metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).290D4, C2.9(C20⋊2D4), (C22×D5).42D4, C22.249(D4×D5), C10.92(C4⋊D4), (C22×C4).105D10, C5⋊5(C23.11D4), C2.7(C20.23D4), C10.54(C4.4D4), (C23×D5).22C22, C23.382(C22×D5), C10.10C42⋊20C2, C10.29(C42⋊2C2), C22.110(C4○D20), (C22×C20).393C22, (C22×C10).357C23, C22.53(Q8⋊2D5), C2.22(D10.13D4), C22.104(D4⋊2D5), C10.64(C22.D4), (C22×Dic5).62C22, C2.15(C23.23D10), (C2×C4⋊C4)⋊11D5, (C10×C4⋊C4)⋊27C2, (C2×C10).337(C2×D4), (C2×C4).43(C5⋊D4), C2.14(C4⋊C4⋊D5), C22.142(C2×C5⋊D4), (C2×D10⋊C4).17C2, (C2×C10).192(C4○D4), SmallGroup(320,620)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20).290D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab9, dcd=ac-1 >
Subgroups: 726 in 170 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.11D4, D10⋊C4, C5×C4⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C10.10C42, C2×D10⋊C4, C2×D10⋊C4, C10×C4⋊C4, (C2×C20).290D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C5⋊D4, C22×D5, C23.11D4, C4○D20, D4×D5, D4⋊2D5, Q8⋊2D5, C2×C5⋊D4, D10.13D4, C4⋊C4⋊D5, C23.23D10, C20⋊2D4, C20.23D4, (C2×C20).290D4
►On 160 points
Generators in S160
(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 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(61 137)(62 138)(63 139)(64 140)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)
(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 85 153 76)(2 37 154 131)(3 83 155 74)(4 35 156 129)(5 81 157 72)(6 33 158 127)(7 99 159 70)(8 31 160 125)(9 97 141 68)(10 29 142 123)(11 95 143 66)(12 27 144 121)(13 93 145 64)(14 25 146 139)(15 91 147 62)(16 23 148 137)(17 89 149 80)(18 21 150 135)(19 87 151 78)(20 39 152 133)(22 119 136 45)(24 117 138 43)(26 115 140 41)(28 113 122 59)(30 111 124 57)(32 109 126 55)(34 107 128 53)(36 105 130 51)(38 103 132 49)(40 101 134 47)(42 92 116 63)(44 90 118 61)(46 88 120 79)(48 86 102 77)(50 84 104 75)(52 82 106 73)(54 100 108 71)(56 98 110 69)(58 96 112 67)(60 94 114 65)
(2 58)(3 19)(4 56)(5 17)(6 54)(7 15)(8 52)(9 13)(10 50)(12 48)(14 46)(16 44)(18 42)(20 60)(21 139)(22 72)(23 137)(24 70)(25 135)(26 68)(27 133)(28 66)(29 131)(30 64)(31 129)(32 62)(33 127)(34 80)(35 125)(36 78)(37 123)(38 76)(39 121)(40 74)(41 57)(43 55)(45 53)(47 51)(61 90)(63 88)(65 86)(67 84)(69 82)(71 100)(73 98)(75 96)(77 94)(79 92)(81 136)(83 134)(85 132)(87 130)(89 128)(91 126)(93 124)(95 122)(97 140)(99 138)(101 105)(102 144)(104 142)(106 160)(107 119)(108 158)(109 117)(110 156)(111 115)(112 154)(114 152)(116 150)(118 148)(120 146)(141 145)(147 159)(149 157)(151 155)
G:=sub<Sym(160)| (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)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(61,137)(62,138)(63,139)(64,140)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150), (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), (1,85,153,76)(2,37,154,131)(3,83,155,74)(4,35,156,129)(5,81,157,72)(6,33,158,127)(7,99,159,70)(8,31,160,125)(9,97,141,68)(10,29,142,123)(11,95,143,66)(12,27,144,121)(13,93,145,64)(14,25,146,139)(15,91,147,62)(16,23,148,137)(17,89,149,80)(18,21,150,135)(19,87,151,78)(20,39,152,133)(22,119,136,45)(24,117,138,43)(26,115,140,41)(28,113,122,59)(30,111,124,57)(32,109,126,55)(34,107,128,53)(36,105,130,51)(38,103,132,49)(40,101,134,47)(42,92,116,63)(44,90,118,61)(46,88,120,79)(48,86,102,77)(50,84,104,75)(52,82,106,73)(54,100,108,71)(56,98,110,69)(58,96,112,67)(60,94,114,65), (2,58)(3,19)(4,56)(5,17)(6,54)(7,15)(8,52)(9,13)(10,50)(12,48)(14,46)(16,44)(18,42)(20,60)(21,139)(22,72)(23,137)(24,70)(25,135)(26,68)(27,133)(28,66)(29,131)(30,64)(31,129)(32,62)(33,127)(34,80)(35,125)(36,78)(37,123)(38,76)(39,121)(40,74)(41,57)(43,55)(45,53)(47,51)(61,90)(63,88)(65,86)(67,84)(69,82)(71,100)(73,98)(75,96)(77,94)(79,92)(81,136)(83,134)(85,132)(87,130)(89,128)(91,126)(93,124)(95,122)(97,140)(99,138)(101,105)(102,144)(104,142)(106,160)(107,119)(108,158)(109,117)(110,156)(111,115)(112,154)(114,152)(116,150)(118,148)(120,146)(141,145)(147,159)(149,157)(151,155)>;
G:=Group( (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)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(61,137)(62,138)(63,139)(64,140)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150), (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), (1,85,153,76)(2,37,154,131)(3,83,155,74)(4,35,156,129)(5,81,157,72)(6,33,158,127)(7,99,159,70)(8,31,160,125)(9,97,141,68)(10,29,142,123)(11,95,143,66)(12,27,144,121)(13,93,145,64)(14,25,146,139)(15,91,147,62)(16,23,148,137)(17,89,149,80)(18,21,150,135)(19,87,151,78)(20,39,152,133)(22,119,136,45)(24,117,138,43)(26,115,140,41)(28,113,122,59)(30,111,124,57)(32,109,126,55)(34,107,128,53)(36,105,130,51)(38,103,132,49)(40,101,134,47)(42,92,116,63)(44,90,118,61)(46,88,120,79)(48,86,102,77)(50,84,104,75)(52,82,106,73)(54,100,108,71)(56,98,110,69)(58,96,112,67)(60,94,114,65), (2,58)(3,19)(4,56)(5,17)(6,54)(7,15)(8,52)(9,13)(10,50)(12,48)(14,46)(16,44)(18,42)(20,60)(21,139)(22,72)(23,137)(24,70)(25,135)(26,68)(27,133)(28,66)(29,131)(30,64)(31,129)(32,62)(33,127)(34,80)(35,125)(36,78)(37,123)(38,76)(39,121)(40,74)(41,57)(43,55)(45,53)(47,51)(61,90)(63,88)(65,86)(67,84)(69,82)(71,100)(73,98)(75,96)(77,94)(79,92)(81,136)(83,134)(85,132)(87,130)(89,128)(91,126)(93,124)(95,122)(97,140)(99,138)(101,105)(102,144)(104,142)(106,160)(107,119)(108,158)(109,117)(110,156)(111,115)(112,154)(114,152)(116,150)(118,148)(120,146)(141,145)(147,159)(149,157)(151,155) );
G=PermutationGroup([[(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),(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(61,137),(62,138),(63,139),(64,140),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136),(101,151),(102,152),(103,153),(104,154),(105,155),(106,156),(107,157),(108,158),(109,159),(110,160),(111,141),(112,142),(113,143),(114,144),(115,145),(116,146),(117,147),(118,148),(119,149),(120,150)], [(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)], [(1,85,153,76),(2,37,154,131),(3,83,155,74),(4,35,156,129),(5,81,157,72),(6,33,158,127),(7,99,159,70),(8,31,160,125),(9,97,141,68),(10,29,142,123),(11,95,143,66),(12,27,144,121),(13,93,145,64),(14,25,146,139),(15,91,147,62),(16,23,148,137),(17,89,149,80),(18,21,150,135),(19,87,151,78),(20,39,152,133),(22,119,136,45),(24,117,138,43),(26,115,140,41),(28,113,122,59),(30,111,124,57),(32,109,126,55),(34,107,128,53),(36,105,130,51),(38,103,132,49),(40,101,134,47),(42,92,116,63),(44,90,118,61),(46,88,120,79),(48,86,102,77),(50,84,104,75),(52,82,106,73),(54,100,108,71),(56,98,110,69),(58,96,112,67),(60,94,114,65)], [(2,58),(3,19),(4,56),(5,17),(6,54),(7,15),(8,52),(9,13),(10,50),(12,48),(14,46),(16,44),(18,42),(20,60),(21,139),(22,72),(23,137),(24,70),(25,135),(26,68),(27,133),(28,66),(29,131),(30,64),(31,129),(32,62),(33,127),(34,80),(35,125),(36,78),(37,123),(38,76),(39,121),(40,74),(41,57),(43,55),(45,53),(47,51),(61,90),(63,88),(65,86),(67,84),(69,82),(71,100),(73,98),(75,96),(77,94),(79,92),(81,136),(83,134),(85,132),(87,130),(89,128),(91,126),(93,124),(95,122),(97,140),(99,138),(101,105),(102,144),(104,142),(106,160),(107,119),(108,158),(109,117),(110,156),(111,115),(112,154),(114,152),(116,150),(118,148),(120,146),(141,145),(147,159),(149,157),(151,155)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | C5⋊D4 | C4○D20 | D4×D5 | D4⋊2D5 | Q8⋊2D5 |
| kernel | (C2×C20).290D4 | C10.10C42 | C2×D10⋊C4 | C10×C4⋊C4 | C2×C20 | C22×D5 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 2 | 2 | 10 | 6 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of (C2×C20).290D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 14 | 22 | 0 | 0 | 0 | 0 |
| 19 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 |
| 0 | 0 | 18 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 0 | 0 | 0 | 0 | 36 | 35 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 28 | 0 | 0 |
| 0 | 0 | 32 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 3 |
| 0 | 0 | 0 | 0 | 3 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 7 |
| 0 | 0 | 0 | 0 | 36 | 35 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,19,0,0,0,0,22,27,0,0,0,0,0,0,35,18,0,0,0,0,23,20,0,0,0,0,0,0,40,36,0,0,0,0,40,35],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,32,0,0,0,0,28,28,0,0,0,0,0,0,20,3,0,0,0,0,3,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,6,36,0,0,0,0,7,35] >;
(C2×C20).290D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})._{290}D_4 % in TeX
G:=Group("(C2xC20).290D4"); // GroupNames label
G:=SmallGroup(320,620);
// by ID
G=gap.SmallGroup(320,620);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,254,387,100,12550]);
// 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,c*b*c^-1=a*b^-1,d*b*d=a*b^9,d*c*d=a*c^-1>;
// generators/relations