metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.185D10, C8⋊C4⋊9D5, (C2×C8).158D10, C10.47(C8○D4), C20.8Q8⋊37C2, D10⋊1C8.16C2, D10⋊C4.18C4, C4.131(C4○D20), C20.247(C4○D4), (C4×C20).230C22, (C2×C20).815C23, (C2×C40).312C22, C10.D4.18C4, C42⋊D5.12C2, C2.8(D20.2C4), C2.13(C42⋊D5), C5⋊4(C42.7C22), C10.29(C42⋊C2), (C4×Dic5).201C22, (C4×C5⋊2C8)⋊22C2, (C5×C8⋊C4)⋊18C2, (C2×C4).62(C4×D5), C22.100(C2×C4×D5), (C2×C20).322(C2×C4), (C2×C4×D5).229C22, (C2×Dic5).17(C2×C4), (C22×D5).16(C2×C4), (C2×C4).757(C22×D5), (C2×C10).171(C22×C4), (C2×C5⋊2C8).305C22, SmallGroup(320,336)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.185D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=a2c9 >
Subgroups: 302 in 96 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C42.7C22, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C4×C5⋊2C8, C20.8Q8, D10⋊1C8, C5×C8⋊C4, C42⋊D5, C42.185D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C8○D4, C4×D5, C22×D5, C42.7C22, C2×C4×D5, C4○D20, C42⋊D5, D20.2C4, C42.185D10
►On 160 points
Generators in S160
(1 117 74 131)(2 98 75 152)(3 119 76 133)(4 100 77 154)(5 81 78 135)(6 102 79 156)(7 83 80 137)(8 104 41 158)(9 85 42 139)(10 106 43 160)(11 87 44 141)(12 108 45 122)(13 89 46 143)(14 110 47 124)(15 91 48 145)(16 112 49 126)(17 93 50 147)(18 114 51 128)(19 95 52 149)(20 116 53 130)(21 97 54 151)(22 118 55 132)(23 99 56 153)(24 120 57 134)(25 101 58 155)(26 82 59 136)(27 103 60 157)(28 84 61 138)(29 105 62 159)(30 86 63 140)(31 107 64 121)(32 88 65 142)(33 109 66 123)(34 90 67 144)(35 111 68 125)(36 92 69 146)(37 113 70 127)(38 94 71 148)(39 115 72 129)(40 96 73 150)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 44 53 21 30 64 73)(2 52 45 29 22 72 65 9)(3 28 46 71 23 8 66 51)(4 70 47 7 24 50 67 27)(5 6 48 49 25 26 68 69)(11 20 54 63 31 40 74 43)(12 62 55 39 32 42 75 19)(13 38 56 41 33 18 76 61)(14 80 57 17 34 60 77 37)(15 16 58 59 35 36 78 79)(81 102 145 126 101 82 125 146)(83 120 147 144 103 100 127 124)(84 143 148 99 104 123 128 119)(85 98 149 122 105 118 129 142)(86 121 150 117 106 141 130 97)(87 116 151 140 107 96 131 160)(88 139 152 95 108 159 132 115)(89 94 153 158 109 114 133 138)(90 157 154 113 110 137 134 93)(91 112 155 136 111 92 135 156)
G:=sub<Sym(160)| (1,117,74,131)(2,98,75,152)(3,119,76,133)(4,100,77,154)(5,81,78,135)(6,102,79,156)(7,83,80,137)(8,104,41,158)(9,85,42,139)(10,106,43,160)(11,87,44,141)(12,108,45,122)(13,89,46,143)(14,110,47,124)(15,91,48,145)(16,112,49,126)(17,93,50,147)(18,114,51,128)(19,95,52,149)(20,116,53,130)(21,97,54,151)(22,118,55,132)(23,99,56,153)(24,120,57,134)(25,101,58,155)(26,82,59,136)(27,103,60,157)(28,84,61,138)(29,105,62,159)(30,86,63,140)(31,107,64,121)(32,88,65,142)(33,109,66,123)(34,90,67,144)(35,111,68,125)(36,92,69,146)(37,113,70,127)(38,94,71,148)(39,115,72,129)(40,96,73,150), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,44,53,21,30,64,73)(2,52,45,29,22,72,65,9)(3,28,46,71,23,8,66,51)(4,70,47,7,24,50,67,27)(5,6,48,49,25,26,68,69)(11,20,54,63,31,40,74,43)(12,62,55,39,32,42,75,19)(13,38,56,41,33,18,76,61)(14,80,57,17,34,60,77,37)(15,16,58,59,35,36,78,79)(81,102,145,126,101,82,125,146)(83,120,147,144,103,100,127,124)(84,143,148,99,104,123,128,119)(85,98,149,122,105,118,129,142)(86,121,150,117,106,141,130,97)(87,116,151,140,107,96,131,160)(88,139,152,95,108,159,132,115)(89,94,153,158,109,114,133,138)(90,157,154,113,110,137,134,93)(91,112,155,136,111,92,135,156)>;
G:=Group( (1,117,74,131)(2,98,75,152)(3,119,76,133)(4,100,77,154)(5,81,78,135)(6,102,79,156)(7,83,80,137)(8,104,41,158)(9,85,42,139)(10,106,43,160)(11,87,44,141)(12,108,45,122)(13,89,46,143)(14,110,47,124)(15,91,48,145)(16,112,49,126)(17,93,50,147)(18,114,51,128)(19,95,52,149)(20,116,53,130)(21,97,54,151)(22,118,55,132)(23,99,56,153)(24,120,57,134)(25,101,58,155)(26,82,59,136)(27,103,60,157)(28,84,61,138)(29,105,62,159)(30,86,63,140)(31,107,64,121)(32,88,65,142)(33,109,66,123)(34,90,67,144)(35,111,68,125)(36,92,69,146)(37,113,70,127)(38,94,71,148)(39,115,72,129)(40,96,73,150), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,44,53,21,30,64,73)(2,52,45,29,22,72,65,9)(3,28,46,71,23,8,66,51)(4,70,47,7,24,50,67,27)(5,6,48,49,25,26,68,69)(11,20,54,63,31,40,74,43)(12,62,55,39,32,42,75,19)(13,38,56,41,33,18,76,61)(14,80,57,17,34,60,77,37)(15,16,58,59,35,36,78,79)(81,102,145,126,101,82,125,146)(83,120,147,144,103,100,127,124)(84,143,148,99,104,123,128,119)(85,98,149,122,105,118,129,142)(86,121,150,117,106,141,130,97)(87,116,151,140,107,96,131,160)(88,139,152,95,108,159,132,115)(89,94,153,158,109,114,133,138)(90,157,154,113,110,137,134,93)(91,112,155,136,111,92,135,156) );
G=PermutationGroup([[(1,117,74,131),(2,98,75,152),(3,119,76,133),(4,100,77,154),(5,81,78,135),(6,102,79,156),(7,83,80,137),(8,104,41,158),(9,85,42,139),(10,106,43,160),(11,87,44,141),(12,108,45,122),(13,89,46,143),(14,110,47,124),(15,91,48,145),(16,112,49,126),(17,93,50,147),(18,114,51,128),(19,95,52,149),(20,116,53,130),(21,97,54,151),(22,118,55,132),(23,99,56,153),(24,120,57,134),(25,101,58,155),(26,82,59,136),(27,103,60,157),(28,84,61,138),(29,105,62,159),(30,86,63,140),(31,107,64,121),(32,88,65,142),(33,109,66,123),(34,90,67,144),(35,111,68,125),(36,92,69,146),(37,113,70,127),(38,94,71,148),(39,115,72,129),(40,96,73,150)], [(1,11,21,31),(2,12,22,32),(3,13,23,33),(4,14,24,34),(5,15,25,35),(6,16,26,36),(7,17,27,37),(8,18,28,38),(9,19,29,39),(10,20,30,40),(41,51,61,71),(42,52,62,72),(43,53,63,73),(44,54,64,74),(45,55,65,75),(46,56,66,76),(47,57,67,77),(48,58,68,78),(49,59,69,79),(50,60,70,80),(81,91,101,111),(82,92,102,112),(83,93,103,113),(84,94,104,114),(85,95,105,115),(86,96,106,116),(87,97,107,117),(88,98,108,118),(89,99,109,119),(90,100,110,120),(121,131,141,151),(122,132,142,152),(123,133,143,153),(124,134,144,154),(125,135,145,155),(126,136,146,156),(127,137,147,157),(128,138,148,158),(129,139,149,159),(130,140,150,160)], [(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)], [(1,10,44,53,21,30,64,73),(2,52,45,29,22,72,65,9),(3,28,46,71,23,8,66,51),(4,70,47,7,24,50,67,27),(5,6,48,49,25,26,68,69),(11,20,54,63,31,40,74,43),(12,62,55,39,32,42,75,19),(13,38,56,41,33,18,76,61),(14,80,57,17,34,60,77,37),(15,16,58,59,35,36,78,79),(81,102,145,126,101,82,125,146),(83,120,147,144,103,100,127,124),(84,143,148,99,104,123,128,119),(85,98,149,122,105,118,129,142),(86,121,150,117,106,141,130,97),(87,116,151,140,107,96,131,160),(88,139,152,95,108,159,132,115),(89,94,153,158,109,114,133,138),(90,157,154,113,110,137,134,93),(91,112,155,136,111,92,135,156)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D5 | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C4○D20 | D20.2C4 |
| kernel | C42.185D10 | C4×C5⋊2C8 | C20.8Q8 | D10⋊1C8 | C5×C8⋊C4 | C42⋊D5 | C10.D4 | D10⋊C4 | C8⋊C4 | C20 | C42 | C2×C8 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 2 | 4 | 8 | 8 | 16 | 8 |
Matrix representation of C42.185D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 31 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 35 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 30 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 18 |
| 0 | 0 | 0 | 0 | 1 | 27 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 23 |
| 0 | 0 | 0 | 0 | 0 | 14 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,31,1,0,0,0,0,0,0,9,0,0,0,0,0,35,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,14,30,0,0,0,0,0,27,0,0,0,0,0,0,14,1,0,0,0,0,18,27],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,27,0,0,0,0,0,23,14] >;
C42.185D10 in GAP, Magma, Sage, TeX
C_4^2._{185}D_{10} % in TeX
G:=Group("C4^2.185D10"); // GroupNames label
G:=SmallGroup(320,336);
// by ID
G=gap.SmallGroup(320,336);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,422,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=a^2*b,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations