metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).20D20, (C2×C20).31D4, (C22×D5)⋊1Q8, C10.6C22≀C2, C5⋊1(C23⋊Q8), C22.42(Q8×D5), (C2×Dic5).20D4, (C22×C4).72D10, C22.157(D4×D5), C22.82(C2×D20), C2.9(C22⋊D20), C2.8(D10⋊2Q8), (C22×Dic10)⋊1C2, C2.7(C4.D20), C2.C42⋊12D5, C10.26(C22⋊Q8), (C23×D5).5C22, C10.10C42⋊5C2, C2.10(D10⋊Q8), C10.20(C4.4D4), C22.90(C4○D20), (C22×C20).17C22, C23.361(C22×D5), C22.88(D4⋊2D5), (C22×C10).298C23, C2.10(Dic5.5D4), (C22×Dic5).20C22, (C2×C10).69(C2×Q8), (C2×C10).206(C2×D4), (C2×D10⋊C4).8C2, (C2×C10).60(C4○D4), (C5×C2.C42)⋊10C2, SmallGroup(320,300)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).20D20
G = < a,b,c,d | a2=b20=c4=1, d2=b10, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b10c-1 >
Subgroups: 886 in 202 conjugacy classes, 61 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], Q8 [×8], C23, C23 [×8], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×Q8 [×6], C24, Dic5 [×5], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×2], C2×C22⋊C4 [×3], C22×Q8, Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C23⋊Q8, D10⋊C4 [×6], C2×Dic10 [×6], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42 [×2], C5×C2.C42, C2×D10⋊C4, C2×D10⋊C4 [×2], C22×Dic10, (C2×C4).20D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], D20 [×2], C22×D5, C23⋊Q8, C2×D20, C4○D20 [×2], D4×D5 [×2], D4⋊2D5, Q8×D5, C4.D20, C22⋊D20, Dic5.5D4 [×2], D10⋊Q8 [×2], D10⋊2Q8, (C2×C4).20D20
►On 160 points
Generators in S160
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 41)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 101)(38 102)(39 103)(40 104)(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)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
(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 66 29 148)(2 93 30 140)(3 68 31 150)(4 95 32 122)(5 70 33 152)(6 97 34 124)(7 72 35 154)(8 99 36 126)(9 74 37 156)(10 81 38 128)(11 76 39 158)(12 83 40 130)(13 78 21 160)(14 85 22 132)(15 80 23 142)(16 87 24 134)(17 62 25 144)(18 89 26 136)(19 64 27 146)(20 91 28 138)(41 65 112 147)(42 92 113 139)(43 67 114 149)(44 94 115 121)(45 69 116 151)(46 96 117 123)(47 71 118 153)(48 98 119 125)(49 73 120 155)(50 100 101 127)(51 75 102 157)(52 82 103 129)(53 77 104 159)(54 84 105 131)(55 79 106 141)(56 86 107 133)(57 61 108 143)(58 88 109 135)(59 63 110 145)(60 90 111 137)
(1 158 11 148)(2 157 12 147)(3 156 13 146)(4 155 14 145)(5 154 15 144)(6 153 16 143)(7 152 17 142)(8 151 18 141)(9 150 19 160)(10 149 20 159)(21 64 31 74)(22 63 32 73)(23 62 33 72)(24 61 34 71)(25 80 35 70)(26 79 36 69)(27 78 37 68)(28 77 38 67)(29 76 39 66)(30 75 40 65)(41 130 51 140)(42 129 52 139)(43 128 53 138)(44 127 54 137)(45 126 55 136)(46 125 56 135)(47 124 57 134)(48 123 58 133)(49 122 59 132)(50 121 60 131)(81 104 91 114)(82 103 92 113)(83 102 93 112)(84 101 94 111)(85 120 95 110)(86 119 96 109)(87 118 97 108)(88 117 98 107)(89 116 99 106)(90 115 100 105)
G:=sub<Sym(160)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,101)(38,102)(39,103)(40,104)(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)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (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), (1,66,29,148)(2,93,30,140)(3,68,31,150)(4,95,32,122)(5,70,33,152)(6,97,34,124)(7,72,35,154)(8,99,36,126)(9,74,37,156)(10,81,38,128)(11,76,39,158)(12,83,40,130)(13,78,21,160)(14,85,22,132)(15,80,23,142)(16,87,24,134)(17,62,25,144)(18,89,26,136)(19,64,27,146)(20,91,28,138)(41,65,112,147)(42,92,113,139)(43,67,114,149)(44,94,115,121)(45,69,116,151)(46,96,117,123)(47,71,118,153)(48,98,119,125)(49,73,120,155)(50,100,101,127)(51,75,102,157)(52,82,103,129)(53,77,104,159)(54,84,105,131)(55,79,106,141)(56,86,107,133)(57,61,108,143)(58,88,109,135)(59,63,110,145)(60,90,111,137), (1,158,11,148)(2,157,12,147)(3,156,13,146)(4,155,14,145)(5,154,15,144)(6,153,16,143)(7,152,17,142)(8,151,18,141)(9,150,19,160)(10,149,20,159)(21,64,31,74)(22,63,32,73)(23,62,33,72)(24,61,34,71)(25,80,35,70)(26,79,36,69)(27,78,37,68)(28,77,38,67)(29,76,39,66)(30,75,40,65)(41,130,51,140)(42,129,52,139)(43,128,53,138)(44,127,54,137)(45,126,55,136)(46,125,56,135)(47,124,57,134)(48,123,58,133)(49,122,59,132)(50,121,60,131)(81,104,91,114)(82,103,92,113)(83,102,93,112)(84,101,94,111)(85,120,95,110)(86,119,96,109)(87,118,97,108)(88,117,98,107)(89,116,99,106)(90,115,100,105)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,101)(38,102)(39,103)(40,104)(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)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (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), (1,66,29,148)(2,93,30,140)(3,68,31,150)(4,95,32,122)(5,70,33,152)(6,97,34,124)(7,72,35,154)(8,99,36,126)(9,74,37,156)(10,81,38,128)(11,76,39,158)(12,83,40,130)(13,78,21,160)(14,85,22,132)(15,80,23,142)(16,87,24,134)(17,62,25,144)(18,89,26,136)(19,64,27,146)(20,91,28,138)(41,65,112,147)(42,92,113,139)(43,67,114,149)(44,94,115,121)(45,69,116,151)(46,96,117,123)(47,71,118,153)(48,98,119,125)(49,73,120,155)(50,100,101,127)(51,75,102,157)(52,82,103,129)(53,77,104,159)(54,84,105,131)(55,79,106,141)(56,86,107,133)(57,61,108,143)(58,88,109,135)(59,63,110,145)(60,90,111,137), (1,158,11,148)(2,157,12,147)(3,156,13,146)(4,155,14,145)(5,154,15,144)(6,153,16,143)(7,152,17,142)(8,151,18,141)(9,150,19,160)(10,149,20,159)(21,64,31,74)(22,63,32,73)(23,62,33,72)(24,61,34,71)(25,80,35,70)(26,79,36,69)(27,78,37,68)(28,77,38,67)(29,76,39,66)(30,75,40,65)(41,130,51,140)(42,129,52,139)(43,128,53,138)(44,127,54,137)(45,126,55,136)(46,125,56,135)(47,124,57,134)(48,123,58,133)(49,122,59,132)(50,121,60,131)(81,104,91,114)(82,103,92,113)(83,102,93,112)(84,101,94,111)(85,120,95,110)(86,119,96,109)(87,118,97,108)(88,117,98,107)(89,116,99,106)(90,115,100,105) );
G=PermutationGroup([(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,41),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,101),(38,102),(39,103),(40,104),(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),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)], [(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)], [(1,66,29,148),(2,93,30,140),(3,68,31,150),(4,95,32,122),(5,70,33,152),(6,97,34,124),(7,72,35,154),(8,99,36,126),(9,74,37,156),(10,81,38,128),(11,76,39,158),(12,83,40,130),(13,78,21,160),(14,85,22,132),(15,80,23,142),(16,87,24,134),(17,62,25,144),(18,89,26,136),(19,64,27,146),(20,91,28,138),(41,65,112,147),(42,92,113,139),(43,67,114,149),(44,94,115,121),(45,69,116,151),(46,96,117,123),(47,71,118,153),(48,98,119,125),(49,73,120,155),(50,100,101,127),(51,75,102,157),(52,82,103,129),(53,77,104,159),(54,84,105,131),(55,79,106,141),(56,86,107,133),(57,61,108,143),(58,88,109,135),(59,63,110,145),(60,90,111,137)], [(1,158,11,148),(2,157,12,147),(3,156,13,146),(4,155,14,145),(5,154,15,144),(6,153,16,143),(7,152,17,142),(8,151,18,141),(9,150,19,160),(10,149,20,159),(21,64,31,74),(22,63,32,73),(23,62,33,72),(24,61,34,71),(25,80,35,70),(26,79,36,69),(27,78,37,68),(28,77,38,67),(29,76,39,66),(30,75,40,65),(41,130,51,140),(42,129,52,139),(43,128,53,138),(44,127,54,137),(45,126,55,136),(46,125,56,135),(47,124,57,134),(48,123,58,133),(49,122,59,132),(50,121,60,131),(81,104,91,114),(82,103,92,113),(83,102,93,112),(84,101,94,111),(85,120,95,110),(86,119,96,109),(87,118,97,108),(88,117,98,107),(89,116,99,106),(90,115,100,105)])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | D20 | C4○D20 | D4×D5 | D4⋊2D5 | Q8×D5 |
| kernel | (C2×C4).20D20 | C10.10C42 | C5×C2.C42 | C2×D10⋊C4 | C22×Dic10 | C2×Dic5 | C2×C20 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 4 | 2 | 2 | 2 | 6 | 6 | 8 | 16 | 4 | 2 | 2 |
Matrix representation of (C2×C4).20D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 14 | 30 | 0 | 0 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 21 | 9 |
| 24 | 1 | 0 | 0 | 0 | 0 |
| 40 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 28 | 0 | 0 |
| 0 | 0 | 22 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 25 |
| 24 | 1 | 0 | 0 | 0 | 0 |
| 38 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 2 | 0 | 0 |
| 0 | 0 | 5 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 15 | 25 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[14,11,0,0,0,0,30,9,0,0,0,0,0,0,34,33,0,0,0,0,1,1,0,0,0,0,0,0,32,21,0,0,0,0,0,9],[24,40,0,0,0,0,1,17,0,0,0,0,0,0,11,22,0,0,0,0,28,30,0,0,0,0,0,0,16,16,0,0,0,0,2,25],[24,38,0,0,0,0,1,17,0,0,0,0,0,0,14,5,0,0,0,0,2,27,0,0,0,0,0,0,16,15,0,0,0,0,2,25] >;
(C2×C4).20D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{20}D_{20} % in TeX
G:=Group("(C2xC4).20D20"); // GroupNames label
G:=SmallGroup(320,300);
// by ID
G=gap.SmallGroup(320,300);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,926,387,268,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations