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, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2.C42, C2×C22⋊C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23⋊Q8, D10⋊C4, C2×Dic10, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×D10⋊C4, C2×D10⋊C4, C22×Dic10, (C2×C4).20D20
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22≀C2, C22⋊Q8, C4.4D4, D20, C22×D5, C23⋊Q8, C2×D20, C4○D20, D4×D5, D4⋊2D5, Q8×D5, C4.D20, C22⋊D20, Dic5.5D4, D10⋊Q8, D10⋊2Q8, (C2×C4).20D20
►On 160 points
Generators in S160
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 157)(11 158)(12 159)(13 160)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)
(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 63 24 92)(2 52 25 121)(3 65 26 94)(4 54 27 123)(5 67 28 96)(6 56 29 125)(7 69 30 98)(8 58 31 127)(9 71 32 100)(10 60 33 129)(11 73 34 82)(12 42 35 131)(13 75 36 84)(14 44 37 133)(15 77 38 86)(16 46 39 135)(17 79 40 88)(18 48 21 137)(19 61 22 90)(20 50 23 139)(41 108 130 158)(43 110 132 160)(45 112 134 142)(47 114 136 144)(49 116 138 146)(51 118 140 148)(53 120 122 150)(55 102 124 152)(57 104 126 154)(59 106 128 156)(62 117 91 147)(64 119 93 149)(66 101 95 151)(68 103 97 153)(70 105 99 155)(72 107 81 157)(74 109 83 159)(76 111 85 141)(78 113 87 143)(80 115 89 145)
(1 82 11 92)(2 81 12 91)(3 100 13 90)(4 99 14 89)(5 98 15 88)(6 97 16 87)(7 96 17 86)(8 95 18 85)(9 94 19 84)(10 93 20 83)(21 76 31 66)(22 75 32 65)(23 74 33 64)(24 73 34 63)(25 72 35 62)(26 71 36 61)(27 70 37 80)(28 69 38 79)(29 68 39 78)(30 67 40 77)(41 108 51 118)(42 107 52 117)(43 106 53 116)(44 105 54 115)(45 104 55 114)(46 103 56 113)(47 102 57 112)(48 101 58 111)(49 120 59 110)(50 119 60 109)(121 147 131 157)(122 146 132 156)(123 145 133 155)(124 144 134 154)(125 143 135 153)(126 142 136 152)(127 141 137 151)(128 160 138 150)(129 159 139 149)(130 158 140 148)
G:=sub<Sym(160)| (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,158)(12,159)(13,160)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128), (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), (1,63,24,92)(2,52,25,121)(3,65,26,94)(4,54,27,123)(5,67,28,96)(6,56,29,125)(7,69,30,98)(8,58,31,127)(9,71,32,100)(10,60,33,129)(11,73,34,82)(12,42,35,131)(13,75,36,84)(14,44,37,133)(15,77,38,86)(16,46,39,135)(17,79,40,88)(18,48,21,137)(19,61,22,90)(20,50,23,139)(41,108,130,158)(43,110,132,160)(45,112,134,142)(47,114,136,144)(49,116,138,146)(51,118,140,148)(53,120,122,150)(55,102,124,152)(57,104,126,154)(59,106,128,156)(62,117,91,147)(64,119,93,149)(66,101,95,151)(68,103,97,153)(70,105,99,155)(72,107,81,157)(74,109,83,159)(76,111,85,141)(78,113,87,143)(80,115,89,145), (1,82,11,92)(2,81,12,91)(3,100,13,90)(4,99,14,89)(5,98,15,88)(6,97,16,87)(7,96,17,86)(8,95,18,85)(9,94,19,84)(10,93,20,83)(21,76,31,66)(22,75,32,65)(23,74,33,64)(24,73,34,63)(25,72,35,62)(26,71,36,61)(27,70,37,80)(28,69,38,79)(29,68,39,78)(30,67,40,77)(41,108,51,118)(42,107,52,117)(43,106,53,116)(44,105,54,115)(45,104,55,114)(46,103,56,113)(47,102,57,112)(48,101,58,111)(49,120,59,110)(50,119,60,109)(121,147,131,157)(122,146,132,156)(123,145,133,155)(124,144,134,154)(125,143,135,153)(126,142,136,152)(127,141,137,151)(128,160,138,150)(129,159,139,149)(130,158,140,148)>;
G:=Group( (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,158)(12,159)(13,160)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128), (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), (1,63,24,92)(2,52,25,121)(3,65,26,94)(4,54,27,123)(5,67,28,96)(6,56,29,125)(7,69,30,98)(8,58,31,127)(9,71,32,100)(10,60,33,129)(11,73,34,82)(12,42,35,131)(13,75,36,84)(14,44,37,133)(15,77,38,86)(16,46,39,135)(17,79,40,88)(18,48,21,137)(19,61,22,90)(20,50,23,139)(41,108,130,158)(43,110,132,160)(45,112,134,142)(47,114,136,144)(49,116,138,146)(51,118,140,148)(53,120,122,150)(55,102,124,152)(57,104,126,154)(59,106,128,156)(62,117,91,147)(64,119,93,149)(66,101,95,151)(68,103,97,153)(70,105,99,155)(72,107,81,157)(74,109,83,159)(76,111,85,141)(78,113,87,143)(80,115,89,145), (1,82,11,92)(2,81,12,91)(3,100,13,90)(4,99,14,89)(5,98,15,88)(6,97,16,87)(7,96,17,86)(8,95,18,85)(9,94,19,84)(10,93,20,83)(21,76,31,66)(22,75,32,65)(23,74,33,64)(24,73,34,63)(25,72,35,62)(26,71,36,61)(27,70,37,80)(28,69,38,79)(29,68,39,78)(30,67,40,77)(41,108,51,118)(42,107,52,117)(43,106,53,116)(44,105,54,115)(45,104,55,114)(46,103,56,113)(47,102,57,112)(48,101,58,111)(49,120,59,110)(50,119,60,109)(121,147,131,157)(122,146,132,156)(123,145,133,155)(124,144,134,154)(125,143,135,153)(126,142,136,152)(127,141,137,151)(128,160,138,150)(129,159,139,149)(130,158,140,148) );
G=PermutationGroup([[(1,148),(2,149),(3,150),(4,151),(5,152),(6,153),(7,154),(8,155),(9,156),(10,157),(11,158),(12,159),(13,160),(14,141),(15,142),(16,143),(17,144),(18,145),(19,146),(20,147),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128)], [(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)], [(1,63,24,92),(2,52,25,121),(3,65,26,94),(4,54,27,123),(5,67,28,96),(6,56,29,125),(7,69,30,98),(8,58,31,127),(9,71,32,100),(10,60,33,129),(11,73,34,82),(12,42,35,131),(13,75,36,84),(14,44,37,133),(15,77,38,86),(16,46,39,135),(17,79,40,88),(18,48,21,137),(19,61,22,90),(20,50,23,139),(41,108,130,158),(43,110,132,160),(45,112,134,142),(47,114,136,144),(49,116,138,146),(51,118,140,148),(53,120,122,150),(55,102,124,152),(57,104,126,154),(59,106,128,156),(62,117,91,147),(64,119,93,149),(66,101,95,151),(68,103,97,153),(70,105,99,155),(72,107,81,157),(74,109,83,159),(76,111,85,141),(78,113,87,143),(80,115,89,145)], [(1,82,11,92),(2,81,12,91),(3,100,13,90),(4,99,14,89),(5,98,15,88),(6,97,16,87),(7,96,17,86),(8,95,18,85),(9,94,19,84),(10,93,20,83),(21,76,31,66),(22,75,32,65),(23,74,33,64),(24,73,34,63),(25,72,35,62),(26,71,36,61),(27,70,37,80),(28,69,38,79),(29,68,39,78),(30,67,40,77),(41,108,51,118),(42,107,52,117),(43,106,53,116),(44,105,54,115),(45,104,55,114),(46,103,56,113),(47,102,57,112),(48,101,58,111),(49,120,59,110),(50,119,60,109),(121,147,131,157),(122,146,132,156),(123,145,133,155),(124,144,134,154),(125,143,135,153),(126,142,136,152),(127,141,137,151),(128,160,138,150),(129,159,139,149),(130,158,140,148)]])
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