direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C20, C4⋊1(C22×C20), (C23×C20)⋊6C2, C23⋊4(C2×C20), (C2×C42)⋊7C10, (C23×C4)⋊3C10, C42⋊16(C2×C10), C20⋊10(C22×C4), (C4×C20)⋊57C22, C2.4(C23×C20), C24.30(C2×C10), C10.77(C23×C4), C22⋊1(C22×C20), C22.59(D4×C10), (C2×C20).707C23, (C2×C10).335C24, (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), (C10×C22⋊C4)⋊36C2, (C2×C22⋊C4)⋊16C10, C22⋊C4⋊17(C2×C10), (C22×C4)⋊16(C2×C10), (C22×C10)⋊20(C2×C4), (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
Subgroups: 578 in 426 conjugacy classes, 274 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×6], C22, C22 [×14], C22 [×24], C5, C2×C4 [×18], C2×C4 [×22], D4 [×16], C23, C23 [×12], C23 [×8], C10 [×3], C10 [×4], C10 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×12], C24 [×2], C20 [×8], C20 [×6], C2×C10, C2×C10 [×14], C2×C10 [×24], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C2×C20 [×18], C2×C20 [×22], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×8], C2×C4×D4, C4×C20 [×4], C5×C22⋊C4 [×8], C5×C4⋊C4 [×4], C22×C20 [×3], C22×C20 [×10], C22×C20 [×8], D4×C10 [×12], C23×C10 [×2], C2×C4×C20, C10×C22⋊C4 [×2], C10×C4⋊C4, D4×C20 [×8], C23×C20 [×2], D4×C2×C10, D4×C2×C20
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], D4 [×4], C23 [×15], C10 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C20 [×8], C2×C10 [×35], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C20 [×28], C5×D4 [×4], C22×C10 [×15], C2×C4×D4, C22×C20 [×14], D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C20 [×4], C23×C20, D4×C2×C10, C10×C4○D4, D4×C2×C20
Generators and relations
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(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)(20 121)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 144)(102 145)(103 146)(104 147)(105 148)(106 149)(107 150)(108 151)(109 152)(110 153)(111 154)(112 155)(113 156)(114 157)(115 158)(116 159)(117 160)(118 141)(119 142)(120 143)
(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)
(1 110 37 68)(2 111 38 69)(3 112 39 70)(4 113 40 71)(5 114 21 72)(6 115 22 73)(7 116 23 74)(8 117 24 75)(9 118 25 76)(10 119 26 77)(11 120 27 78)(12 101 28 79)(13 102 29 80)(14 103 30 61)(15 104 31 62)(16 105 32 63)(17 106 33 64)(18 107 34 65)(19 108 35 66)(20 109 36 67)(41 86 134 145)(42 87 135 146)(43 88 136 147)(44 89 137 148)(45 90 138 149)(46 91 139 150)(47 92 140 151)(48 93 121 152)(49 94 122 153)(50 95 123 154)(51 96 124 155)(52 97 125 156)(53 98 126 157)(54 99 127 158)(55 100 128 159)(56 81 129 160)(57 82 130 141)(58 83 131 142)(59 84 132 143)(60 85 133 144)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 126)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 121)(37 122)(38 123)(39 124)(40 125)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 144)(102 145)(103 146)(104 147)(105 148)(106 149)(107 150)(108 151)(109 152)(110 153)(111 154)(112 155)(113 156)(114 157)(115 158)(116 159)(117 160)(118 141)(119 142)(120 143)
G:=sub<Sym(160)| (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)(20,121)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,141)(119,142)(120,143), (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), (1,110,37,68)(2,111,38,69)(3,112,39,70)(4,113,40,71)(5,114,21,72)(6,115,22,73)(7,116,23,74)(8,117,24,75)(9,118,25,76)(10,119,26,77)(11,120,27,78)(12,101,28,79)(13,102,29,80)(14,103,30,61)(15,104,31,62)(16,105,32,63)(17,106,33,64)(18,107,34,65)(19,108,35,66)(20,109,36,67)(41,86,134,145)(42,87,135,146)(43,88,136,147)(44,89,137,148)(45,90,138,149)(46,91,139,150)(47,92,140,151)(48,93,121,152)(49,94,122,153)(50,95,123,154)(51,96,124,155)(52,97,125,156)(53,98,126,157)(54,99,127,158)(55,100,128,159)(56,81,129,160)(57,82,130,141)(58,83,131,142)(59,84,132,143)(60,85,133,144), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,121)(37,122)(38,123)(39,124)(40,125)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,141)(119,142)(120,143)>;
G:=Group( (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)(20,121)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,141)(119,142)(120,143), (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), (1,110,37,68)(2,111,38,69)(3,112,39,70)(4,113,40,71)(5,114,21,72)(6,115,22,73)(7,116,23,74)(8,117,24,75)(9,118,25,76)(10,119,26,77)(11,120,27,78)(12,101,28,79)(13,102,29,80)(14,103,30,61)(15,104,31,62)(16,105,32,63)(17,106,33,64)(18,107,34,65)(19,108,35,66)(20,109,36,67)(41,86,134,145)(42,87,135,146)(43,88,136,147)(44,89,137,148)(45,90,138,149)(46,91,139,150)(47,92,140,151)(48,93,121,152)(49,94,122,153)(50,95,123,154)(51,96,124,155)(52,97,125,156)(53,98,126,157)(54,99,127,158)(55,100,128,159)(56,81,129,160)(57,82,130,141)(58,83,131,142)(59,84,132,143)(60,85,133,144), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,121)(37,122)(38,123)(39,124)(40,125)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,141)(119,142)(120,143) );
G=PermutationGroup([(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),(20,121),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(101,144),(102,145),(103,146),(104,147),(105,148),(106,149),(107,150),(108,151),(109,152),(110,153),(111,154),(112,155),(113,156),(114,157),(115,158),(116,159),(117,160),(118,141),(119,142),(120,143)], [(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)], [(1,110,37,68),(2,111,38,69),(3,112,39,70),(4,113,40,71),(5,114,21,72),(6,115,22,73),(7,116,23,74),(8,117,24,75),(9,118,25,76),(10,119,26,77),(11,120,27,78),(12,101,28,79),(13,102,29,80),(14,103,30,61),(15,104,31,62),(16,105,32,63),(17,106,33,64),(18,107,34,65),(19,108,35,66),(20,109,36,67),(41,86,134,145),(42,87,135,146),(43,88,136,147),(44,89,137,148),(45,90,138,149),(46,91,139,150),(47,92,140,151),(48,93,121,152),(49,94,122,153),(50,95,123,154),(51,96,124,155),(52,97,125,156),(53,98,126,157),(54,99,127,158),(55,100,128,159),(56,81,129,160),(57,82,130,141),(58,83,131,142),(59,84,132,143),(60,85,133,144)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,126),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,121),(37,122),(38,123),(39,124),(40,125),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(101,144),(102,145),(103,146),(104,147),(105,148),(106,149),(107,150),(108,151),(109,152),(110,153),(111,154),(112,155),(113,156),(114,157),(115,158),(116,159),(117,160),(118,141),(119,142),(120,143)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀