metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.182D10, D10.5M4(2), Dic5.8M4(2), C8⋊C4⋊7D5, (C2×C8).156D10, C2.9(D5×M4(2)), C5⋊5(C42.6C4), C20.8Q8⋊36C2, (C4×Dic5).18C4, (D5×C42).14C2, D10⋊1C8.15C2, C20.245(C4○D4), C4.129(C4○D20), (C2×C40).311C22, (C4×C20).227C22, (C2×C20).811C23, C42.D5⋊18C2, C10.50(C2×M4(2)), C2.12(C42⋊D5), C10.28(C42⋊C2), (C4×Dic5).298C22, (C2×C4×D5).19C4, (C5×C8⋊C4)⋊17C2, C22.98(C2×C4×D5), (C2×C4).127(C4×D5), (C2×C20).319(C2×C4), (C2×C4×D5).339C22, (C22×D5).96(C2×C4), (C2×C4).753(C22×D5), (C2×C10).167(C22×C4), (C2×C5⋊2C8).193C22, (C2×Dic5).136(C2×C4), SmallGroup(320,332)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.182D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2c9 >
Subgroups: 350 in 110 conjugacy classes, 51 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.6C4, C2×C5⋊2C8, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C42.D5, C20.8Q8, D10⋊1C8, C5×C8⋊C4, D5×C42, C42.182D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, C4○D4, D10, C42⋊C2, C2×M4(2), C4×D5, C22×D5, C42.6C4, C2×C4×D5, C4○D20, C42⋊D5, D5×M4(2), C42.182D10
►On 160 points
Generators in S160
(1 66 120 145)(2 47 81 126)(3 68 82 147)(4 49 83 128)(5 70 84 149)(6 51 85 130)(7 72 86 151)(8 53 87 132)(9 74 88 153)(10 55 89 134)(11 76 90 155)(12 57 91 136)(13 78 92 157)(14 59 93 138)(15 80 94 159)(16 61 95 140)(17 42 96 121)(18 63 97 142)(19 44 98 123)(20 65 99 144)(21 46 100 125)(22 67 101 146)(23 48 102 127)(24 69 103 148)(25 50 104 129)(26 71 105 150)(27 52 106 131)(28 73 107 152)(29 54 108 133)(30 75 109 154)(31 56 110 135)(32 77 111 156)(33 58 112 137)(34 79 113 158)(35 60 114 139)(36 41 115 160)(37 62 116 141)(38 43 117 122)(39 64 118 143)(40 45 119 124)
(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 90 99 21 30 110 119)(2 98 91 29 22 118 111 9)(3 28 92 117 23 8 112 97)(4 116 93 7 24 96 113 27)(5 6 94 95 25 26 114 115)(11 20 100 109 31 40 120 89)(12 108 101 39 32 88 81 19)(13 38 102 87 33 18 82 107)(14 86 103 17 34 106 83 37)(15 16 104 105 35 36 84 85)(41 129 130 60 61 149 150 80)(42 59 131 148 62 79 151 128)(43 147 132 78 63 127 152 58)(44 77 133 126 64 57 153 146)(45 125 134 56 65 145 154 76)(46 55 135 144 66 75 155 124)(47 143 136 74 67 123 156 54)(48 73 137 122 68 53 157 142)(49 121 138 52 69 141 158 72)(50 51 139 140 70 71 159 160)
G:=sub<Sym(160)| (1,66,120,145)(2,47,81,126)(3,68,82,147)(4,49,83,128)(5,70,84,149)(6,51,85,130)(7,72,86,151)(8,53,87,132)(9,74,88,153)(10,55,89,134)(11,76,90,155)(12,57,91,136)(13,78,92,157)(14,59,93,138)(15,80,94,159)(16,61,95,140)(17,42,96,121)(18,63,97,142)(19,44,98,123)(20,65,99,144)(21,46,100,125)(22,67,101,146)(23,48,102,127)(24,69,103,148)(25,50,104,129)(26,71,105,150)(27,52,106,131)(28,73,107,152)(29,54,108,133)(30,75,109,154)(31,56,110,135)(32,77,111,156)(33,58,112,137)(34,79,113,158)(35,60,114,139)(36,41,115,160)(37,62,116,141)(38,43,117,122)(39,64,118,143)(40,45,119,124), (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), (1,10,90,99,21,30,110,119)(2,98,91,29,22,118,111,9)(3,28,92,117,23,8,112,97)(4,116,93,7,24,96,113,27)(5,6,94,95,25,26,114,115)(11,20,100,109,31,40,120,89)(12,108,101,39,32,88,81,19)(13,38,102,87,33,18,82,107)(14,86,103,17,34,106,83,37)(15,16,104,105,35,36,84,85)(41,129,130,60,61,149,150,80)(42,59,131,148,62,79,151,128)(43,147,132,78,63,127,152,58)(44,77,133,126,64,57,153,146)(45,125,134,56,65,145,154,76)(46,55,135,144,66,75,155,124)(47,143,136,74,67,123,156,54)(48,73,137,122,68,53,157,142)(49,121,138,52,69,141,158,72)(50,51,139,140,70,71,159,160)>;
G:=Group( (1,66,120,145)(2,47,81,126)(3,68,82,147)(4,49,83,128)(5,70,84,149)(6,51,85,130)(7,72,86,151)(8,53,87,132)(9,74,88,153)(10,55,89,134)(11,76,90,155)(12,57,91,136)(13,78,92,157)(14,59,93,138)(15,80,94,159)(16,61,95,140)(17,42,96,121)(18,63,97,142)(19,44,98,123)(20,65,99,144)(21,46,100,125)(22,67,101,146)(23,48,102,127)(24,69,103,148)(25,50,104,129)(26,71,105,150)(27,52,106,131)(28,73,107,152)(29,54,108,133)(30,75,109,154)(31,56,110,135)(32,77,111,156)(33,58,112,137)(34,79,113,158)(35,60,114,139)(36,41,115,160)(37,62,116,141)(38,43,117,122)(39,64,118,143)(40,45,119,124), (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), (1,10,90,99,21,30,110,119)(2,98,91,29,22,118,111,9)(3,28,92,117,23,8,112,97)(4,116,93,7,24,96,113,27)(5,6,94,95,25,26,114,115)(11,20,100,109,31,40,120,89)(12,108,101,39,32,88,81,19)(13,38,102,87,33,18,82,107)(14,86,103,17,34,106,83,37)(15,16,104,105,35,36,84,85)(41,129,130,60,61,149,150,80)(42,59,131,148,62,79,151,128)(43,147,132,78,63,127,152,58)(44,77,133,126,64,57,153,146)(45,125,134,56,65,145,154,76)(46,55,135,144,66,75,155,124)(47,143,136,74,67,123,156,54)(48,73,137,122,68,53,157,142)(49,121,138,52,69,141,158,72)(50,51,139,140,70,71,159,160) );
G=PermutationGroup([[(1,66,120,145),(2,47,81,126),(3,68,82,147),(4,49,83,128),(5,70,84,149),(6,51,85,130),(7,72,86,151),(8,53,87,132),(9,74,88,153),(10,55,89,134),(11,76,90,155),(12,57,91,136),(13,78,92,157),(14,59,93,138),(15,80,94,159),(16,61,95,140),(17,42,96,121),(18,63,97,142),(19,44,98,123),(20,65,99,144),(21,46,100,125),(22,67,101,146),(23,48,102,127),(24,69,103,148),(25,50,104,129),(26,71,105,150),(27,52,106,131),(28,73,107,152),(29,54,108,133),(30,75,109,154),(31,56,110,135),(32,77,111,156),(33,58,112,137),(34,79,113,158),(35,60,114,139),(36,41,115,160),(37,62,116,141),(38,43,117,122),(39,64,118,143),(40,45,119,124)], [(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)], [(1,10,90,99,21,30,110,119),(2,98,91,29,22,118,111,9),(3,28,92,117,23,8,112,97),(4,116,93,7,24,96,113,27),(5,6,94,95,25,26,114,115),(11,20,100,109,31,40,120,89),(12,108,101,39,32,88,81,19),(13,38,102,87,33,18,82,107),(14,86,103,17,34,106,83,37),(15,16,104,105,35,36,84,85),(41,129,130,60,61,149,150,80),(42,59,131,148,62,79,151,128),(43,147,132,78,63,127,152,58),(44,77,133,126,64,57,153,146),(45,125,134,56,65,145,154,76),(46,55,135,144,66,75,155,124),(47,143,136,74,67,123,156,54),(48,73,137,122,68,53,157,142),(49,121,138,52,69,141,158,72),(50,51,139,140,70,71,159,160)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 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 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D5 | M4(2) | C4○D4 | M4(2) | D10 | D10 | C4×D5 | C4○D20 | D5×M4(2) |
| kernel | C42.182D10 | C42.D5 | C20.8Q8 | D10⋊1C8 | C5×C8⋊C4 | D5×C42 | C4×Dic5 | C2×C4×D5 | C8⋊C4 | Dic5 | C20 | D10 | C42 | C2×C8 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 16 | 8 |
Matrix representation of C42.182D10 ►in GL4(𝔽41) generated by
| 32 | 38 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 13 | 1 | 0 | 0 |
| 4 | 28 | 0 | 0 |
| 0 | 0 | 14 | 27 |
| 0 | 0 | 14 | 30 |
| 13 | 17 | 0 | 0 |
| 4 | 28 | 0 | 0 |
| 0 | 0 | 27 | 14 |
| 0 | 0 | 30 | 14 |
G:=sub<GL(4,GF(41))| [32,0,0,0,38,9,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[13,4,0,0,1,28,0,0,0,0,14,14,0,0,27,30],[13,4,0,0,17,28,0,0,0,0,27,30,0,0,14,14] >;
C42.182D10 in GAP, Magma, Sage, TeX
C_4^2._{182}D_{10} % in TeX
G:=Group("C4^2.182D10"); // GroupNames label
G:=SmallGroup(320,332);
// by ID
G=gap.SmallGroup(320,332);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,422,387,58,136,12550]);
// 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=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations