)(67,104,140,94)(68,105,121,95)(69,106,122,96)(70,107,123,97)(71,108,124,98)(72,109,125,99)(73,110,126,100)(74,111,127,81)(75,112,128,82)(76,113,129,83)(77,114,130,84)(78,115,131,85)(79,116,132,86)(80,117,133,87), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (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), (1,132,28,69)(2,78,29,121)(3,130,30,67)(4,76,31,139)(5,128,32,65)(6,74,33,137)(7,126,34,63)(8,72,35,135)(9,124,36,61)(10,70,37,133)(11,122,38,79)(12,68,39,131)(13,140,40,77)(14,66,21,129)(15,138,22,75)(16,64,23,127)(17,136,24,73)(18,62,25,125)(19,134,26,71)(20,80,27,123)(41,83,141,103)(42,112,142,92)(43,81,143,101)(44,110,144,90)(45,99,145,119)(46,108,146,88)(47,97,147,117)(48,106,148,86)(49,95,149,115)(50,104,150,84)(51,93,151,113)(52,102,152,82)(53,91,153,111)(54,120,154,100)(55,89,155,109)(56,118,156,98)(57,87,157,107)(58,116,158,96)(59,85,159,105)(60,114,160,94)>;
G:=Group( (1,58,38,148)(2,59,39,149)(3,60,40,150)(4,41,21,151)(5,42,22,152)(6,43,23,153)(7,44,24,154)(8,45,25,155)(9,46,26,156)(10,47,27,157)(11,48,28,158)(12,49,29,159)(13,50,30,160)(14,51,31,141)(15,52,32,142)(16,53,33,143)(17,54,34,144)(18,55,35,145)(19,56,36,146)(20,57,37,147)(61,118,134,88)(62,119,135,89)(63,120,136,90)(64,101,137,91)(65,102,138,92)(66,103,139,93)(67,104,140,94)(68,105,121,95)(69,106,122,96)(70,107,123,97)(71,108,124,98)(72,109,125,99)(73,110,126,100)(74,111,127,81)(75,112,128,82)(76,113,129,83)(77,114,130,84)(78,115,131,85)(79,116,132,86)(80,117,133,87), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (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), (1,132,28,69)(2,78,29,121)(3,130,30,67)(4,76,31,139)(5,128,32,65)(6,74,33,137)(7,126,34,63)(8,72,35,135)(9,124,36,61)(10,70,37,133)(11,122,38,79)(12,68,39,131)(13,140,40,77)(14,66,21,129)(15,138,22,75)(16,64,23,127)(17,136,24,73)(18,62,25,125)(19,134,26,71)(20,80,27,123)(41,83,141,103)(42,112,142,92)(43,81,143,101)(44,110,144,90)(45,99,145,119)(46,108,146,88)(47,97,147,117)(48,106,148,86)(49,95,149,115)(50,104,150,84)(51,93,151,113)(52,102,152,82)(53,91,153,111)(54,120,154,100)(55,89,155,109)(56,118,156,98)(57,87,157,107)(58,116,158,96)(59,85,159,105)(60,114,160,94) );
G=PermutationGroup([(1,58,38,148),(2,59,39,149),(3,60,40,150),(4,41,21,151),(5,42,22,152),(6,43,23,153),(7,44,24,154),(8,45,25,155),(9,46,26,156),(10,47,27,157),(11,48,28,158),(12,49,29,159),(13,50,30,160),(14,51,31,141),(15,52,32,142),(16,53,33,143),(17,54,34,144),(18,55,35,145),(19,56,36,146),(20,57,37,147),(61,118,134,88),(62,119,135,89),(63,120,136,90),(64,101,137,91),(65,102,138,92),(66,103,139,93),(67,104,140,94),(68,105,121,95),(69,106,122,96),(70,107,123,97),(71,108,124,98),(72,109,125,99),(73,110,126,100),(74,111,127,81),(75,112,128,82),(76,113,129,83),(77,114,130,84),(78,115,131,85),(79,116,132,86),(80,117,133,87)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,141),(22,142),(23,143),(24,144),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140)], [(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)], [(1,132,28,69),(2,78,29,121),(3,130,30,67),(4,76,31,139),(5,128,32,65),(6,74,33,137),(7,126,34,63),(8,72,35,135),(9,124,36,61),(10,70,37,133),(11,122,38,79),(12,68,39,131),(13,140,40,77),(14,66,21,129),(15,138,22,75),(16,64,23,127),(17,136,24,73),(18,62,25,125),(19,134,26,71),(20,80,27,123),(41,83,141,103),(42,112,142,92),(43,81,143,101),(44,110,144,90),(45,99,145,119),(46,108,146,88),(47,97,147,117),(48,106,148,86),(49,95,149,115),(50,104,150,84),(51,93,151,113),(52,102,152,82),(53,91,153,111),(54,120,154,100),(55,89,155,109),(56,118,156,98),(57,87,157,107),(58,116,158,96),(59,85,159,105),(60,114,160,94)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | Dic10 | C4○D20 | C8⋊C22 | D4.D10 | D4.8D10 |
| kernel | D4.3Dic10 | C20⋊3C8 | C10.D8 | C20.Q8 | D4⋊Dic5 | C20.6Q8 | D4×C20 | C2×C20 | C5×D4 | C4×D4 | C20 | C42 | C4⋊C4 | C2×D4 | C10 | C2×C4 | D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of D4.3Dic10 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 28 | 39 |
| 0 | 0 | 2 | 16 |
| 12 | 29 | 0 | 0 |
| 29 | 29 | 0 | 0 |
| 0 | 0 | 27 | 39 |
| 0 | 0 | 37 | 14 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,28,2,0,0,39,16],[12,29,0,0,29,29,0,0,0,0,27,37,0,0,39,14] >;
D4.3Dic10 in GAP, Magma, Sage, TeX
D_4._3{\rm Dic}_{10} % in TeX
G:=Group("D4.3Dic10"); // GroupNames label
G:=SmallGroup(320,636);
// by ID
G=gap.SmallGroup(320,636);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2*c^10,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations