metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.19D10, C8⋊C4⋊10D5, (C2×C4).26D20, (C2×C20).37D4, (C2×C8).159D10, C20⋊4D4.4C2, (C4×C20).4C22, C20.6Q8⋊2C2, D20⋊5C4⋊38C2, C2.8(C8⋊D10), C10.5(C8⋊C22), (C2×D20).9C22, C22.98(C2×D20), C4⋊Dic5.9C22, C4.108(C4○D20), C20.224(C4○D4), (C2×C40).313C22, (C2×C20).734C23, C10.8(C4.4D4), C2.13(C4.D20), C5⋊1(C42.29C22), (C5×C8⋊C4)⋊19C2, (C2×C10).117(C2×D4), (C2×C4).678(C22×D5), SmallGroup(320,340)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.19D10
G = < a,b,c,d | a4=b4=1, c10=a2b, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=bc9 >
Subgroups: 638 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], D5 [×2], C10, C10 [×2], C42, C4⋊C4 [×4], C2×C8 [×2], C2×D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C8⋊C4, D4⋊C4 [×4], C42.C2, C4⋊1D4, C40 [×2], D20 [×8], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C42.29C22, C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C2×C40 [×2], C2×D20 [×2], C2×D20 [×2], D20⋊5C4 [×4], C5×C8⋊C4, C20.6Q8, C20⋊4D4, C42.19D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22 [×2], D20 [×2], C22×D5, C42.29C22, C2×D20, C4○D20 [×2], C4.D20, C8⋊D10 [×2], C42.19D10
►On 160 points
Generators in S160
(1 91 48 151)(2 112 49 132)(3 93 50 153)(4 114 51 134)(5 95 52 155)(6 116 53 136)(7 97 54 157)(8 118 55 138)(9 99 56 159)(10 120 57 140)(11 101 58 121)(12 82 59 142)(13 103 60 123)(14 84 61 144)(15 105 62 125)(16 86 63 146)(17 107 64 127)(18 88 65 148)(19 109 66 129)(20 90 67 150)(21 111 68 131)(22 92 69 152)(23 113 70 133)(24 94 71 154)(25 115 72 135)(26 96 73 156)(27 117 74 137)(28 98 75 158)(29 119 76 139)(30 100 77 160)(31 81 78 141)(32 102 79 122)(33 83 80 143)(34 104 41 124)(35 85 42 145)(36 106 43 126)(37 87 44 147)(38 108 45 128)(39 89 46 149)(40 110 47 130)
(1 58 21 78)(2 59 22 79)(3 60 23 80)(4 61 24 41)(5 62 25 42)(6 63 26 43)(7 64 27 44)(8 65 28 45)(9 66 29 46)(10 67 30 47)(11 68 31 48)(12 69 32 49)(13 70 33 50)(14 71 34 51)(15 72 35 52)(16 73 36 53)(17 74 37 54)(18 75 38 55)(19 76 39 56)(20 77 40 57)(81 151 101 131)(82 152 102 132)(83 153 103 133)(84 154 104 134)(85 155 105 135)(86 156 106 136)(87 157 107 137)(88 158 108 138)(89 159 109 139)(90 160 110 140)(91 121 111 141)(92 122 112 142)(93 123 113 143)(94 124 114 144)(95 125 115 145)(96 126 116 146)(97 127 117 147)(98 128 118 148)(99 129 119 149)(100 130 120 150)
(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 68 77)(2 76 69 9)(3 8 70 75)(4 74 71 7)(5 6 72 73)(11 40 78 67)(12 66 79 39)(13 38 80 65)(14 64 41 37)(15 36 42 63)(16 62 43 35)(17 34 44 61)(18 60 45 33)(19 32 46 59)(20 58 47 31)(21 30 48 57)(22 56 49 29)(23 28 50 55)(24 54 51 27)(25 26 52 53)(81 130 121 90)(82 89 122 129)(83 128 123 88)(84 87 124 127)(85 126 125 86)(91 160 131 120)(92 119 132 159)(93 158 133 118)(94 117 134 157)(95 156 135 116)(96 115 136 155)(97 154 137 114)(98 113 138 153)(99 152 139 112)(100 111 140 151)(101 150 141 110)(102 109 142 149)(103 148 143 108)(104 107 144 147)(105 146 145 106)
G:=sub<Sym(160)| (1,91,48,151)(2,112,49,132)(3,93,50,153)(4,114,51,134)(5,95,52,155)(6,116,53,136)(7,97,54,157)(8,118,55,138)(9,99,56,159)(10,120,57,140)(11,101,58,121)(12,82,59,142)(13,103,60,123)(14,84,61,144)(15,105,62,125)(16,86,63,146)(17,107,64,127)(18,88,65,148)(19,109,66,129)(20,90,67,150)(21,111,68,131)(22,92,69,152)(23,113,70,133)(24,94,71,154)(25,115,72,135)(26,96,73,156)(27,117,74,137)(28,98,75,158)(29,119,76,139)(30,100,77,160)(31,81,78,141)(32,102,79,122)(33,83,80,143)(34,104,41,124)(35,85,42,145)(36,106,43,126)(37,87,44,147)(38,108,45,128)(39,89,46,149)(40,110,47,130), (1,58,21,78)(2,59,22,79)(3,60,23,80)(4,61,24,41)(5,62,25,42)(6,63,26,43)(7,64,27,44)(8,65,28,45)(9,66,29,46)(10,67,30,47)(11,68,31,48)(12,69,32,49)(13,70,33,50)(14,71,34,51)(15,72,35,52)(16,73,36,53)(17,74,37,54)(18,75,38,55)(19,76,39,56)(20,77,40,57)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,121,111,141)(92,122,112,142)(93,123,113,143)(94,124,114,144)(95,125,115,145)(96,126,116,146)(97,127,117,147)(98,128,118,148)(99,129,119,149)(100,130,120,150), (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), (1,10,68,77)(2,76,69,9)(3,8,70,75)(4,74,71,7)(5,6,72,73)(11,40,78,67)(12,66,79,39)(13,38,80,65)(14,64,41,37)(15,36,42,63)(16,62,43,35)(17,34,44,61)(18,60,45,33)(19,32,46,59)(20,58,47,31)(21,30,48,57)(22,56,49,29)(23,28,50,55)(24,54,51,27)(25,26,52,53)(81,130,121,90)(82,89,122,129)(83,128,123,88)(84,87,124,127)(85,126,125,86)(91,160,131,120)(92,119,132,159)(93,158,133,118)(94,117,134,157)(95,156,135,116)(96,115,136,155)(97,154,137,114)(98,113,138,153)(99,152,139,112)(100,111,140,151)(101,150,141,110)(102,109,142,149)(103,148,143,108)(104,107,144,147)(105,146,145,106)>;
G:=Group( (1,91,48,151)(2,112,49,132)(3,93,50,153)(4,114,51,134)(5,95,52,155)(6,116,53,136)(7,97,54,157)(8,118,55,138)(9,99,56,159)(10,120,57,140)(11,101,58,121)(12,82,59,142)(13,103,60,123)(14,84,61,144)(15,105,62,125)(16,86,63,146)(17,107,64,127)(18,88,65,148)(19,109,66,129)(20,90,67,150)(21,111,68,131)(22,92,69,152)(23,113,70,133)(24,94,71,154)(25,115,72,135)(26,96,73,156)(27,117,74,137)(28,98,75,158)(29,119,76,139)(30,100,77,160)(31,81,78,141)(32,102,79,122)(33,83,80,143)(34,104,41,124)(35,85,42,145)(36,106,43,126)(37,87,44,147)(38,108,45,128)(39,89,46,149)(40,110,47,130), (1,58,21,78)(2,59,22,79)(3,60,23,80)(4,61,24,41)(5,62,25,42)(6,63,26,43)(7,64,27,44)(8,65,28,45)(9,66,29,46)(10,67,30,47)(11,68,31,48)(12,69,32,49)(13,70,33,50)(14,71,34,51)(15,72,35,52)(16,73,36,53)(17,74,37,54)(18,75,38,55)(19,76,39,56)(20,77,40,57)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,121,111,141)(92,122,112,142)(93,123,113,143)(94,124,114,144)(95,125,115,145)(96,126,116,146)(97,127,117,147)(98,128,118,148)(99,129,119,149)(100,130,120,150), (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), (1,10,68,77)(2,76,69,9)(3,8,70,75)(4,74,71,7)(5,6,72,73)(11,40,78,67)(12,66,79,39)(13,38,80,65)(14,64,41,37)(15,36,42,63)(16,62,43,35)(17,34,44,61)(18,60,45,33)(19,32,46,59)(20,58,47,31)(21,30,48,57)(22,56,49,29)(23,28,50,55)(24,54,51,27)(25,26,52,53)(81,130,121,90)(82,89,122,129)(83,128,123,88)(84,87,124,127)(85,126,125,86)(91,160,131,120)(92,119,132,159)(93,158,133,118)(94,117,134,157)(95,156,135,116)(96,115,136,155)(97,154,137,114)(98,113,138,153)(99,152,139,112)(100,111,140,151)(101,150,141,110)(102,109,142,149)(103,148,143,108)(104,107,144,147)(105,146,145,106) );
G=PermutationGroup([(1,91,48,151),(2,112,49,132),(3,93,50,153),(4,114,51,134),(5,95,52,155),(6,116,53,136),(7,97,54,157),(8,118,55,138),(9,99,56,159),(10,120,57,140),(11,101,58,121),(12,82,59,142),(13,103,60,123),(14,84,61,144),(15,105,62,125),(16,86,63,146),(17,107,64,127),(18,88,65,148),(19,109,66,129),(20,90,67,150),(21,111,68,131),(22,92,69,152),(23,113,70,133),(24,94,71,154),(25,115,72,135),(26,96,73,156),(27,117,74,137),(28,98,75,158),(29,119,76,139),(30,100,77,160),(31,81,78,141),(32,102,79,122),(33,83,80,143),(34,104,41,124),(35,85,42,145),(36,106,43,126),(37,87,44,147),(38,108,45,128),(39,89,46,149),(40,110,47,130)], [(1,58,21,78),(2,59,22,79),(3,60,23,80),(4,61,24,41),(5,62,25,42),(6,63,26,43),(7,64,27,44),(8,65,28,45),(9,66,29,46),(10,67,30,47),(11,68,31,48),(12,69,32,49),(13,70,33,50),(14,71,34,51),(15,72,35,52),(16,73,36,53),(17,74,37,54),(18,75,38,55),(19,76,39,56),(20,77,40,57),(81,151,101,131),(82,152,102,132),(83,153,103,133),(84,154,104,134),(85,155,105,135),(86,156,106,136),(87,157,107,137),(88,158,108,138),(89,159,109,139),(90,160,110,140),(91,121,111,141),(92,122,112,142),(93,123,113,143),(94,124,114,144),(95,125,115,145),(96,126,116,146),(97,127,117,147),(98,128,118,148),(99,129,119,149),(100,130,120,150)], [(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)], [(1,10,68,77),(2,76,69,9),(3,8,70,75),(4,74,71,7),(5,6,72,73),(11,40,78,67),(12,66,79,39),(13,38,80,65),(14,64,41,37),(15,36,42,63),(16,62,43,35),(17,34,44,61),(18,60,45,33),(19,32,46,59),(20,58,47,31),(21,30,48,57),(22,56,49,29),(23,28,50,55),(24,54,51,27),(25,26,52,53),(81,130,121,90),(82,89,122,129),(83,128,123,88),(84,87,124,127),(85,126,125,86),(91,160,131,120),(92,119,132,159),(93,158,133,118),(94,117,134,157),(95,156,135,116),(96,115,136,155),(97,154,137,114),(98,113,138,153),(99,152,139,112),(100,111,140,151),(101,150,141,110),(102,109,142,149),(103,148,143,108),(104,107,144,147),(105,146,145,106)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 40 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C4○D20 | C8⋊C22 | C8⋊D10 |
| kernel | C42.19D10 | D20⋊5C4 | C5×C8⋊C4 | C20.6Q8 | C20⋊4D4 | C2×C20 | C8⋊C4 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 16 | 2 | 8 |
Matrix representation of C42.19D10 ►in GL6(𝔽41)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 13 | 0 | 0 |
| 0 | 0 | 19 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 19 | 11 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 9 | 16 | 0 | 0 |
| 0 | 0 | 36 | 14 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 32 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 25 |
| 0 | 0 | 0 | 0 | 25 | 27 |
| 0 | 0 | 27 | 16 | 0 | 0 |
| 0 | 0 | 16 | 14 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,19,0,0,0,0,13,11,0,0,0,0,0,0,30,19,0,0,0,0,13,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,36,0,0,0,0,16,14,0,0,0,7,0,0,0,0,35,34,0,0],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,0,0,27,16,0,0,0,0,16,14,0,0,14,25,0,0,0,0,25,27,0,0] >;
C42.19D10 in GAP, Magma, Sage, TeX
C_4^2._{19}D_{10} % in TeX
G:=Group("C4^2.19D10"); // GroupNames label
G:=SmallGroup(320,340);
// by ID
G=gap.SmallGroup(320,340);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations