metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.13D20, Q8.13D20, C20.63C24, C40.12C23, D20.26C23, D40.14C22, M4(2).28D10, Dic20.10C22, Dic10.26C23, C8○D4⋊5D5, C5⋊1(Q8○D8), (C5×D4).25D4, C4.29(C2×D20), C20.75(C2×D4), (C5×Q8).25D4, C4○D4.40D10, (C2×C8).102D10, D40⋊7C2⋊13C2, C4.60(C23×D5), C22.5(C2×D20), C8.54(C22×D5), (C2×Dic20)⋊15C2, C8.D10⋊12C2, (C2×C40).70C22, C40⋊C2.2C22, C2.32(C22×D20), C10.30(C22×D4), D4.10D10⋊4C2, (C2×C20).517C23, C4○D20.27C22, (C5×M4(2)).30C22, (C2×Dic10).200C22, (C5×C8○D4)⋊5C2, (C2×C10).10(C2×D4), (C5×C4○D4).47C22, (C2×C4).228(C22×D5), SmallGroup(320,1425)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.13D20
G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c19 >
Subgroups: 902 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C40, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, Q8○D8, C40⋊C2, D40, Dic20, C2×C40, C5×M4(2), C2×Dic10, C4○D20, D4⋊2D5, Q8×D5, C5×C4○D4, D40⋊7C2, C2×Dic20, C8.D10, C5×C8○D4, D4.10D10, D4.13D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, D20, C22×D5, Q8○D8, C2×D20, C23×D5, C22×D20, D4.13D20
►On 160 points
Generators in S160
(1 57 21 77)(2 58 22 78)(3 59 23 79)(4 60 24 80)(5 61 25 41)(6 62 26 42)(7 63 27 43)(8 64 28 44)(9 65 29 45)(10 66 30 46)(11 67 31 47)(12 68 32 48)(13 69 33 49)(14 70 34 50)(15 71 35 51)(16 72 36 52)(17 73 37 53)(18 74 38 54)(19 75 39 55)(20 76 40 56)(81 139 101 159)(82 140 102 160)(83 141 103 121)(84 142 104 122)(85 143 105 123)(86 144 106 124)(87 145 107 125)(88 146 108 126)(89 147 109 127)(90 148 110 128)(91 149 111 129)(92 150 112 130)(93 151 113 131)(94 152 114 132)(95 153 115 133)(96 154 116 134)(97 155 117 135)(98 156 118 136)(99 157 119 137)(100 158 120 138)
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 157)(37 158)(38 159)(39 160)(40 121)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(57 104)(58 105)(59 106)(60 107)(61 108)(62 109)(63 110)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 120)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(73 80)(74 79)(75 78)(76 77)(81 106)(82 105)(83 104)(84 103)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 142)(122 141)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)
G:=sub<Sym(160)| (1,57,21,77)(2,58,22,78)(3,59,23,79)(4,60,24,80)(5,61,25,41)(6,62,26,42)(7,63,27,43)(8,64,28,44)(9,65,29,45)(10,66,30,46)(11,67,31,47)(12,68,32,48)(13,69,33,49)(14,70,34,50)(15,71,35,51)(16,72,36,52)(17,73,37,53)(18,74,38,54)(19,75,39,55)(20,76,40,56)(81,139,101,159)(82,140,102,160)(83,141,103,121)(84,142,104,122)(85,143,105,123)(86,144,106,124)(87,145,107,125)(88,146,108,126)(89,147,109,127)(90,148,110,128)(91,149,111,129)(92,150,112,130)(93,151,113,131)(94,152,114,132)(95,153,115,133)(96,154,116,134)(97,155,117,135)(98,156,118,136)(99,157,119,137)(100,158,120,138), (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,121)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,108)(62,109)(63,110)(64,111)(65,112)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119)(73,120)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(73,80)(74,79)(75,78)(76,77)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)>;
G:=Group( (1,57,21,77)(2,58,22,78)(3,59,23,79)(4,60,24,80)(5,61,25,41)(6,62,26,42)(7,63,27,43)(8,64,28,44)(9,65,29,45)(10,66,30,46)(11,67,31,47)(12,68,32,48)(13,69,33,49)(14,70,34,50)(15,71,35,51)(16,72,36,52)(17,73,37,53)(18,74,38,54)(19,75,39,55)(20,76,40,56)(81,139,101,159)(82,140,102,160)(83,141,103,121)(84,142,104,122)(85,143,105,123)(86,144,106,124)(87,145,107,125)(88,146,108,126)(89,147,109,127)(90,148,110,128)(91,149,111,129)(92,150,112,130)(93,151,113,131)(94,152,114,132)(95,153,115,133)(96,154,116,134)(97,155,117,135)(98,156,118,136)(99,157,119,137)(100,158,120,138), (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,121)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,108)(62,109)(63,110)(64,111)(65,112)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119)(73,120)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(73,80)(74,79)(75,78)(76,77)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152) );
G=PermutationGroup([[(1,57,21,77),(2,58,22,78),(3,59,23,79),(4,60,24,80),(5,61,25,41),(6,62,26,42),(7,63,27,43),(8,64,28,44),(9,65,29,45),(10,66,30,46),(11,67,31,47),(12,68,32,48),(13,69,33,49),(14,70,34,50),(15,71,35,51),(16,72,36,52),(17,73,37,53),(18,74,38,54),(19,75,39,55),(20,76,40,56),(81,139,101,159),(82,140,102,160),(83,141,103,121),(84,142,104,122),(85,143,105,123),(86,144,106,124),(87,145,107,125),(88,146,108,126),(89,147,109,127),(90,148,110,128),(91,149,111,129),(92,150,112,130),(93,151,113,131),(94,152,114,132),(95,153,115,133),(96,154,116,134),(97,155,117,135),(98,156,118,136),(99,157,119,137),(100,158,120,138)], [(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,141),(21,142),(22,143),(23,144),(24,145),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,153),(33,154),(34,155),(35,156),(36,157),(37,158),(38,159),(39,160),(40,121),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103),(57,104),(58,105),(59,106),(60,107),(61,108),(62,109),(63,110),(64,111),(65,112),(66,113),(67,114),(68,115),(69,116),(70,117),(71,118),(72,119),(73,120),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,72),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57),(73,80),(74,79),(75,78),(76,77),(81,106),(82,105),(83,104),(84,103),(85,102),(86,101),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(121,142),(122,141),(123,140),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133),(131,132),(143,160),(144,159),(145,158),(146,157),(147,156),(148,155),(149,154),(150,153),(151,152)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | Q8○D8 | D4.13D20 |
| kernel | D4.13D20 | D40⋊7C2 | C2×Dic20 | C8.D10 | C5×C8○D4 | D4.10D10 | C5×D4 | C5×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 3 | 1 | 2 | 6 | 6 | 2 | 12 | 4 | 2 | 8 |
Matrix representation of D4.13D20 ►in GL4(𝔽41) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 33 | 14 | 15 | 25 |
| 27 | 8 | 16 | 26 |
| 15 | 25 | 8 | 27 |
| 16 | 26 | 14 | 33 |
| 35 | 3 | 0 | 0 |
| 38 | 15 | 0 | 0 |
| 0 | 0 | 35 | 3 |
| 0 | 0 | 38 | 15 |
| 20 | 39 | 0 | 0 |
| 15 | 21 | 0 | 0 |
| 0 | 0 | 20 | 39 |
| 0 | 0 | 15 | 21 |
G:=sub<GL(4,GF(41))| [0,0,40,0,0,0,0,40,1,0,0,0,0,1,0,0],[33,27,15,16,14,8,25,26,15,16,8,14,25,26,27,33],[35,38,0,0,3,15,0,0,0,0,35,38,0,0,3,15],[20,15,0,0,39,21,0,0,0,0,20,15,0,0,39,21] >;
D4.13D20 in GAP, Magma, Sage, TeX
D_4._{13}D_{20} % in TeX
G:=Group("D4.13D20"); // GroupNames label
G:=SmallGroup(320,1425);
// by ID
G=gap.SmallGroup(320,1425);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,192,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^19>;
// generators/relations