metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.159D10, C10.982- 1+4, C20⋊Q8⋊39C2, C42⋊2C2.D5, C4⋊C4.116D10, C20.6Q8⋊8C2, (C4×Dic10)⋊13C2, (C2×C20).93C23, (C4×C20).31C22, C22⋊C4.39D10, C4.Dic10⋊38C2, Dic5⋊3Q8⋊39C2, (C2×C10).245C24, C4⋊Dic5.53C22, C23.51(C22×D5), Dic5.20(C4○D4), Dic5.Q8⋊36C2, (C22×C10).59C23, C23.D10.3C2, C22.266(C23×D5), C23.D5.61C22, Dic5.14D4.4C2, C5⋊6(C22.35C24), (C2×Dic5).127C23, (C4×Dic5).237C22, C23.11D10.3C2, C2.62(D4.10D10), (C2×Dic10).261C22, C10.D4.126C22, (C22×Dic5).148C22, C2.92(D5×C4○D4), C10.203(C2×C4○D4), (C5×C4⋊C4).200C22, (C5×C42⋊2C2).1C2, (C2×C4).302(C22×D5), (C5×C22⋊C4).70C22, SmallGroup(320,1373)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.159D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 558 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C42⋊2C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22.35C24, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C4×Dic10, C20.6Q8, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊3Q8, C20⋊Q8, Dic5.Q8, C4.Dic10, C5×C42⋊2C2, C42.159D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.35C24, C23×D5, D5×C4○D4, D4.10D10, C42.159D10
►On 160 points
Generators in S160
(1 46 63 112)(2 57 64 103)(3 48 65 114)(4 59 66 105)(5 50 67 116)(6 41 68 107)(7 52 69 118)(8 43 70 109)(9 54 71 120)(10 45 72 111)(11 56 73 102)(12 47 74 113)(13 58 75 104)(14 49 76 115)(15 60 77 106)(16 51 78 117)(17 42 79 108)(18 53 80 119)(19 44 61 110)(20 55 62 101)(21 157 125 92)(22 148 126 83)(23 159 127 94)(24 150 128 85)(25 141 129 96)(26 152 130 87)(27 143 131 98)(28 154 132 89)(29 145 133 100)(30 156 134 91)(31 147 135 82)(32 158 136 93)(33 149 137 84)(34 160 138 95)(35 151 139 86)(36 142 140 97)(37 153 121 88)(38 144 122 99)(39 155 123 90)(40 146 124 81)
(1 51 11 41)(2 108 12 118)(3 53 13 43)(4 110 14 120)(5 55 15 45)(6 112 16 102)(7 57 17 47)(8 114 18 104)(9 59 19 49)(10 116 20 106)(21 142 31 152)(22 88 32 98)(23 144 33 154)(24 90 34 100)(25 146 35 156)(26 92 36 82)(27 148 37 158)(28 94 38 84)(29 150 39 160)(30 96 40 86)(42 74 52 64)(44 76 54 66)(46 78 56 68)(48 80 58 70)(50 62 60 72)(61 115 71 105)(63 117 73 107)(65 119 75 109)(67 101 77 111)(69 103 79 113)(81 139 91 129)(83 121 93 131)(85 123 95 133)(87 125 97 135)(89 127 99 137)(122 149 132 159)(124 151 134 141)(126 153 136 143)(128 155 138 145)(130 157 140 147)
(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)
(1 150 63 85)(2 159 64 94)(3 148 65 83)(4 157 66 92)(5 146 67 81)(6 155 68 90)(7 144 69 99)(8 153 70 88)(9 142 71 97)(10 151 72 86)(11 160 73 95)(12 149 74 84)(13 158 75 93)(14 147 76 82)(15 156 77 91)(16 145 78 100)(17 154 79 89)(18 143 80 98)(19 152 61 87)(20 141 62 96)(21 49 125 115)(22 58 126 104)(23 47 127 113)(24 56 128 102)(25 45 129 111)(26 54 130 120)(27 43 131 109)(28 52 132 118)(29 41 133 107)(30 50 134 116)(31 59 135 105)(32 48 136 114)(33 57 137 103)(34 46 138 112)(35 55 139 101)(36 44 140 110)(37 53 121 119)(38 42 122 108)(39 51 123 117)(40 60 124 106)
G:=sub<Sym(160)| (1,46,63,112)(2,57,64,103)(3,48,65,114)(4,59,66,105)(5,50,67,116)(6,41,68,107)(7,52,69,118)(8,43,70,109)(9,54,71,120)(10,45,72,111)(11,56,73,102)(12,47,74,113)(13,58,75,104)(14,49,76,115)(15,60,77,106)(16,51,78,117)(17,42,79,108)(18,53,80,119)(19,44,61,110)(20,55,62,101)(21,157,125,92)(22,148,126,83)(23,159,127,94)(24,150,128,85)(25,141,129,96)(26,152,130,87)(27,143,131,98)(28,154,132,89)(29,145,133,100)(30,156,134,91)(31,147,135,82)(32,158,136,93)(33,149,137,84)(34,160,138,95)(35,151,139,86)(36,142,140,97)(37,153,121,88)(38,144,122,99)(39,155,123,90)(40,146,124,81), (1,51,11,41)(2,108,12,118)(3,53,13,43)(4,110,14,120)(5,55,15,45)(6,112,16,102)(7,57,17,47)(8,114,18,104)(9,59,19,49)(10,116,20,106)(21,142,31,152)(22,88,32,98)(23,144,33,154)(24,90,34,100)(25,146,35,156)(26,92,36,82)(27,148,37,158)(28,94,38,84)(29,150,39,160)(30,96,40,86)(42,74,52,64)(44,76,54,66)(46,78,56,68)(48,80,58,70)(50,62,60,72)(61,115,71,105)(63,117,73,107)(65,119,75,109)(67,101,77,111)(69,103,79,113)(81,139,91,129)(83,121,93,131)(85,123,95,133)(87,125,97,135)(89,127,99,137)(122,149,132,159)(124,151,134,141)(126,153,136,143)(128,155,138,145)(130,157,140,147), (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), (1,150,63,85)(2,159,64,94)(3,148,65,83)(4,157,66,92)(5,146,67,81)(6,155,68,90)(7,144,69,99)(8,153,70,88)(9,142,71,97)(10,151,72,86)(11,160,73,95)(12,149,74,84)(13,158,75,93)(14,147,76,82)(15,156,77,91)(16,145,78,100)(17,154,79,89)(18,143,80,98)(19,152,61,87)(20,141,62,96)(21,49,125,115)(22,58,126,104)(23,47,127,113)(24,56,128,102)(25,45,129,111)(26,54,130,120)(27,43,131,109)(28,52,132,118)(29,41,133,107)(30,50,134,116)(31,59,135,105)(32,48,136,114)(33,57,137,103)(34,46,138,112)(35,55,139,101)(36,44,140,110)(37,53,121,119)(38,42,122,108)(39,51,123,117)(40,60,124,106)>;
G:=Group( (1,46,63,112)(2,57,64,103)(3,48,65,114)(4,59,66,105)(5,50,67,116)(6,41,68,107)(7,52,69,118)(8,43,70,109)(9,54,71,120)(10,45,72,111)(11,56,73,102)(12,47,74,113)(13,58,75,104)(14,49,76,115)(15,60,77,106)(16,51,78,117)(17,42,79,108)(18,53,80,119)(19,44,61,110)(20,55,62,101)(21,157,125,92)(22,148,126,83)(23,159,127,94)(24,150,128,85)(25,141,129,96)(26,152,130,87)(27,143,131,98)(28,154,132,89)(29,145,133,100)(30,156,134,91)(31,147,135,82)(32,158,136,93)(33,149,137,84)(34,160,138,95)(35,151,139,86)(36,142,140,97)(37,153,121,88)(38,144,122,99)(39,155,123,90)(40,146,124,81), (1,51,11,41)(2,108,12,118)(3,53,13,43)(4,110,14,120)(5,55,15,45)(6,112,16,102)(7,57,17,47)(8,114,18,104)(9,59,19,49)(10,116,20,106)(21,142,31,152)(22,88,32,98)(23,144,33,154)(24,90,34,100)(25,146,35,156)(26,92,36,82)(27,148,37,158)(28,94,38,84)(29,150,39,160)(30,96,40,86)(42,74,52,64)(44,76,54,66)(46,78,56,68)(48,80,58,70)(50,62,60,72)(61,115,71,105)(63,117,73,107)(65,119,75,109)(67,101,77,111)(69,103,79,113)(81,139,91,129)(83,121,93,131)(85,123,95,133)(87,125,97,135)(89,127,99,137)(122,149,132,159)(124,151,134,141)(126,153,136,143)(128,155,138,145)(130,157,140,147), (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), (1,150,63,85)(2,159,64,94)(3,148,65,83)(4,157,66,92)(5,146,67,81)(6,155,68,90)(7,144,69,99)(8,153,70,88)(9,142,71,97)(10,151,72,86)(11,160,73,95)(12,149,74,84)(13,158,75,93)(14,147,76,82)(15,156,77,91)(16,145,78,100)(17,154,79,89)(18,143,80,98)(19,152,61,87)(20,141,62,96)(21,49,125,115)(22,58,126,104)(23,47,127,113)(24,56,128,102)(25,45,129,111)(26,54,130,120)(27,43,131,109)(28,52,132,118)(29,41,133,107)(30,50,134,116)(31,59,135,105)(32,48,136,114)(33,57,137,103)(34,46,138,112)(35,55,139,101)(36,44,140,110)(37,53,121,119)(38,42,122,108)(39,51,123,117)(40,60,124,106) );
G=PermutationGroup([[(1,46,63,112),(2,57,64,103),(3,48,65,114),(4,59,66,105),(5,50,67,116),(6,41,68,107),(7,52,69,118),(8,43,70,109),(9,54,71,120),(10,45,72,111),(11,56,73,102),(12,47,74,113),(13,58,75,104),(14,49,76,115),(15,60,77,106),(16,51,78,117),(17,42,79,108),(18,53,80,119),(19,44,61,110),(20,55,62,101),(21,157,125,92),(22,148,126,83),(23,159,127,94),(24,150,128,85),(25,141,129,96),(26,152,130,87),(27,143,131,98),(28,154,132,89),(29,145,133,100),(30,156,134,91),(31,147,135,82),(32,158,136,93),(33,149,137,84),(34,160,138,95),(35,151,139,86),(36,142,140,97),(37,153,121,88),(38,144,122,99),(39,155,123,90),(40,146,124,81)], [(1,51,11,41),(2,108,12,118),(3,53,13,43),(4,110,14,120),(5,55,15,45),(6,112,16,102),(7,57,17,47),(8,114,18,104),(9,59,19,49),(10,116,20,106),(21,142,31,152),(22,88,32,98),(23,144,33,154),(24,90,34,100),(25,146,35,156),(26,92,36,82),(27,148,37,158),(28,94,38,84),(29,150,39,160),(30,96,40,86),(42,74,52,64),(44,76,54,66),(46,78,56,68),(48,80,58,70),(50,62,60,72),(61,115,71,105),(63,117,73,107),(65,119,75,109),(67,101,77,111),(69,103,79,113),(81,139,91,129),(83,121,93,131),(85,123,95,133),(87,125,97,135),(89,127,99,137),(122,149,132,159),(124,151,134,141),(126,153,136,143),(128,155,138,145),(130,157,140,147)], [(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)], [(1,150,63,85),(2,159,64,94),(3,148,65,83),(4,157,66,92),(5,146,67,81),(6,155,68,90),(7,144,69,99),(8,153,70,88),(9,142,71,97),(10,151,72,86),(11,160,73,95),(12,149,74,84),(13,158,75,93),(14,147,76,82),(15,156,77,91),(16,145,78,100),(17,154,79,89),(18,143,80,98),(19,152,61,87),(20,141,62,96),(21,49,125,115),(22,58,126,104),(23,47,127,113),(24,56,128,102),(25,45,129,111),(26,54,130,120),(27,43,131,109),(28,52,132,118),(29,41,133,107),(30,50,134,116),(31,59,135,105),(32,48,136,114),(33,57,137,103),(34,46,138,112),(35,55,139,101),(36,44,140,110),(37,53,121,119),(38,42,122,108),(39,51,123,117),(40,60,124,106)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- 1+4 | D5×C4○D4 | D4.10D10 |
| kernel | C42.159D10 | C4×Dic10 | C20.6Q8 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊3Q8 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 1 | 2 | 4 | 2 | 6 | 6 | 2 | 4 | 8 |
Matrix representation of C42.159D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 | 14 |
| 0 | 0 | 0 | 1 | 11 | 11 |
| 0 | 0 | 28 | 24 | 40 | 0 |
| 0 | 0 | 13 | 28 | 0 | 40 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 13 |
| 0 | 0 | 0 | 0 | 28 | 2 |
| 32 | 23 | 0 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 25 | 0 | 7 |
| 0 | 0 | 16 | 2 | 34 | 0 |
| 0 | 0 | 0 | 0 | 39 | 16 |
| 0 | 0 | 0 | 0 | 25 | 25 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 14 | 0 | 0 |
| 0 | 0 | 26 | 39 | 0 | 0 |
| 0 | 0 | 24 | 16 | 15 | 10 |
| 0 | 0 | 16 | 3 | 2 | 26 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,28,13,0,0,0,1,24,28,0,0,11,11,40,0,0,0,14,11,0,40],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,39,28,0,0,0,0,13,2],[32,9,0,0,0,0,23,9,0,0,0,0,0,0,16,16,0,0,0,0,25,2,0,0,0,0,0,34,39,25,0,0,7,0,16,25],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,2,26,24,16,0,0,14,39,16,3,0,0,0,0,15,2,0,0,0,0,10,26] >;
C42.159D10 in GAP, Magma, Sage, TeX
C_4^2._{159}D_{10} % in TeX
G:=Group("C4^2.159D10"); // GroupNames label
G:=SmallGroup(320,1373);
// by ID
G=gap.SmallGroup(320,1373);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations