direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C20, C4⋊1(C22×C20), (C23×C20)⋊6C2, C23⋊4(C2×C20), (C23×C4)⋊3C10, (C2×C42)⋊7C10, C20⋊10(C22×C4), C42⋊16(C2×C10), (C4×C20)⋊57C22, C2.4(C23×C20), C24.30(C2×C10), C10.77(C23×C4), C22⋊1(C22×C20), C22.59(D4×C10), (C2×C10).335C24, (C2×C20).707C23, (C22×C20)⋊58C22, (C22×D4).13C10, C10.179(C22×D4), C22.8(C23×C10), (D4×C10).330C22, C23.28(C22×C10), (C23×C10).90C22, (C22×C10).252C23, (C2×C4×C20)⋊20C2, C2.3(D4×C2×C10), (C2×C4)⋊7(C2×C20), (C2×C4⋊C4)⋊25C10, (C10×C4⋊C4)⋊52C2, C4⋊C4⋊19(C2×C10), (C2×C20)⋊45(C2×C4), (D4×C2×C10).26C2, C2.2(C10×C4○D4), (C5×C4⋊C4)⋊76C22, (C2×C10)⋊8(C22×C4), C22⋊C4⋊17(C2×C10), (C2×C22⋊C4)⋊16C10, (C10×C22⋊C4)⋊36C2, (C22×C10)⋊20(C2×C4), (C22×C4)⋊16(C2×C10), (C2×D4).76(C2×C10), C10.221(C2×C4○D4), (C2×C10).681(C2×D4), C22.27(C5×C4○D4), (C5×C22⋊C4)⋊71C22, (C2×C4).54(C22×C10), (C2×C10).227(C4○D4), SmallGroup(320,1517)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C2×C20
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 578 in 426 conjugacy classes, 274 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C4×D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C22×C20, D4×C10, C23×C10, C2×C4×C20, C10×C22⋊C4, C10×C4⋊C4, D4×C20, C23×C20, D4×C2×C10, D4×C2×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22×C4, C2×D4, C4○D4, C24, C20, C2×C10, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C20, C5×D4, C22×C10, C2×C4×D4, C22×C20, D4×C10, C5×C4○D4, C23×C10, D4×C20, C23×C20, D4×C2×C10, C10×C4○D4, D4×C2×C20
►On 160 points
Generators in S160
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 81)(77 82)(78 83)(79 84)(80 85)(101 139)(102 140)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)
(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 88 135 158)(2 89 136 159)(3 90 137 160)(4 91 138 141)(5 92 139 142)(6 93 140 143)(7 94 121 144)(8 95 122 145)(9 96 123 146)(10 97 124 147)(11 98 125 148)(12 99 126 149)(13 100 127 150)(14 81 128 151)(15 82 129 152)(16 83 130 153)(17 84 131 154)(18 85 132 155)(19 86 133 156)(20 87 134 157)(21 75 109 56)(22 76 110 57)(23 77 111 58)(24 78 112 59)(25 79 113 60)(26 80 114 41)(27 61 115 42)(28 62 116 43)(29 63 117 44)(30 64 118 45)(31 65 119 46)(32 66 120 47)(33 67 101 48)(34 68 102 49)(35 69 103 50)(36 70 104 51)(37 71 105 52)(38 72 106 53)(39 73 107 54)(40 74 108 55)
(1 117)(2 118)(3 119)(4 120)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 81)(77 82)(78 83)(79 84)(80 85)
G:=sub<Sym(160)| (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,81)(77,82)(78,83)(79,84)(80,85)(101,139)(102,140)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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), (1,88,135,158)(2,89,136,159)(3,90,137,160)(4,91,138,141)(5,92,139,142)(6,93,140,143)(7,94,121,144)(8,95,122,145)(9,96,123,146)(10,97,124,147)(11,98,125,148)(12,99,126,149)(13,100,127,150)(14,81,128,151)(15,82,129,152)(16,83,130,153)(17,84,131,154)(18,85,132,155)(19,86,133,156)(20,87,134,157)(21,75,109,56)(22,76,110,57)(23,77,111,58)(24,78,112,59)(25,79,113,60)(26,80,114,41)(27,61,115,42)(28,62,116,43)(29,63,117,44)(30,64,118,45)(31,65,119,46)(32,66,120,47)(33,67,101,48)(34,68,102,49)(35,69,103,50)(36,70,104,51)(37,71,105,52)(38,72,106,53)(39,73,107,54)(40,74,108,55), (1,117)(2,118)(3,119)(4,120)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,81)(77,82)(78,83)(79,84)(80,85)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,81)(77,82)(78,83)(79,84)(80,85)(101,139)(102,140)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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), (1,88,135,158)(2,89,136,159)(3,90,137,160)(4,91,138,141)(5,92,139,142)(6,93,140,143)(7,94,121,144)(8,95,122,145)(9,96,123,146)(10,97,124,147)(11,98,125,148)(12,99,126,149)(13,100,127,150)(14,81,128,151)(15,82,129,152)(16,83,130,153)(17,84,131,154)(18,85,132,155)(19,86,133,156)(20,87,134,157)(21,75,109,56)(22,76,110,57)(23,77,111,58)(24,78,112,59)(25,79,113,60)(26,80,114,41)(27,61,115,42)(28,62,116,43)(29,63,117,44)(30,64,118,45)(31,65,119,46)(32,66,120,47)(33,67,101,48)(34,68,102,49)(35,69,103,50)(36,70,104,51)(37,71,105,52)(38,72,106,53)(39,73,107,54)(40,74,108,55), (1,117)(2,118)(3,119)(4,120)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,81)(77,82)(78,83)(79,84)(80,85) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28),(41,155),(42,156),(43,157),(44,158),(45,159),(46,160),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,153),(60,154),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,81),(77,82),(78,83),(79,84),(80,85),(101,139),(102,140),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138)], [(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)], [(1,88,135,158),(2,89,136,159),(3,90,137,160),(4,91,138,141),(5,92,139,142),(6,93,140,143),(7,94,121,144),(8,95,122,145),(9,96,123,146),(10,97,124,147),(11,98,125,148),(12,99,126,149),(13,100,127,150),(14,81,128,151),(15,82,129,152),(16,83,130,153),(17,84,131,154),(18,85,132,155),(19,86,133,156),(20,87,134,157),(21,75,109,56),(22,76,110,57),(23,77,111,58),(24,78,112,59),(25,79,113,60),(26,80,114,41),(27,61,115,42),(28,62,116,43),(29,63,117,44),(30,64,118,45),(31,65,119,46),(32,66,120,47),(33,67,101,48),(34,68,102,49),(35,69,103,50),(36,70,104,51),(37,71,105,52),(38,72,106,53),(39,73,107,54),(40,74,108,55)], [(1,117),(2,118),(3,119),(4,120),(5,101),(6,102),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,113),(18,114),(19,115),(20,116),(21,127),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,137),(32,138),(33,139),(34,140),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,155),(42,156),(43,157),(44,158),(45,159),(46,160),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,153),(60,154),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,81),(77,82),(78,83),(79,84),(80,85)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4X | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10BH | 20A | ··· | 20AF | 20AG | ··· | 20CR |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | D4×C2×C20 | C2×C4×C20 | C10×C22⋊C4 | C10×C4⋊C4 | D4×C20 | C23×C20 | D4×C2×C10 | D4×C10 | C2×C4×D4 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C4×D4 | C23×C4 | C22×D4 | C2×D4 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 16 | 4 | 4 | 8 | 4 | 32 | 8 | 4 | 64 | 4 | 4 | 16 | 16 |
Matrix representation of D4×C2×C20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 2 |
| 0 | 0 | 40 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 40 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[20,0,0,0,0,9,0,0,0,0,2,0,0,0,0,2],[40,0,0,0,0,40,0,0,0,0,40,40,0,0,2,1],[40,0,0,0,0,1,0,0,0,0,40,40,0,0,0,1] >;
D4×C2×C20 in GAP, Magma, Sage, TeX
D_4\times C_2\times C_{20} % in TeX
G:=Group("D4xC2xC20"); // GroupNames label
G:=SmallGroup(320,1517);
// by ID
G=gap.SmallGroup(320,1517);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,856]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations