direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40.9C4, C10⋊4M5(2), C40.69C23, C5⋊6(C2×M5(2)), C20.79(C2×C8), (C2×C20).17C8, (C2×C40).51C4, C40.120(C2×C4), (C2×C8).326D10, C8.26(C2×Dic5), (C2×C8).15Dic5, (C22×C8).14D5, C8.63(C22×D5), C5⋊2C16⋊12C22, (C22×C40).22C2, (C22×C10).11C8, (C22×C20).55C4, C10.51(C22×C8), C23.3(C5⋊2C8), C20.235(C22×C4), (C2×C40).412C22, (C22×C4).17Dic5, C4.30(C22×Dic5), C4.9(C2×C5⋊2C8), (C2×C5⋊2C16)⋊12C2, (C2×C10).65(C2×C8), (C2×C4).6(C5⋊2C8), C22.6(C2×C5⋊2C8), C2.7(C22×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×C40.9C4
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 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C5, C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×4], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C16 [×2], M5(2) [×4], C22×C8, C40 [×2], C40 [×2], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×M5(2), C5⋊2C16 [×4], C2×C40 [×2], C2×C40 [×4], C22×C20, C2×C5⋊2C16 [×2], C40.9C4 [×4], C22×C40, C2×C40.9C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], C22×C4, Dic5 [×4], D10 [×3], M5(2) [×2], C22×C8, C5⋊2C8 [×4], C2×Dic5 [×6], C22×D5, C2×M5(2), C2×C5⋊2C8 [×6], C22×Dic5, C40.9C4 [×2], C22×C5⋊2C8, C2×C40.9C4
►On 160 points
Generators in S160
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(81 135)(82 136)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)
(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)
(1 118 26 83 11 88 36 93 21 98 6 103 31 108 16 113)(2 107 27 112 12 117 37 82 22 87 7 92 32 97 17 102)(3 96 28 101 13 106 38 111 23 116 8 81 33 86 18 91)(4 85 29 90 14 95 39 100 24 105 9 110 34 115 19 120)(5 114 30 119 15 84 40 89 25 94 10 99 35 104 20 109)(41 151 66 156 51 121 76 126 61 131 46 136 71 141 56 146)(42 140 67 145 52 150 77 155 62 160 47 125 72 130 57 135)(43 129 68 134 53 139 78 144 63 149 48 154 73 159 58 124)(44 158 69 123 54 128 79 133 64 138 49 143 74 148 59 153)(45 147 70 152 55 157 80 122 65 127 50 132 75 137 60 142)
G:=sub<Sym(160)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(81,135)(82,136)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134), (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), (1,118,26,83,11,88,36,93,21,98,6,103,31,108,16,113)(2,107,27,112,12,117,37,82,22,87,7,92,32,97,17,102)(3,96,28,101,13,106,38,111,23,116,8,81,33,86,18,91)(4,85,29,90,14,95,39,100,24,105,9,110,34,115,19,120)(5,114,30,119,15,84,40,89,25,94,10,99,35,104,20,109)(41,151,66,156,51,121,76,126,61,131,46,136,71,141,56,146)(42,140,67,145,52,150,77,155,62,160,47,125,72,130,57,135)(43,129,68,134,53,139,78,144,63,149,48,154,73,159,58,124)(44,158,69,123,54,128,79,133,64,138,49,143,74,148,59,153)(45,147,70,152,55,157,80,122,65,127,50,132,75,137,60,142)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(81,135)(82,136)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134), (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), (1,118,26,83,11,88,36,93,21,98,6,103,31,108,16,113)(2,107,27,112,12,117,37,82,22,87,7,92,32,97,17,102)(3,96,28,101,13,106,38,111,23,116,8,81,33,86,18,91)(4,85,29,90,14,95,39,100,24,105,9,110,34,115,19,120)(5,114,30,119,15,84,40,89,25,94,10,99,35,104,20,109)(41,151,66,156,51,121,76,126,61,131,46,136,71,141,56,146)(42,140,67,145,52,150,77,155,62,160,47,125,72,130,57,135)(43,129,68,134,53,139,78,144,63,149,48,154,73,159,58,124)(44,158,69,123,54,128,79,133,64,138,49,143,74,148,59,153)(45,147,70,152,55,157,80,122,65,127,50,132,75,137,60,142) );
G=PermutationGroup([(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,41),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49),(81,135),(82,136),(83,137),(84,138),(85,139),(86,140),(87,141),(88,142),(89,143),(90,144),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134)], [(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)], [(1,118,26,83,11,88,36,93,21,98,6,103,31,108,16,113),(2,107,27,112,12,117,37,82,22,87,7,92,32,97,17,102),(3,96,28,101,13,106,38,111,23,116,8,81,33,86,18,91),(4,85,29,90,14,95,39,100,24,105,9,110,34,115,19,120),(5,114,30,119,15,84,40,89,25,94,10,99,35,104,20,109),(41,151,66,156,51,121,76,126,61,131,46,136,71,141,56,146),(42,140,67,145,52,150,77,155,62,160,47,125,72,130,57,135),(43,129,68,134,53,139,78,144,63,149,48,154,73,159,58,124),(44,158,69,123,54,128,79,133,64,138,49,143,74,148,59,153),(45,147,70,152,55,157,80,122,65,127,50,132,75,137,60,142)])
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 | C40.9C4 |
| kernel | C2×C40.9C4 | C2×C5⋊2C16 | C40.9C4 | 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×C40.9C4 ►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×C40.9C4 in GAP, Magma, Sage, TeX
C_2\times C_{40}._9C_4 % in TeX
G:=Group("C2xC40.9C4"); // 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