direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.4C8, C10⋊4M5(2), C40.69C23, C5⋊6(C2×M5(2)), (C2×C20).17C8, C20.79(C2×C8), (C2×C40).51C4, C40.120(C2×C4), (C2×C8).326D10, C8.26(C2×Dic5), (C2×C8).15Dic5, C8.63(C22×D5), (C22×C8).14D5, C5⋊2C16⋊12C22, (C22×C20).55C4, C10.51(C22×C8), (C22×C10).11C8, (C22×C40).22C2, C23.3(C5⋊2C8), C20.235(C22×C4), (C2×C40).412C22, C4.30(C22×Dic5), (C22×C4).17Dic5, C4.9(C2×C5⋊2C8), (C2×C5⋊2C16)⋊12C2, (C2×C10).65(C2×C8), (C2×C4).6(C5⋊2C8), C2.7(C22×C5⋊2C8), C22.6(C2×C5⋊2C8), (C2×C20).478(C2×C4), (C2×C4).101(C2×Dic5), SmallGroup(320,724)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.4C8
G = < a,b,c | a2=b40=1, c4=b10, ab=ba, ac=ca, cbc-1=b29 >
Subgroups: 142 in 90 conjugacy classes, 71 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C10, C16, C2×C8, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, M5(2), C22×C8, C40, C40, C2×C20, C2×C20, C22×C10, C2×M5(2), C5⋊2C16, C2×C40, C2×C40, C22×C20, C2×C5⋊2C16, C20.4C8, C22×C40, C2×C20.4C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, Dic5, D10, M5(2), C22×C8, C5⋊2C8, C2×Dic5, C22×D5, C2×M5(2), C2×C5⋊2C8, C22×Dic5, C20.4C8, C22×C5⋊2C8, C2×C20.4C8
►On 160 points
Generators in S160
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(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 94 26 99 11 104 36 109 21 114 6 119 31 84 16 89)(2 83 27 88 12 93 37 98 22 103 7 108 32 113 17 118)(3 112 28 117 13 82 38 87 23 92 8 97 33 102 18 107)(4 101 29 106 14 111 39 116 24 81 9 86 34 91 19 96)(5 90 30 95 15 100 40 105 25 110 10 115 35 120 20 85)(41 150 66 155 51 160 76 125 61 130 46 135 71 140 56 145)(42 139 67 144 52 149 77 154 62 159 47 124 72 129 57 134)(43 128 68 133 53 138 78 143 63 148 48 153 73 158 58 123)(44 157 69 122 54 127 79 132 64 137 49 142 74 147 59 152)(45 146 70 151 55 156 80 121 65 126 50 131 75 136 60 141)
G:=sub<Sym(160)| (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,94,26,99,11,104,36,109,21,114,6,119,31,84,16,89)(2,83,27,88,12,93,37,98,22,103,7,108,32,113,17,118)(3,112,28,117,13,82,38,87,23,92,8,97,33,102,18,107)(4,101,29,106,14,111,39,116,24,81,9,86,34,91,19,96)(5,90,30,95,15,100,40,105,25,110,10,115,35,120,20,85)(41,150,66,155,51,160,76,125,61,130,46,135,71,140,56,145)(42,139,67,144,52,149,77,154,62,159,47,124,72,129,57,134)(43,128,68,133,53,138,78,143,63,148,48,153,73,158,58,123)(44,157,69,122,54,127,79,132,64,137,49,142,74,147,59,152)(45,146,70,151,55,156,80,121,65,126,50,131,75,136,60,141)>;
G:=Group( (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,94,26,99,11,104,36,109,21,114,6,119,31,84,16,89)(2,83,27,88,12,93,37,98,22,103,7,108,32,113,17,118)(3,112,28,117,13,82,38,87,23,92,8,97,33,102,18,107)(4,101,29,106,14,111,39,116,24,81,9,86,34,91,19,96)(5,90,30,95,15,100,40,105,25,110,10,115,35,120,20,85)(41,150,66,155,51,160,76,125,61,130,46,135,71,140,56,145)(42,139,67,144,52,149,77,154,62,159,47,124,72,129,57,134)(43,128,68,133,53,138,78,143,63,148,48,153,73,158,58,123)(44,157,69,122,54,127,79,132,64,137,49,142,74,147,59,152)(45,146,70,151,55,156,80,121,65,126,50,131,75,136,60,141) );
G=PermutationGroup([[(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)], [(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)], [(1,94,26,99,11,104,36,109,21,114,6,119,31,84,16,89),(2,83,27,88,12,93,37,98,22,103,7,108,32,113,17,118),(3,112,28,117,13,82,38,87,23,92,8,97,33,102,18,107),(4,101,29,106,14,111,39,116,24,81,9,86,34,91,19,96),(5,90,30,95,15,100,40,105,25,110,10,115,35,120,20,85),(41,150,66,155,51,160,76,125,61,130,46,135,71,140,56,145),(42,139,67,144,52,149,77,154,62,159,47,124,72,129,57,134),(43,128,68,133,53,138,78,143,63,148,48,153,73,158,58,123),(44,157,69,122,54,127,79,132,64,137,49,142,74,147,59,152),(45,146,70,151,55,156,80,121,65,126,50,131,75,136,60,141)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10N | 16A | ··· | 16P | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D5 | Dic5 | D10 | Dic5 | M5(2) | C5⋊2C8 | C5⋊2C8 | C20.4C8 |
| kernel | C2×C20.4C8 | C2×C5⋊2C16 | C20.4C8 | C22×C40 | C2×C40 | C22×C20 | C2×C20 | C22×C10 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 12 | 4 | 2 | 6 | 6 | 2 | 8 | 12 | 4 | 32 |
Matrix representation of C2×C20.4C8 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 240 | 0 | 0 |
| 0 | 125 | 0 |
| 0 | 0 | 41 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 30 | 0 |
G:=sub<GL(3,GF(241))| [240,0,0,0,1,0,0,0,1],[240,0,0,0,125,0,0,0,41],[1,0,0,0,0,30,0,1,0] >;
C2×C20.4C8 in GAP, Magma, Sage, TeX
C_2\times C_{20}._4C_8 % in TeX
G:=Group("C2xC20.4C8"); // GroupNames label
G:=SmallGroup(320,724);
// by ID
G=gap.SmallGroup(320,724);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,80,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^2=b^40=1,c^4=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^29>;
// generators/relations