metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.1D20, C42.50D10, (C4×D4)⋊4D5, (D4×C20)⋊4C2, C20⋊3C8⋊24C2, (C5×D4).18D4, C20.18(C2×D4), C4.14(C2×D20), (C2×C20).61D4, C4⋊C4.244D10, C5⋊4(D4.2D4), D20⋊6C4⋊31C2, (C2×D4).191D10, C4.10(C4○D20), C10.89(C4○D8), C20.51(C4○D4), C4.D20⋊13C2, C10.Q16⋊29C2, (C4×C20).87C22, C2.13(C20⋊7D4), C10.65(C4⋊D4), C10.87(C8⋊C22), (C2×C20).338C23, (C2×D20).98C22, C2.9(D4.D10), (D4×C10).233C22, C2.11(D4.8D10), (C2×Dic10).103C22, (C2×D4⋊D5).5C2, (C2×D4.D5)⋊7C2, (C2×C10).469(C2×D4), (C2×C4).218(C5⋊D4), (C5×C4⋊C4).275C22, (C2×C5⋊2C8).94C22, (C2×C4).438(C22×D5), C22.150(C2×C5⋊D4), SmallGroup(320,643)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.1D20
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 502 in 124 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], D4 [×2], D4 [×3], Q8 [×2], C23 [×2], D5, C10 [×3], C10 [×2], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×4], D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8 [×2], Dic10 [×2], D20 [×2], C2×Dic5, C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×D5, C22×C10, D4.2D4, C2×C5⋊2C8 [×2], D10⋊C4 [×2], D4⋊D5 [×2], D4.D5 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, C22×C20, D4×C10, C20⋊3C8, D20⋊6C4, C10.Q16, C4.D20, C2×D4⋊D5, C2×D4.D5, D4×C20, D4.1D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, D20 [×2], C5⋊D4 [×2], C22×D5, D4.2D4, C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, D4.D10, D4.8D10, D4.1D20
(1 80 113 145)(2 61 114 146)(3 62 115 147)(4 63 116 148)(5 64 117 149)(6 65 118 150)(7 66 119 151)(8 67 120 152)(9 68 101 153)(10 69 102 154)(11 70 103 155)(12 71 104 156)(13 72 105 157)(14 73 106 158)(15 74 107 159)(16 75 108 160)(17 76 109 141)(18 77 110 142)(19 78 111 143)(20 79 112 144)(21 133 85 42)(22 134 86 43)(23 135 87 44)(24 136 88 45)(25 137 89 46)(26 138 90 47)(27 139 91 48)(28 140 92 49)(29 121 93 50)(30 122 94 51)(31 123 95 52)(32 124 96 53)(33 125 97 54)(34 126 98 55)(35 127 99 56)(36 128 100 57)(37 129 81 58)(38 130 82 59)(39 131 83 60)(40 132 84 41)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(61 98)(62 99)(63 100)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)
(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)
(1 112 113 20)(2 19 114 111)(3 110 115 18)(4 17 116 109)(5 108 117 16)(6 15 118 107)(7 106 119 14)(8 13 120 105)(9 104 101 12)(10 11 102 103)(21 45 85 136)(22 135 86 44)(23 43 87 134)(24 133 88 42)(25 41 89 132)(26 131 90 60)(27 59 91 130)(28 129 92 58)(29 57 93 128)(30 127 94 56)(31 55 95 126)(32 125 96 54)(33 53 97 124)(34 123 98 52)(35 51 99 122)(36 121 100 50)(37 49 81 140)(38 139 82 48)(39 47 83 138)(40 137 84 46)(61 143 146 78)(62 77 147 142)(63 141 148 76)(64 75 149 160)(65 159 150 74)(66 73 151 158)(67 157 152 72)(68 71 153 156)(69 155 154 70)(79 145 144 80)
G:=sub<Sym(160)| (1,80,113,145)(2,61,114,146)(3,62,115,147)(4,63,116,148)(5,64,117,149)(6,65,118,150)(7,66,119,151)(8,67,120,152)(9,68,101,153)(10,69,102,154)(11,70,103,155)(12,71,104,156)(13,72,105,157)(14,73,106,158)(15,74,107,159)(16,75,108,160)(17,76,109,141)(18,77,110,142)(19,78,111,143)(20,79,112,144)(21,133,85,42)(22,134,86,43)(23,135,87,44)(24,136,88,45)(25,137,89,46)(26,138,90,47)(27,139,91,48)(28,140,92,49)(29,121,93,50)(30,122,94,51)(31,123,95,52)(32,124,96,53)(33,125,97,54)(34,126,98,55)(35,127,99,56)(36,128,100,57)(37,129,81,58)(38,130,82,59)(39,131,83,60)(40,132,84,41), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(61,98)(62,99)(63,100)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132), (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), (1,112,113,20)(2,19,114,111)(3,110,115,18)(4,17,116,109)(5,108,117,16)(6,15,118,107)(7,106,119,14)(8,13,120,105)(9,104,101,12)(10,11,102,103)(21,45,85,136)(22,135,86,44)(23,43,87,134)(24,133,88,42)(25,41,89,132)(26,131,90,60)(27,59,91,130)(28,129,92,58)(29,57,93,128)(30,127,94,56)(31,55,95,126)(32,125,96,54)(33,53,97,124)(34,123,98,52)(35,51,99,122)(36,121,100,50)(37,49,81,140)(38,139,82,48)(39,47,83,138)(40,137,84,46)(61,143,146,78)(62,77,147,142)(63,141,148,76)(64,75,149,160)(65,159,150,74)(66,73,151,158)(67,157,152,72)(68,71,153,156)(69,155,154,70)(79,145,144,80)>;
G:=Group( (1,80,113,145)(2,61,114,146)(3,62,115,147)(4,63,116,148)(5,64,117,149)(6,65,118,150)(7,66,119,151)(8,67,120,152)(9,68,101,153)(10,69,102,154)(11,70,103,155)(12,71,104,156)(13,72,105,157)(14,73,106,158)(15,74,107,159)(16,75,108,160)(17,76,109,141)(18,77,110,142)(19,78,111,143)(20,79,112,144)(21,133,85,42)(22,134,86,43)(23,135,87,44)(24,136,88,45)(25,137,89,46)(26,138,90,47)(27,139,91,48)(28,140,92,49)(29,121,93,50)(30,122,94,51)(31,123,95,52)(32,124,96,53)(33,125,97,54)(34,126,98,55)(35,127,99,56)(36,128,100,57)(37,129,81,58)(38,130,82,59)(39,131,83,60)(40,132,84,41), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(61,98)(62,99)(63,100)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132), (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), (1,112,113,20)(2,19,114,111)(3,110,115,18)(4,17,116,109)(5,108,117,16)(6,15,118,107)(7,106,119,14)(8,13,120,105)(9,104,101,12)(10,11,102,103)(21,45,85,136)(22,135,86,44)(23,43,87,134)(24,133,88,42)(25,41,89,132)(26,131,90,60)(27,59,91,130)(28,129,92,58)(29,57,93,128)(30,127,94,56)(31,55,95,126)(32,125,96,54)(33,53,97,124)(34,123,98,52)(35,51,99,122)(36,121,100,50)(37,49,81,140)(38,139,82,48)(39,47,83,138)(40,137,84,46)(61,143,146,78)(62,77,147,142)(63,141,148,76)(64,75,149,160)(65,159,150,74)(66,73,151,158)(67,157,152,72)(68,71,153,156)(69,155,154,70)(79,145,144,80) );
G=PermutationGroup([(1,80,113,145),(2,61,114,146),(3,62,115,147),(4,63,116,148),(5,64,117,149),(6,65,118,150),(7,66,119,151),(8,67,120,152),(9,68,101,153),(10,69,102,154),(11,70,103,155),(12,71,104,156),(13,72,105,157),(14,73,106,158),(15,74,107,159),(16,75,108,160),(17,76,109,141),(18,77,110,142),(19,78,111,143),(20,79,112,144),(21,133,85,42),(22,134,86,43),(23,135,87,44),(24,136,88,45),(25,137,89,46),(26,138,90,47),(27,139,91,48),(28,140,92,49),(29,121,93,50),(30,122,94,51),(31,123,95,52),(32,124,96,53),(33,125,97,54),(34,126,98,55),(35,127,99,56),(36,128,100,57),(37,129,81,58),(38,130,82,59),(39,131,83,60),(40,132,84,41)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,153),(22,154),(23,155),(24,156),(25,157),(26,158),(27,159),(28,160),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(61,98),(62,99),(63,100),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(101,133),(102,134),(103,135),(104,136),(105,137),(106,138),(107,139),(108,140),(109,121),(110,122),(111,123),(112,124),(113,125),(114,126),(115,127),(116,128),(117,129),(118,130),(119,131),(120,132)], [(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)], [(1,112,113,20),(2,19,114,111),(3,110,115,18),(4,17,116,109),(5,108,117,16),(6,15,118,107),(7,106,119,14),(8,13,120,105),(9,104,101,12),(10,11,102,103),(21,45,85,136),(22,135,86,44),(23,43,87,134),(24,133,88,42),(25,41,89,132),(26,131,90,60),(27,59,91,130),(28,129,92,58),(29,57,93,128),(30,127,94,56),(31,55,95,126),(32,125,96,54),(33,53,97,124),(34,123,98,52),(35,51,99,122),(36,121,100,50),(37,49,81,140),(38,139,82,48),(39,47,83,138),(40,137,84,46),(61,143,146,78),(62,77,147,142),(63,141,148,76),(64,75,149,160),(65,159,150,74),(66,73,151,158),(67,157,152,72),(68,71,153,156),(69,155,154,70),(79,145,144,80)])
59 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 40 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | D20 | C4○D20 | C8⋊C22 | D4.D10 | D4.8D10 |
kernel | D4.1D20 | C20⋊3C8 | D20⋊6C4 | C10.Q16 | C4.D20 | C2×D4⋊D5 | C2×D4.D5 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of D4.1D20 ►in GL4(𝔽41) generated by
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 21 |
0 | 0 | 37 | 40 |
40 | 0 | 0 | 0 |
0 | 40 | 0 | 0 |
0 | 0 | 0 | 35 |
0 | 0 | 34 | 0 |
16 | 30 | 0 | 0 |
27 | 2 | 0 | 0 |
0 | 0 | 32 | 0 |
0 | 0 | 0 | 32 |
39 | 30 | 0 | 0 |
4 | 2 | 0 | 0 |
0 | 0 | 32 | 0 |
0 | 0 | 36 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,37,0,0,21,40],[40,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0],[16,27,0,0,30,2,0,0,0,0,32,0,0,0,0,32],[39,4,0,0,30,2,0,0,0,0,32,36,0,0,0,9] >;
D4.1D20 in GAP, Magma, Sage, TeX
D_4._1D_{20}
% in TeX
G:=Group("D4.1D20");
// GroupNames label
G:=SmallGroup(320,643);
// by ID
G=gap.SmallGroup(320,643);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2,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