metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.3Q8, C42.38D10, C4⋊C8⋊7D5, C5⋊3(D4.Q8), C4.45(Q8×D5), C40⋊5C4⋊13C2, C40⋊6C4⋊17C2, (C2×C4).40D20, (C4×D20).13C2, (C2×C20).246D4, (C2×C8).132D10, C20.104(C2×Q8), D20⋊5C4.4C2, C10.14(C4○D8), (C2×C40).26C22, (C4×C20).73C22, C20.6Q8⋊10C2, C20.288(C4○D4), C2.19(C8⋊D10), C10.16(C8⋊C22), (C2×C20).757C23, C22.120(C2×D20), C10.32(C22⋊Q8), C4⋊Dic5.20C22, C4.112(D4⋊2D5), C2.16(D40⋊7C2), C2.13(D10⋊2Q8), (C2×D20).203C22, (C5×C4⋊C8)⋊9C2, (C2×C10).140(C2×D4), (C2×C4).702(C22×D5), SmallGroup(320,474)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.3Q8
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5c2, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a15c3 >
Subgroups: 470 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4.Q8, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C40⋊6C4, C40⋊5C4, D20⋊5C4, C5×C4⋊C8, C20.6Q8, C4×D20, D20.3Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, D20, C22×D5, D4.Q8, C2×D20, D4⋊2D5, Q8×D5, D10⋊2Q8, D40⋊7C2, C8⋊D10, D20.3Q8
►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)
(1 149)(2 148)(3 147)(4 146)(5 145)(6 144)(7 143)(8 142)(9 141)(10 160)(11 159)(12 158)(13 157)(14 156)(15 155)(16 154)(17 153)(18 152)(19 151)(20 150)(21 124)(22 123)(23 122)(24 121)(25 140)(26 139)(27 138)(28 137)(29 136)(30 135)(31 134)(32 133)(33 132)(34 131)(35 130)(36 129)(37 128)(38 127)(39 126)(40 125)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 100)(58 99)(59 98)(60 97)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)
(1 35 145 121 11 25 155 131)(2 36 146 122 12 26 156 132)(3 37 147 123 13 27 157 133)(4 38 148 124 14 28 158 134)(5 39 149 125 15 29 159 135)(6 40 150 126 16 30 160 136)(7 21 151 127 17 31 141 137)(8 22 152 128 18 32 142 138)(9 23 153 129 19 33 143 139)(10 24 154 130 20 34 144 140)(41 105 92 67 51 115 82 77)(42 106 93 68 52 116 83 78)(43 107 94 69 53 117 84 79)(44 108 95 70 54 118 85 80)(45 109 96 71 55 119 86 61)(46 110 97 72 56 120 87 62)(47 111 98 73 57 101 88 63)(48 112 99 74 58 102 89 64)(49 113 100 75 59 103 90 65)(50 114 81 76 60 104 91 66)
(1 97 150 41)(2 98 151 42)(3 99 152 43)(4 100 153 44)(5 81 154 45)(6 82 155 46)(7 83 156 47)(8 84 157 48)(9 85 158 49)(10 86 159 50)(11 87 160 51)(12 88 141 52)(13 89 142 53)(14 90 143 54)(15 91 144 55)(16 92 145 56)(17 93 146 57)(18 94 147 58)(19 95 148 59)(20 96 149 60)(21 101 132 68)(22 102 133 69)(23 103 134 70)(24 104 135 71)(25 105 136 72)(26 106 137 73)(27 107 138 74)(28 108 139 75)(29 109 140 76)(30 110 121 77)(31 111 122 78)(32 112 123 79)(33 113 124 80)(34 114 125 61)(35 115 126 62)(36 116 127 63)(37 117 128 64)(38 118 129 65)(39 119 130 66)(40 120 131 67)
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), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,124)(22,123)(23,122)(24,121)(25,140)(26,139)(27,138)(28,137)(29,136)(30,135)(31,134)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,100)(58,99)(59,98)(60,97)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101), (1,35,145,121,11,25,155,131)(2,36,146,122,12,26,156,132)(3,37,147,123,13,27,157,133)(4,38,148,124,14,28,158,134)(5,39,149,125,15,29,159,135)(6,40,150,126,16,30,160,136)(7,21,151,127,17,31,141,137)(8,22,152,128,18,32,142,138)(9,23,153,129,19,33,143,139)(10,24,154,130,20,34,144,140)(41,105,92,67,51,115,82,77)(42,106,93,68,52,116,83,78)(43,107,94,69,53,117,84,79)(44,108,95,70,54,118,85,80)(45,109,96,71,55,119,86,61)(46,110,97,72,56,120,87,62)(47,111,98,73,57,101,88,63)(48,112,99,74,58,102,89,64)(49,113,100,75,59,103,90,65)(50,114,81,76,60,104,91,66), (1,97,150,41)(2,98,151,42)(3,99,152,43)(4,100,153,44)(5,81,154,45)(6,82,155,46)(7,83,156,47)(8,84,157,48)(9,85,158,49)(10,86,159,50)(11,87,160,51)(12,88,141,52)(13,89,142,53)(14,90,143,54)(15,91,144,55)(16,92,145,56)(17,93,146,57)(18,94,147,58)(19,95,148,59)(20,96,149,60)(21,101,132,68)(22,102,133,69)(23,103,134,70)(24,104,135,71)(25,105,136,72)(26,106,137,73)(27,107,138,74)(28,108,139,75)(29,109,140,76)(30,110,121,77)(31,111,122,78)(32,112,123,79)(33,113,124,80)(34,114,125,61)(35,115,126,62)(36,116,127,63)(37,117,128,64)(38,118,129,65)(39,119,130,66)(40,120,131,67)>;
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), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,124)(22,123)(23,122)(24,121)(25,140)(26,139)(27,138)(28,137)(29,136)(30,135)(31,134)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,100)(58,99)(59,98)(60,97)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101), (1,35,145,121,11,25,155,131)(2,36,146,122,12,26,156,132)(3,37,147,123,13,27,157,133)(4,38,148,124,14,28,158,134)(5,39,149,125,15,29,159,135)(6,40,150,126,16,30,160,136)(7,21,151,127,17,31,141,137)(8,22,152,128,18,32,142,138)(9,23,153,129,19,33,143,139)(10,24,154,130,20,34,144,140)(41,105,92,67,51,115,82,77)(42,106,93,68,52,116,83,78)(43,107,94,69,53,117,84,79)(44,108,95,70,54,118,85,80)(45,109,96,71,55,119,86,61)(46,110,97,72,56,120,87,62)(47,111,98,73,57,101,88,63)(48,112,99,74,58,102,89,64)(49,113,100,75,59,103,90,65)(50,114,81,76,60,104,91,66), (1,97,150,41)(2,98,151,42)(3,99,152,43)(4,100,153,44)(5,81,154,45)(6,82,155,46)(7,83,156,47)(8,84,157,48)(9,85,158,49)(10,86,159,50)(11,87,160,51)(12,88,141,52)(13,89,142,53)(14,90,143,54)(15,91,144,55)(16,92,145,56)(17,93,146,57)(18,94,147,58)(19,95,148,59)(20,96,149,60)(21,101,132,68)(22,102,133,69)(23,103,134,70)(24,104,135,71)(25,105,136,72)(26,106,137,73)(27,107,138,74)(28,108,139,75)(29,109,140,76)(30,110,121,77)(31,111,122,78)(32,112,123,79)(33,113,124,80)(34,114,125,61)(35,115,126,62)(36,116,127,63)(37,117,128,64)(38,118,129,65)(39,119,130,66)(40,120,131,67) );
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)], [(1,149),(2,148),(3,147),(4,146),(5,145),(6,144),(7,143),(8,142),(9,141),(10,160),(11,159),(12,158),(13,157),(14,156),(15,155),(16,154),(17,153),(18,152),(19,151),(20,150),(21,124),(22,123),(23,122),(24,121),(25,140),(26,139),(27,138),(28,137),(29,136),(30,135),(31,134),(32,133),(33,132),(34,131),(35,130),(36,129),(37,128),(38,127),(39,126),(40,125),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,100),(58,99),(59,98),(60,97),(61,120),(62,119),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101)], [(1,35,145,121,11,25,155,131),(2,36,146,122,12,26,156,132),(3,37,147,123,13,27,157,133),(4,38,148,124,14,28,158,134),(5,39,149,125,15,29,159,135),(6,40,150,126,16,30,160,136),(7,21,151,127,17,31,141,137),(8,22,152,128,18,32,142,138),(9,23,153,129,19,33,143,139),(10,24,154,130,20,34,144,140),(41,105,92,67,51,115,82,77),(42,106,93,68,52,116,83,78),(43,107,94,69,53,117,84,79),(44,108,95,70,54,118,85,80),(45,109,96,71,55,119,86,61),(46,110,97,72,56,120,87,62),(47,111,98,73,57,101,88,63),(48,112,99,74,58,102,89,64),(49,113,100,75,59,103,90,65),(50,114,81,76,60,104,91,66)], [(1,97,150,41),(2,98,151,42),(3,99,152,43),(4,100,153,44),(5,81,154,45),(6,82,155,46),(7,83,156,47),(8,84,157,48),(9,85,158,49),(10,86,159,50),(11,87,160,51),(12,88,141,52),(13,89,142,53),(14,90,143,54),(15,91,144,55),(16,92,145,56),(17,93,146,57),(18,94,147,58),(19,95,148,59),(20,96,149,60),(21,101,132,68),(22,102,133,69),(23,103,134,70),(24,104,135,71),(25,105,136,72),(26,106,137,73),(27,107,138,74),(28,108,139,75),(29,109,140,76),(30,110,121,77),(31,111,122,78),(32,112,123,79),(33,113,124,80),(34,114,125,61),(35,115,126,62),(36,116,127,63),(37,117,128,64),(38,118,129,65),(39,119,130,66),(40,120,131,67)]])
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 | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | D20 | D40⋊7C2 | C8⋊C22 | D4⋊2D5 | Q8×D5 | C8⋊D10 |
| kernel | D20.3Q8 | C40⋊6C4 | C40⋊5C4 | D20⋊5C4 | C5×C4⋊C8 | C20.6Q8 | C4×D20 | D20 | C2×C20 | C4⋊C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C2 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of D20.3Q8 ►in GL4(𝔽41) generated by
| 25 | 2 | 0 | 0 |
| 39 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 25 | 0 | 0 |
| 13 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 26 | 31 | 0 | 0 |
| 10 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 35 | 2 |
| 0 | 0 | 2 | 6 |
G:=sub<GL(4,GF(41))| [25,39,0,0,2,13,0,0,0,0,1,0,0,0,0,1],[2,13,0,0,25,39,0,0,0,0,40,0,0,0,0,40],[26,10,0,0,31,4,0,0,0,0,0,40,0,0,1,0],[9,0,0,0,0,9,0,0,0,0,35,2,0,0,2,6] >;
D20.3Q8 in GAP, Magma, Sage, TeX
D_{20}._3Q_8 % in TeX
G:=Group("D20.3Q8"); // GroupNames label
G:=SmallGroup(320,474);
// by ID
G=gap.SmallGroup(320,474);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5*c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^15*c^3>;
// generators/relations