direct product, abelian, monomial, 2-elementary
Aliases: C2×C4×C20, SmallGroup(160,175)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C4×C20 |
| C1 — C2×C4×C20 |
| C1 — C2×C4×C20 |
Generators and relations for C2×C4×C20
G = < a,b,c | a2=b4=c20=1, ab=ba, ac=ca, bc=cb >
Subgroups: 108, all normal (8 characteristic)
C1, C2, C4, C22, C22, C5, C2×C4, C23, C10, C42, C22×C4, C20, C2×C10, C2×C10, C2×C42, C2×C20, C22×C10, C4×C20, C22×C20, C2×C4×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, C22×C4, C20, C2×C10, C2×C42, C2×C20, C22×C10, C4×C20, C22×C20, C2×C4×C20
►Regular action on 160 points
Generators in S160
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 101)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 141)(97 142)(98 143)(99 144)(100 145)
(1 142 121 102)(2 143 122 103)(3 144 123 104)(4 145 124 105)(5 146 125 106)(6 147 126 107)(7 148 127 108)(8 149 128 109)(9 150 129 110)(10 151 130 111)(11 152 131 112)(12 153 132 113)(13 154 133 114)(14 155 134 115)(15 156 135 116)(16 157 136 117)(17 158 137 118)(18 159 138 119)(19 160 139 120)(20 141 140 101)(21 49 97 68)(22 50 98 69)(23 51 99 70)(24 52 100 71)(25 53 81 72)(26 54 82 73)(27 55 83 74)(28 56 84 75)(29 57 85 76)(30 58 86 77)(31 59 87 78)(32 60 88 79)(33 41 89 80)(34 42 90 61)(35 43 91 62)(36 44 92 63)(37 45 93 64)(38 46 94 65)(39 47 95 66)(40 48 96 67)
(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)
G:=sub<Sym(160)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,101)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,141)(97,142)(98,143)(99,144)(100,145), (1,142,121,102)(2,143,122,103)(3,144,123,104)(4,145,124,105)(5,146,125,106)(6,147,126,107)(7,148,127,108)(8,149,128,109)(9,150,129,110)(10,151,130,111)(11,152,131,112)(12,153,132,113)(13,154,133,114)(14,155,134,115)(15,156,135,116)(16,157,136,117)(17,158,137,118)(18,159,138,119)(19,160,139,120)(20,141,140,101)(21,49,97,68)(22,50,98,69)(23,51,99,70)(24,52,100,71)(25,53,81,72)(26,54,82,73)(27,55,83,74)(28,56,84,75)(29,57,85,76)(30,58,86,77)(31,59,87,78)(32,60,88,79)(33,41,89,80)(34,42,90,61)(35,43,91,62)(36,44,92,63)(37,45,93,64)(38,46,94,65)(39,47,95,66)(40,48,96,67), (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)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,101)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,141)(97,142)(98,143)(99,144)(100,145), (1,142,121,102)(2,143,122,103)(3,144,123,104)(4,145,124,105)(5,146,125,106)(6,147,126,107)(7,148,127,108)(8,149,128,109)(9,150,129,110)(10,151,130,111)(11,152,131,112)(12,153,132,113)(13,154,133,114)(14,155,134,115)(15,156,135,116)(16,157,136,117)(17,158,137,118)(18,159,138,119)(19,160,139,120)(20,141,140,101)(21,49,97,68)(22,50,98,69)(23,51,99,70)(24,52,100,71)(25,53,81,72)(26,54,82,73)(27,55,83,74)(28,56,84,75)(29,57,85,76)(30,58,86,77)(31,59,87,78)(32,60,88,79)(33,41,89,80)(34,42,90,61)(35,43,91,62)(36,44,92,63)(37,45,93,64)(38,46,94,65)(39,47,95,66)(40,48,96,67), (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) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,101),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(75,128),(76,129),(77,130),(78,131),(79,132),(80,133),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,141),(97,142),(98,143),(99,144),(100,145)], [(1,142,121,102),(2,143,122,103),(3,144,123,104),(4,145,124,105),(5,146,125,106),(6,147,126,107),(7,148,127,108),(8,149,128,109),(9,150,129,110),(10,151,130,111),(11,152,131,112),(12,153,132,113),(13,154,133,114),(14,155,134,115),(15,156,135,116),(16,157,136,117),(17,158,137,118),(18,159,138,119),(19,160,139,120),(20,141,140,101),(21,49,97,68),(22,50,98,69),(23,51,99,70),(24,52,100,71),(25,53,81,72),(26,54,82,73),(27,55,83,74),(28,56,84,75),(29,57,85,76),(30,58,86,77),(31,59,87,78),(32,60,88,79),(33,41,89,80),(34,42,90,61),(35,43,91,62),(36,44,92,63),(37,45,93,64),(38,46,94,65),(39,47,95,66),(40,48,96,67)], [(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)]])
C2×C4×C20 is a maximal subgroup of
C42⋊6Dic5 (C2×C20)⋊8C8 C20⋊13M4(2) C42.6Dic5 C42.7Dic5 C20⋊7(C4⋊C4) (C2×C20)⋊10Q8 C42⋊4Dic5 C10.92(C4×D4) C42⋊8Dic5 C42⋊9Dic5 C42⋊5Dic5 (C2×C4)⋊6D20 (C2×C42)⋊D5 C42.274D10 C42.276D10 C42.277D10
160 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20CR |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 |
| kernel | C2×C4×C20 | C4×C20 | C22×C20 | C2×C20 | C2×C42 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 3 | 24 | 4 | 16 | 12 | 96 |
Matrix representation of C2×C4×C20 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 9 |
| 36 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 40 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[1,0,0,0,40,0,0,0,9],[36,0,0,0,8,0,0,0,40] >;
C2×C4×C20 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{20} % in TeX
G:=Group("C2xC4xC20"); // GroupNames label
G:=SmallGroup(160,175);
// by ID
G=gap.SmallGroup(160,175);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,487]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^20=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations