metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2xC20).31D4, (C2xC4).20D20, (C22xD5):1Q8, C10.6C22wrC2, C5:1(C23:Q8), C22.42(Q8xD5), (C2xDic5).20D4, C22.82(C2xD20), (C22xC4).72D10, C22.157(D4xD5), C2.9(C22:D20), C2.8(D10:2Q8), C2.7(C4.D20), (C22xDic10):1C2, C2.C42:12D5, C10.26(C22:Q8), (C23xD5).5C22, C10.10C42:5C2, C2.10(D10:Q8), C10.20(C4.4D4), C22.90(C4oD20), (C22xC20).17C22, C23.361(C22xD5), C22.88(D4:2D5), (C22xC10).298C23, C2.10(Dic5.5D4), (C22xDic5).20C22, (C2xC10).69(C2xQ8), (C2xC10).206(C2xD4), (C2xD10:C4).8C2, (C2xC10).60(C4oD4), (C5xC2.C42):10C2, SmallGroup(320,300)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2xC20).31D4
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 [x3], C2 [x4], C2 [x2], C4 [x9], C22 [x3], C22 [x4], C22 [x10], C5, C2xC4 [x2], C2xC4 [x19], Q8 [x8], C23, C23 [x8], D5 [x2], C10 [x3], C10 [x4], C22:C4 [x6], C22xC4, C22xC4 [x2], C22xC4 [x3], C2xQ8 [x6], C24, Dic5 [x5], C20 [x4], D10 [x10], C2xC10 [x3], C2xC10 [x4], C2.C42, C2.C42 [x2], C2xC22:C4 [x3], C22xQ8, Dic10 [x8], C2xDic5 [x4], C2xDic5 [x7], C2xC20 [x2], C2xC20 [x8], C22xD5 [x2], C22xD5 [x6], C22xC10, C23:Q8, D10:C4 [x6], C2xDic10 [x6], C22xDic5, C22xDic5 [x2], C22xC20, C22xC20 [x2], C23xD5, C10.10C42 [x2], C5xC2.C42, C2xD10:C4, C2xD10:C4 [x2], C22xDic10, (C2xC20).31D4
Quotients: C1, C2 [x7], C22 [x7], D4 [x6], Q8 [x2], C23, D5, C2xD4 [x3], C2xQ8, C4oD4 [x3], D10 [x3], C22wrC2, C22:Q8 [x3], C4.4D4 [x3], D20 [x2], C22xD5, C23:Q8, C2xD20, C4oD20 [x2], D4xD5 [x2], D4:2D5, Q8xD5, C4.D20, C22:D20, Dic5.5D4 [x2], D10:Q8 [x2], D10:2Q8, (C2xC20).31D4
►On 160 points
Generators in S160
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 121)(99 122)(100 123)
(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)
(1 117 95 26)(2 58 96 159)(3 119 97 28)(4 60 98 141)(5 101 99 30)(6 42 100 143)(7 103 81 32)(8 44 82 145)(9 105 83 34)(10 46 84 147)(11 107 85 36)(12 48 86 149)(13 109 87 38)(14 50 88 151)(15 111 89 40)(16 52 90 153)(17 113 91 22)(18 54 92 155)(19 115 93 24)(20 56 94 157)(21 64 112 133)(23 66 114 135)(25 68 116 137)(27 70 118 139)(29 72 120 121)(31 74 102 123)(33 76 104 125)(35 78 106 127)(37 80 108 129)(39 62 110 131)(41 122 142 73)(43 124 144 75)(45 126 146 77)(47 128 148 79)(49 130 150 61)(51 132 152 63)(53 134 154 65)(55 136 156 67)(57 138 158 69)(59 140 160 71)
(1 36 11 26)(2 35 12 25)(3 34 13 24)(4 33 14 23)(5 32 15 22)(6 31 16 21)(7 30 17 40)(8 29 18 39)(9 28 19 38)(10 27 20 37)(41 134 51 124)(42 133 52 123)(43 132 53 122)(44 131 54 121)(45 130 55 140)(46 129 56 139)(47 128 57 138)(48 127 58 137)(49 126 59 136)(50 125 60 135)(61 156 71 146)(62 155 72 145)(63 154 73 144)(64 153 74 143)(65 152 75 142)(66 151 76 141)(67 150 77 160)(68 149 78 159)(69 148 79 158)(70 147 80 157)(81 101 91 111)(82 120 92 110)(83 119 93 109)(84 118 94 108)(85 117 95 107)(86 116 96 106)(87 115 97 105)(88 114 98 104)(89 113 99 103)(90 112 100 102)
G:=sub<Sym(160)| (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123), (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), (1,117,95,26)(2,58,96,159)(3,119,97,28)(4,60,98,141)(5,101,99,30)(6,42,100,143)(7,103,81,32)(8,44,82,145)(9,105,83,34)(10,46,84,147)(11,107,85,36)(12,48,86,149)(13,109,87,38)(14,50,88,151)(15,111,89,40)(16,52,90,153)(17,113,91,22)(18,54,92,155)(19,115,93,24)(20,56,94,157)(21,64,112,133)(23,66,114,135)(25,68,116,137)(27,70,118,139)(29,72,120,121)(31,74,102,123)(33,76,104,125)(35,78,106,127)(37,80,108,129)(39,62,110,131)(41,122,142,73)(43,124,144,75)(45,126,146,77)(47,128,148,79)(49,130,150,61)(51,132,152,63)(53,134,154,65)(55,136,156,67)(57,138,158,69)(59,140,160,71), (1,36,11,26)(2,35,12,25)(3,34,13,24)(4,33,14,23)(5,32,15,22)(6,31,16,21)(7,30,17,40)(8,29,18,39)(9,28,19,38)(10,27,20,37)(41,134,51,124)(42,133,52,123)(43,132,53,122)(44,131,54,121)(45,130,55,140)(46,129,56,139)(47,128,57,138)(48,127,58,137)(49,126,59,136)(50,125,60,135)(61,156,71,146)(62,155,72,145)(63,154,73,144)(64,153,74,143)(65,152,75,142)(66,151,76,141)(67,150,77,160)(68,149,78,159)(69,148,79,158)(70,147,80,157)(81,101,91,111)(82,120,92,110)(83,119,93,109)(84,118,94,108)(85,117,95,107)(86,116,96,106)(87,115,97,105)(88,114,98,104)(89,113,99,103)(90,112,100,102)>;
G:=Group( (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123), (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), (1,117,95,26)(2,58,96,159)(3,119,97,28)(4,60,98,141)(5,101,99,30)(6,42,100,143)(7,103,81,32)(8,44,82,145)(9,105,83,34)(10,46,84,147)(11,107,85,36)(12,48,86,149)(13,109,87,38)(14,50,88,151)(15,111,89,40)(16,52,90,153)(17,113,91,22)(18,54,92,155)(19,115,93,24)(20,56,94,157)(21,64,112,133)(23,66,114,135)(25,68,116,137)(27,70,118,139)(29,72,120,121)(31,74,102,123)(33,76,104,125)(35,78,106,127)(37,80,108,129)(39,62,110,131)(41,122,142,73)(43,124,144,75)(45,126,146,77)(47,128,148,79)(49,130,150,61)(51,132,152,63)(53,134,154,65)(55,136,156,67)(57,138,158,69)(59,140,160,71), (1,36,11,26)(2,35,12,25)(3,34,13,24)(4,33,14,23)(5,32,15,22)(6,31,16,21)(7,30,17,40)(8,29,18,39)(9,28,19,38)(10,27,20,37)(41,134,51,124)(42,133,52,123)(43,132,53,122)(44,131,54,121)(45,130,55,140)(46,129,56,139)(47,128,57,138)(48,127,58,137)(49,126,59,136)(50,125,60,135)(61,156,71,146)(62,155,72,145)(63,154,73,144)(64,153,74,143)(65,152,75,142)(66,151,76,141)(67,150,77,160)(68,149,78,159)(69,148,79,158)(70,147,80,157)(81,101,91,111)(82,120,92,110)(83,119,93,109)(84,118,94,108)(85,117,95,107)(86,116,96,106)(87,115,97,105)(88,114,98,104)(89,113,99,103)(90,112,100,102) );
G=PermutationGroup([(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,153),(22,154),(23,155),(24,156),(25,157),(26,158),(27,159),(28,160),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139),(97,140),(98,121),(99,122),(100,123)], [(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)], [(1,117,95,26),(2,58,96,159),(3,119,97,28),(4,60,98,141),(5,101,99,30),(6,42,100,143),(7,103,81,32),(8,44,82,145),(9,105,83,34),(10,46,84,147),(11,107,85,36),(12,48,86,149),(13,109,87,38),(14,50,88,151),(15,111,89,40),(16,52,90,153),(17,113,91,22),(18,54,92,155),(19,115,93,24),(20,56,94,157),(21,64,112,133),(23,66,114,135),(25,68,116,137),(27,70,118,139),(29,72,120,121),(31,74,102,123),(33,76,104,125),(35,78,106,127),(37,80,108,129),(39,62,110,131),(41,122,142,73),(43,124,144,75),(45,126,146,77),(47,128,148,79),(49,130,150,61),(51,132,152,63),(53,134,154,65),(55,136,156,67),(57,138,158,69),(59,140,160,71)], [(1,36,11,26),(2,35,12,25),(3,34,13,24),(4,33,14,23),(5,32,15,22),(6,31,16,21),(7,30,17,40),(8,29,18,39),(9,28,19,38),(10,27,20,37),(41,134,51,124),(42,133,52,123),(43,132,53,122),(44,131,54,121),(45,130,55,140),(46,129,56,139),(47,128,57,138),(48,127,58,137),(49,126,59,136),(50,125,60,135),(61,156,71,146),(62,155,72,145),(63,154,73,144),(64,153,74,143),(65,152,75,142),(66,151,76,141),(67,150,77,160),(68,149,78,159),(69,148,79,158),(70,147,80,157),(81,101,91,111),(82,120,92,110),(83,119,93,109),(84,118,94,108),(85,117,95,107),(86,116,96,106),(87,115,97,105),(88,114,98,104),(89,113,99,103),(90,112,100,102)])
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 | C4oD4 | D10 | D20 | C4oD20 | D4xD5 | D4:2D5 | Q8xD5 |
| kernel | (C2xC20).31D4 | C10.10C42 | C5xC2.C42 | C2xD10:C4 | C22xDic10 | C2xDic5 | C2xC20 | C22xD5 | C2.C42 | C2xC10 | C22xC4 | C2xC4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 4 | 2 | 2 | 2 | 6 | 6 | 8 | 16 | 4 | 2 | 2 |
Matrix representation of (C2xC20).31D4 ►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] >;
(C2xC20).31D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})._{31}D_4 % in TeX
G:=Group("(C2xC20).31D4"); // 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