metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.20D10, C8⋊C4⋊11D5, C20⋊2Q8⋊3C2, (C2×C20).38D4, (C2×C4).27D20, (C2×C8).160D10, (C4×C20).5C22, C2.9(C8⋊D10), C10.6(C8⋊C22), C4.D20.4C2, C22.99(C2×D20), D20⋊5C4.16C2, C20.225(C4○D4), C4.109(C4○D20), C20.44D4⋊38C2, (C2×C20).735C23, (C2×C40).314C22, C10.9(C4.4D4), C2.8(C8.D10), (C2×D20).10C22, C10.4(C8.C22), C4⋊Dic5.10C22, C2.14(C4.D20), C5⋊1(C42.28C22), (C2×Dic10).10C22, (C5×C8⋊C4)⋊20C2, (C2×C10).118(C2×D4), (C2×C4).679(C22×D5), SmallGroup(320,341)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.20D10
G = < a,b,c,d | a4=b4=1, c10=a2b-1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=bc9 >
Subgroups: 494 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×D4, C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C40 [×2], Dic10 [×4], D20 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.28C22, C4⋊Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×Dic10, C2×Dic10, C2×D20, C20.44D4 [×2], D20⋊5C4 [×2], C5×C8⋊C4, C20⋊2Q8, C4.D20, C42.20D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22, C8.C22, D20 [×2], C22×D5, C42.28C22, C2×D20, C4○D20 [×2], C4.D20, C8⋊D10, C8.D10, C42.20D10
►On 160 points
Generators in S160
(1 109 52 159)(2 90 53 140)(3 111 54 121)(4 92 55 142)(5 113 56 123)(6 94 57 144)(7 115 58 125)(8 96 59 146)(9 117 60 127)(10 98 61 148)(11 119 62 129)(12 100 63 150)(13 81 64 131)(14 102 65 152)(15 83 66 133)(16 104 67 154)(17 85 68 135)(18 106 69 156)(19 87 70 137)(20 108 71 158)(21 89 72 139)(22 110 73 160)(23 91 74 141)(24 112 75 122)(25 93 76 143)(26 114 77 124)(27 95 78 145)(28 116 79 126)(29 97 80 147)(30 118 41 128)(31 99 42 149)(32 120 43 130)(33 101 44 151)(34 82 45 132)(35 103 46 153)(36 84 47 134)(37 105 48 155)(38 86 49 136)(39 107 50 157)(40 88 51 138)
(1 42 21 62)(2 43 22 63)(3 44 23 64)(4 45 24 65)(5 46 25 66)(6 47 26 67)(7 48 27 68)(8 49 28 69)(9 50 29 70)(10 51 30 71)(11 52 31 72)(12 53 32 73)(13 54 33 74)(14 55 34 75)(15 56 35 76)(16 57 36 77)(17 58 37 78)(18 59 38 79)(19 60 39 80)(20 61 40 41)(81 121 101 141)(82 122 102 142)(83 123 103 143)(84 124 104 144)(85 125 105 145)(86 126 106 146)(87 127 107 147)(88 128 108 148)(89 129 109 149)(90 130 110 150)(91 131 111 151)(92 132 112 152)(93 133 113 153)(94 134 114 154)(95 135 115 155)(96 136 116 156)(97 137 117 157)(98 138 118 158)(99 139 119 159)(100 140 120 160)
(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 10 52 61)(2 60 53 9)(3 8 54 59)(4 58 55 7)(5 6 56 57)(11 40 62 51)(12 50 63 39)(13 38 64 49)(14 48 65 37)(15 36 66 47)(16 46 67 35)(17 34 68 45)(18 44 69 33)(19 32 70 43)(20 42 71 31)(21 30 72 41)(22 80 73 29)(23 28 74 79)(24 78 75 27)(25 26 76 77)(81 136 131 86)(82 85 132 135)(83 134 133 84)(87 130 137 120)(88 119 138 129)(89 128 139 118)(90 117 140 127)(91 126 141 116)(92 115 142 125)(93 124 143 114)(94 113 144 123)(95 122 145 112)(96 111 146 121)(97 160 147 110)(98 109 148 159)(99 158 149 108)(100 107 150 157)(101 156 151 106)(102 105 152 155)(103 154 153 104)
G:=sub<Sym(160)| (1,109,52,159)(2,90,53,140)(3,111,54,121)(4,92,55,142)(5,113,56,123)(6,94,57,144)(7,115,58,125)(8,96,59,146)(9,117,60,127)(10,98,61,148)(11,119,62,129)(12,100,63,150)(13,81,64,131)(14,102,65,152)(15,83,66,133)(16,104,67,154)(17,85,68,135)(18,106,69,156)(19,87,70,137)(20,108,71,158)(21,89,72,139)(22,110,73,160)(23,91,74,141)(24,112,75,122)(25,93,76,143)(26,114,77,124)(27,95,78,145)(28,116,79,126)(29,97,80,147)(30,118,41,128)(31,99,42,149)(32,120,43,130)(33,101,44,151)(34,82,45,132)(35,103,46,153)(36,84,47,134)(37,105,48,155)(38,86,49,136)(39,107,50,157)(40,88,51,138), (1,42,21,62)(2,43,22,63)(3,44,23,64)(4,45,24,65)(5,46,25,66)(6,47,26,67)(7,48,27,68)(8,49,28,69)(9,50,29,70)(10,51,30,71)(11,52,31,72)(12,53,32,73)(13,54,33,74)(14,55,34,75)(15,56,35,76)(16,57,36,77)(17,58,37,78)(18,59,38,79)(19,60,39,80)(20,61,40,41)(81,121,101,141)(82,122,102,142)(83,123,103,143)(84,124,104,144)(85,125,105,145)(86,126,106,146)(87,127,107,147)(88,128,108,148)(89,129,109,149)(90,130,110,150)(91,131,111,151)(92,132,112,152)(93,133,113,153)(94,134,114,154)(95,135,115,155)(96,136,116,156)(97,137,117,157)(98,138,118,158)(99,139,119,159)(100,140,120,160), (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), (1,10,52,61)(2,60,53,9)(3,8,54,59)(4,58,55,7)(5,6,56,57)(11,40,62,51)(12,50,63,39)(13,38,64,49)(14,48,65,37)(15,36,66,47)(16,46,67,35)(17,34,68,45)(18,44,69,33)(19,32,70,43)(20,42,71,31)(21,30,72,41)(22,80,73,29)(23,28,74,79)(24,78,75,27)(25,26,76,77)(81,136,131,86)(82,85,132,135)(83,134,133,84)(87,130,137,120)(88,119,138,129)(89,128,139,118)(90,117,140,127)(91,126,141,116)(92,115,142,125)(93,124,143,114)(94,113,144,123)(95,122,145,112)(96,111,146,121)(97,160,147,110)(98,109,148,159)(99,158,149,108)(100,107,150,157)(101,156,151,106)(102,105,152,155)(103,154,153,104)>;
G:=Group( (1,109,52,159)(2,90,53,140)(3,111,54,121)(4,92,55,142)(5,113,56,123)(6,94,57,144)(7,115,58,125)(8,96,59,146)(9,117,60,127)(10,98,61,148)(11,119,62,129)(12,100,63,150)(13,81,64,131)(14,102,65,152)(15,83,66,133)(16,104,67,154)(17,85,68,135)(18,106,69,156)(19,87,70,137)(20,108,71,158)(21,89,72,139)(22,110,73,160)(23,91,74,141)(24,112,75,122)(25,93,76,143)(26,114,77,124)(27,95,78,145)(28,116,79,126)(29,97,80,147)(30,118,41,128)(31,99,42,149)(32,120,43,130)(33,101,44,151)(34,82,45,132)(35,103,46,153)(36,84,47,134)(37,105,48,155)(38,86,49,136)(39,107,50,157)(40,88,51,138), (1,42,21,62)(2,43,22,63)(3,44,23,64)(4,45,24,65)(5,46,25,66)(6,47,26,67)(7,48,27,68)(8,49,28,69)(9,50,29,70)(10,51,30,71)(11,52,31,72)(12,53,32,73)(13,54,33,74)(14,55,34,75)(15,56,35,76)(16,57,36,77)(17,58,37,78)(18,59,38,79)(19,60,39,80)(20,61,40,41)(81,121,101,141)(82,122,102,142)(83,123,103,143)(84,124,104,144)(85,125,105,145)(86,126,106,146)(87,127,107,147)(88,128,108,148)(89,129,109,149)(90,130,110,150)(91,131,111,151)(92,132,112,152)(93,133,113,153)(94,134,114,154)(95,135,115,155)(96,136,116,156)(97,137,117,157)(98,138,118,158)(99,139,119,159)(100,140,120,160), (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), (1,10,52,61)(2,60,53,9)(3,8,54,59)(4,58,55,7)(5,6,56,57)(11,40,62,51)(12,50,63,39)(13,38,64,49)(14,48,65,37)(15,36,66,47)(16,46,67,35)(17,34,68,45)(18,44,69,33)(19,32,70,43)(20,42,71,31)(21,30,72,41)(22,80,73,29)(23,28,74,79)(24,78,75,27)(25,26,76,77)(81,136,131,86)(82,85,132,135)(83,134,133,84)(87,130,137,120)(88,119,138,129)(89,128,139,118)(90,117,140,127)(91,126,141,116)(92,115,142,125)(93,124,143,114)(94,113,144,123)(95,122,145,112)(96,111,146,121)(97,160,147,110)(98,109,148,159)(99,158,149,108)(100,107,150,157)(101,156,151,106)(102,105,152,155)(103,154,153,104) );
G=PermutationGroup([(1,109,52,159),(2,90,53,140),(3,111,54,121),(4,92,55,142),(5,113,56,123),(6,94,57,144),(7,115,58,125),(8,96,59,146),(9,117,60,127),(10,98,61,148),(11,119,62,129),(12,100,63,150),(13,81,64,131),(14,102,65,152),(15,83,66,133),(16,104,67,154),(17,85,68,135),(18,106,69,156),(19,87,70,137),(20,108,71,158),(21,89,72,139),(22,110,73,160),(23,91,74,141),(24,112,75,122),(25,93,76,143),(26,114,77,124),(27,95,78,145),(28,116,79,126),(29,97,80,147),(30,118,41,128),(31,99,42,149),(32,120,43,130),(33,101,44,151),(34,82,45,132),(35,103,46,153),(36,84,47,134),(37,105,48,155),(38,86,49,136),(39,107,50,157),(40,88,51,138)], [(1,42,21,62),(2,43,22,63),(3,44,23,64),(4,45,24,65),(5,46,25,66),(6,47,26,67),(7,48,27,68),(8,49,28,69),(9,50,29,70),(10,51,30,71),(11,52,31,72),(12,53,32,73),(13,54,33,74),(14,55,34,75),(15,56,35,76),(16,57,36,77),(17,58,37,78),(18,59,38,79),(19,60,39,80),(20,61,40,41),(81,121,101,141),(82,122,102,142),(83,123,103,143),(84,124,104,144),(85,125,105,145),(86,126,106,146),(87,127,107,147),(88,128,108,148),(89,129,109,149),(90,130,110,150),(91,131,111,151),(92,132,112,152),(93,133,113,153),(94,134,114,154),(95,135,115,155),(96,136,116,156),(97,137,117,157),(98,138,118,158),(99,139,119,159),(100,140,120,160)], [(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)], [(1,10,52,61),(2,60,53,9),(3,8,54,59),(4,58,55,7),(5,6,56,57),(11,40,62,51),(12,50,63,39),(13,38,64,49),(14,48,65,37),(15,36,66,47),(16,46,67,35),(17,34,68,45),(18,44,69,33),(19,32,70,43),(20,42,71,31),(21,30,72,41),(22,80,73,29),(23,28,74,79),(24,78,75,27),(25,26,76,77),(81,136,131,86),(82,85,132,135),(83,134,133,84),(87,130,137,120),(88,119,138,129),(89,128,139,118),(90,117,140,127),(91,126,141,116),(92,115,142,125),(93,124,143,114),(94,113,144,123),(95,122,145,112),(96,111,146,121),(97,160,147,110),(98,109,148,159),(99,158,149,108),(100,107,150,157),(101,156,151,106),(102,105,152,155),(103,154,153,104)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 4 | 4 | 40 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C4○D20 | C8⋊C22 | C8.C22 | C8⋊D10 | C8.D10 |
| kernel | C42.20D10 | C20.44D4 | D20⋊5C4 | C5×C8⋊C4 | C20⋊2Q8 | C4.D20 | C2×C20 | C8⋊C4 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C42.20D10 ►in GL6(𝔽41)
| 40 | 21 | 0 | 0 | 0 | 0 |
| 37 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 13 | 18 | 19 |
| 0 | 0 | 28 | 23 | 15 | 40 |
| 0 | 0 | 22 | 2 | 5 | 28 |
| 0 | 0 | 8 | 20 | 22 | 27 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 0 | 19 | 30 |
| 9 | 16 | 0 | 0 | 0 | 0 |
| 36 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 18 |
| 0 | 0 | 28 | 9 | 20 | 23 |
| 0 | 0 | 21 | 14 | 19 | 28 |
| 0 | 0 | 3 | 7 | 22 | 0 |
| 32 | 25 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 20 | 14 | 7 |
| 0 | 0 | 23 | 20 | 1 | 35 |
| 0 | 0 | 15 | 38 | 3 | 23 |
| 0 | 0 | 23 | 6 | 37 | 38 |
G:=sub<GL(6,GF(41))| [40,37,0,0,0,0,21,1,0,0,0,0,0,0,27,28,22,8,0,0,13,23,2,20,0,0,18,15,5,22,0,0,19,40,28,27],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,11,19,0,0,0,0,13,30],[9,36,0,0,0,0,16,32,0,0,0,0,0,0,13,28,21,3,0,0,13,9,14,7,0,0,0,20,19,22,0,0,18,23,28,0],[32,0,0,0,0,0,25,9,0,0,0,0,0,0,21,23,15,23,0,0,20,20,38,6,0,0,14,1,3,37,0,0,7,35,23,38] >;
C42.20D10 in GAP, Magma, Sage, TeX
C_4^2._{20}D_{10} % in TeX
G:=Group("C4^2.20D10"); // GroupNames label
G:=SmallGroup(320,341);
// by ID
G=gap.SmallGroup(320,341);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^-1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations