metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.48D10, D4⋊D5⋊9C4, D4⋊4(C4×D5), (C4×D4)⋊2D5, (D4×C20)⋊2C2, (C4×D20)⋊20C2, D20⋊21(C2×C4), C5⋊5(D8⋊C4), C4⋊C4.242D10, C10.100(C4×D4), (C2×C20).254D4, D20⋊6C4⋊29C2, (C2×D4).189D10, C20.49(C4○D4), C4.37(C4○D20), D4⋊Dic5⋊11C2, C20.57(C22×C4), (C4×C20).85C22, C20.Q8⋊32C2, C42.D5⋊5C2, C2.3(D4⋊D10), (C2×C20).336C23, C10.108(C8⋊C22), C2.3(D4.D10), (C2×D20).245C22, (D4×C10).231C22, C4⋊Dic5.327C22, C4.22(C2×C4×D5), C5⋊2C8⋊8(C2×C4), (C5×D4)⋊19(C2×C4), (C2×D4⋊D5).4C2, C2.16(C4×C5⋊D4), (C2×C10).467(C2×D4), C22.76(C2×C5⋊D4), (C2×C4).217(C5⋊D4), (C5×C4⋊C4).273C22, (C2×C5⋊2C8).92C22, (C2×C4).436(C22×D5), SmallGroup(320,641)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.48D10
G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c-1 >
Subgroups: 502 in 132 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C4×D4, C2×D8, C5⋊2C8, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, D8⋊C4, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×C20, D4×C10, C42.D5, C20.Q8, D20⋊6C4, D4⋊Dic5, C4×D20, C2×D4⋊D5, D4×C20, C42.48D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8⋊C22, C4×D5, C5⋊D4, C22×D5, D8⋊C4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, D4.D10, D4⋊D10, C42.48D10
►On 160 points
Generators in S160
(1 48 28 38)(2 154 29 144)(3 50 30 40)(4 156 21 146)(5 42 22 32)(6 158 23 148)(7 44 24 34)(8 160 25 150)(9 46 26 36)(10 152 27 142)(11 67 118 57)(12 109 119 99)(13 69 120 59)(14 101 111 91)(15 61 112 51)(16 103 113 93)(17 63 114 53)(18 105 115 95)(19 65 116 55)(20 107 117 97)(31 126 41 136)(33 128 43 138)(35 130 45 140)(37 122 47 132)(39 124 49 134)(52 88 62 72)(54 90 64 74)(56 82 66 76)(58 84 68 78)(60 86 70 80)(71 92 87 102)(73 94 89 104)(75 96 81 106)(77 98 83 108)(79 100 85 110)(121 151 131 141)(123 153 133 143)(125 155 135 145)(127 157 137 147)(129 159 139 149)
(1 60 123 91)(2 92 124 51)(3 52 125 93)(4 94 126 53)(5 54 127 95)(6 96 128 55)(7 56 129 97)(8 98 130 57)(9 58 121 99)(10 100 122 59)(11 160 83 45)(12 46 84 151)(13 152 85 47)(14 48 86 153)(15 154 87 49)(16 50 88 155)(17 156 89 41)(18 42 90 157)(19 158 81 43)(20 44 82 159)(21 104 136 63)(22 64 137 105)(23 106 138 65)(24 66 139 107)(25 108 140 67)(26 68 131 109)(27 110 132 69)(28 70 133 101)(29 102 134 61)(30 62 135 103)(31 114 146 73)(32 74 147 115)(33 116 148 75)(34 76 149 117)(35 118 150 77)(36 78 141 119)(37 120 142 79)(38 80 143 111)(39 112 144 71)(40 72 145 113)
(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 91 59 123 122 60 100)(2 99 51 121 124 58 92 9)(3 8 93 57 125 130 52 98)(4 97 53 129 126 56 94 7)(5 6 95 55 127 128 54 96)(11 50 45 16 83 155 160 88)(12 87 151 154 84 15 46 49)(13 48 47 14 85 153 152 86)(17 44 41 20 89 159 156 82)(18 81 157 158 90 19 42 43)(21 107 63 139 136 66 104 24)(22 23 105 65 137 138 64 106)(25 103 67 135 140 62 108 30)(26 29 109 61 131 134 68 102)(27 101 69 133 132 70 110 28)(31 117 73 149 146 76 114 34)(32 33 115 75 147 148 74 116)(35 113 77 145 150 72 118 40)(36 39 119 71 141 144 78 112)(37 111 79 143 142 80 120 38)
G:=sub<Sym(160)| (1,48,28,38)(2,154,29,144)(3,50,30,40)(4,156,21,146)(5,42,22,32)(6,158,23,148)(7,44,24,34)(8,160,25,150)(9,46,26,36)(10,152,27,142)(11,67,118,57)(12,109,119,99)(13,69,120,59)(14,101,111,91)(15,61,112,51)(16,103,113,93)(17,63,114,53)(18,105,115,95)(19,65,116,55)(20,107,117,97)(31,126,41,136)(33,128,43,138)(35,130,45,140)(37,122,47,132)(39,124,49,134)(52,88,62,72)(54,90,64,74)(56,82,66,76)(58,84,68,78)(60,86,70,80)(71,92,87,102)(73,94,89,104)(75,96,81,106)(77,98,83,108)(79,100,85,110)(121,151,131,141)(123,153,133,143)(125,155,135,145)(127,157,137,147)(129,159,139,149), (1,60,123,91)(2,92,124,51)(3,52,125,93)(4,94,126,53)(5,54,127,95)(6,96,128,55)(7,56,129,97)(8,98,130,57)(9,58,121,99)(10,100,122,59)(11,160,83,45)(12,46,84,151)(13,152,85,47)(14,48,86,153)(15,154,87,49)(16,50,88,155)(17,156,89,41)(18,42,90,157)(19,158,81,43)(20,44,82,159)(21,104,136,63)(22,64,137,105)(23,106,138,65)(24,66,139,107)(25,108,140,67)(26,68,131,109)(27,110,132,69)(28,70,133,101)(29,102,134,61)(30,62,135,103)(31,114,146,73)(32,74,147,115)(33,116,148,75)(34,76,149,117)(35,118,150,77)(36,78,141,119)(37,120,142,79)(38,80,143,111)(39,112,144,71)(40,72,145,113), (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), (1,10,91,59,123,122,60,100)(2,99,51,121,124,58,92,9)(3,8,93,57,125,130,52,98)(4,97,53,129,126,56,94,7)(5,6,95,55,127,128,54,96)(11,50,45,16,83,155,160,88)(12,87,151,154,84,15,46,49)(13,48,47,14,85,153,152,86)(17,44,41,20,89,159,156,82)(18,81,157,158,90,19,42,43)(21,107,63,139,136,66,104,24)(22,23,105,65,137,138,64,106)(25,103,67,135,140,62,108,30)(26,29,109,61,131,134,68,102)(27,101,69,133,132,70,110,28)(31,117,73,149,146,76,114,34)(32,33,115,75,147,148,74,116)(35,113,77,145,150,72,118,40)(36,39,119,71,141,144,78,112)(37,111,79,143,142,80,120,38)>;
G:=Group( (1,48,28,38)(2,154,29,144)(3,50,30,40)(4,156,21,146)(5,42,22,32)(6,158,23,148)(7,44,24,34)(8,160,25,150)(9,46,26,36)(10,152,27,142)(11,67,118,57)(12,109,119,99)(13,69,120,59)(14,101,111,91)(15,61,112,51)(16,103,113,93)(17,63,114,53)(18,105,115,95)(19,65,116,55)(20,107,117,97)(31,126,41,136)(33,128,43,138)(35,130,45,140)(37,122,47,132)(39,124,49,134)(52,88,62,72)(54,90,64,74)(56,82,66,76)(58,84,68,78)(60,86,70,80)(71,92,87,102)(73,94,89,104)(75,96,81,106)(77,98,83,108)(79,100,85,110)(121,151,131,141)(123,153,133,143)(125,155,135,145)(127,157,137,147)(129,159,139,149), (1,60,123,91)(2,92,124,51)(3,52,125,93)(4,94,126,53)(5,54,127,95)(6,96,128,55)(7,56,129,97)(8,98,130,57)(9,58,121,99)(10,100,122,59)(11,160,83,45)(12,46,84,151)(13,152,85,47)(14,48,86,153)(15,154,87,49)(16,50,88,155)(17,156,89,41)(18,42,90,157)(19,158,81,43)(20,44,82,159)(21,104,136,63)(22,64,137,105)(23,106,138,65)(24,66,139,107)(25,108,140,67)(26,68,131,109)(27,110,132,69)(28,70,133,101)(29,102,134,61)(30,62,135,103)(31,114,146,73)(32,74,147,115)(33,116,148,75)(34,76,149,117)(35,118,150,77)(36,78,141,119)(37,120,142,79)(38,80,143,111)(39,112,144,71)(40,72,145,113), (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), (1,10,91,59,123,122,60,100)(2,99,51,121,124,58,92,9)(3,8,93,57,125,130,52,98)(4,97,53,129,126,56,94,7)(5,6,95,55,127,128,54,96)(11,50,45,16,83,155,160,88)(12,87,151,154,84,15,46,49)(13,48,47,14,85,153,152,86)(17,44,41,20,89,159,156,82)(18,81,157,158,90,19,42,43)(21,107,63,139,136,66,104,24)(22,23,105,65,137,138,64,106)(25,103,67,135,140,62,108,30)(26,29,109,61,131,134,68,102)(27,101,69,133,132,70,110,28)(31,117,73,149,146,76,114,34)(32,33,115,75,147,148,74,116)(35,113,77,145,150,72,118,40)(36,39,119,71,141,144,78,112)(37,111,79,143,142,80,120,38) );
G=PermutationGroup([[(1,48,28,38),(2,154,29,144),(3,50,30,40),(4,156,21,146),(5,42,22,32),(6,158,23,148),(7,44,24,34),(8,160,25,150),(9,46,26,36),(10,152,27,142),(11,67,118,57),(12,109,119,99),(13,69,120,59),(14,101,111,91),(15,61,112,51),(16,103,113,93),(17,63,114,53),(18,105,115,95),(19,65,116,55),(20,107,117,97),(31,126,41,136),(33,128,43,138),(35,130,45,140),(37,122,47,132),(39,124,49,134),(52,88,62,72),(54,90,64,74),(56,82,66,76),(58,84,68,78),(60,86,70,80),(71,92,87,102),(73,94,89,104),(75,96,81,106),(77,98,83,108),(79,100,85,110),(121,151,131,141),(123,153,133,143),(125,155,135,145),(127,157,137,147),(129,159,139,149)], [(1,60,123,91),(2,92,124,51),(3,52,125,93),(4,94,126,53),(5,54,127,95),(6,96,128,55),(7,56,129,97),(8,98,130,57),(9,58,121,99),(10,100,122,59),(11,160,83,45),(12,46,84,151),(13,152,85,47),(14,48,86,153),(15,154,87,49),(16,50,88,155),(17,156,89,41),(18,42,90,157),(19,158,81,43),(20,44,82,159),(21,104,136,63),(22,64,137,105),(23,106,138,65),(24,66,139,107),(25,108,140,67),(26,68,131,109),(27,110,132,69),(28,70,133,101),(29,102,134,61),(30,62,135,103),(31,114,146,73),(32,74,147,115),(33,116,148,75),(34,76,149,117),(35,118,150,77),(36,78,141,119),(37,120,142,79),(38,80,143,111),(39,112,144,71),(40,72,145,113)], [(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)], [(1,10,91,59,123,122,60,100),(2,99,51,121,124,58,92,9),(3,8,93,57,125,130,52,98),(4,97,53,129,126,56,94,7),(5,6,95,55,127,128,54,96),(11,50,45,16,83,155,160,88),(12,87,151,154,84,15,46,49),(13,48,47,14,85,153,152,86),(17,44,41,20,89,159,156,82),(18,81,157,158,90,19,42,43),(21,107,63,139,136,66,104,24),(22,23,105,65,137,138,64,106),(25,103,67,135,140,62,108,30),(26,29,109,61,131,134,68,102),(27,101,69,133,132,70,110,28),(31,117,73,149,146,76,114,34),(32,33,115,75,147,148,74,116),(35,113,77,145,150,72,118,40),(36,39,119,71,141,144,78,112),(37,111,79,143,142,80,120,38)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C4×D5 | C4○D20 | C8⋊C22 | D4.D10 | D4⋊D10 |
| kernel | C42.48D10 | C42.D5 | C20.Q8 | D20⋊6C4 | D4⋊Dic5 | C4×D20 | C2×D4⋊D5 | D4×C20 | D4⋊D5 | C2×C20 | C4×D4 | C20 | C42 | C4⋊C4 | C2×D4 | C2×C4 | D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 2 | 4 | 4 |
Matrix representation of C42.48D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 0 | 0 | 0 | 0 | 40 | 17 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 10 | 8 | 0 | 0 | 0 | 0 |
| 3 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 7 | 7 | 23 |
| 0 | 0 | 34 | 15 | 18 | 10 |
| 0 | 0 | 7 | 23 | 18 | 34 |
| 0 | 0 | 18 | 10 | 7 | 26 |
| 31 | 33 | 0 | 0 | 0 | 0 |
| 28 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 23 | 18 | 34 |
| 0 | 0 | 15 | 34 | 31 | 23 |
| 0 | 0 | 23 | 7 | 7 | 23 |
| 0 | 0 | 10 | 18 | 15 | 34 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,24,40,0,0,0,0,1,17,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[10,3,0,0,0,0,8,31,0,0,0,0,0,0,23,34,7,18,0,0,7,15,23,10,0,0,7,18,18,7,0,0,23,10,34,26],[31,28,0,0,0,0,33,10,0,0,0,0,0,0,7,15,23,10,0,0,23,34,7,18,0,0,18,31,7,15,0,0,34,23,23,34] >;
C42.48D10 in GAP, Magma, Sage, TeX
C_4^2._{48}D_{10} % in TeX
G:=Group("C4^2.48D10"); // GroupNames label
G:=SmallGroup(320,641);
// by ID
G=gap.SmallGroup(320,641);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations