metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.16D10, C8⋊1(C4×D5), C40⋊11(C2×C4), C40⋊C2⋊5C4, C8⋊C4⋊1D5, (C4×D20).5C2, C40⋊5C4⋊16C2, C2.14(C4×D20), C10.41(C4×D4), (C2×C8).53D10, (C4×Dic10)⋊2C2, D20.28(C2×C4), (C2×C20).236D4, (C2×C4).114D20, Dic10⋊18(C2×C4), C5⋊2(SD16⋊C4), C2.1(C8⋊D10), C10.2(C8⋊C22), (C2×C40).54C22, (C4×C20).14C22, C22.30(C2×D20), D20⋊5C4.15C2, C4.106(C4○D20), C20.222(C4○D4), C20.44D4⋊37C2, C20.164(C22×C4), (C2×C20).731C23, C2.1(C8.D10), C10.3(C8.C22), (C2×D20).195C22, C4⋊Dic5.265C22, (C2×Dic10).214C22, C4.63(C2×C4×D5), (C5×C8⋊C4)⋊2C2, (C2×C40⋊C2).1C2, (C2×C10).114(C2×D4), (C2×C4).675(C22×D5), SmallGroup(320,337)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.16D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=bc9 >
Subgroups: 518 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C40, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, SD16⋊C4, C40⋊C2, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, C40⋊5C4, D20⋊5C4, C5×C8⋊C4, C4×Dic10, C4×D20, C2×C40⋊C2, C42.16D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8⋊C22, C8.C22, C4×D5, D20, C22×D5, SD16⋊C4, C2×C4×D5, C2×D20, C4○D20, C4×D20, C8⋊D10, C8.D10, C42.16D10
►On 160 points
Generators in S160
(1 57 127 104)(2 78 128 85)(3 59 129 106)(4 80 130 87)(5 61 131 108)(6 42 132 89)(7 63 133 110)(8 44 134 91)(9 65 135 112)(10 46 136 93)(11 67 137 114)(12 48 138 95)(13 69 139 116)(14 50 140 97)(15 71 141 118)(16 52 142 99)(17 73 143 120)(18 54 144 101)(19 75 145 82)(20 56 146 103)(21 77 147 84)(22 58 148 105)(23 79 149 86)(24 60 150 107)(25 41 151 88)(26 62 152 109)(27 43 153 90)(28 64 154 111)(29 45 155 92)(30 66 156 113)(31 47 157 94)(32 68 158 115)(33 49 159 96)(34 70 160 117)(35 51 121 98)(36 72 122 119)(37 53 123 100)(38 74 124 81)(39 55 125 102)(40 76 126 83)
(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 21 30)(2 29 22 9)(3 8 23 28)(4 27 24 7)(5 6 25 26)(11 40 31 20)(12 19 32 39)(13 38 33 18)(14 17 34 37)(15 36 35 16)(41 42 61 62)(43 80 63 60)(44 59 64 79)(45 78 65 58)(46 57 66 77)(47 76 67 56)(48 55 68 75)(49 74 69 54)(50 53 70 73)(51 72 71 52)(81 116 101 96)(82 95 102 115)(83 114 103 94)(84 93 104 113)(85 112 105 92)(86 91 106 111)(87 110 107 90)(88 89 108 109)(97 100 117 120)(98 119 118 99)(121 142 141 122)(123 140 143 160)(124 159 144 139)(125 138 145 158)(126 157 146 137)(127 136 147 156)(128 155 148 135)(129 134 149 154)(130 153 150 133)(131 132 151 152)
G:=sub<Sym(160)| (1,57,127,104)(2,78,128,85)(3,59,129,106)(4,80,130,87)(5,61,131,108)(6,42,132,89)(7,63,133,110)(8,44,134,91)(9,65,135,112)(10,46,136,93)(11,67,137,114)(12,48,138,95)(13,69,139,116)(14,50,140,97)(15,71,141,118)(16,52,142,99)(17,73,143,120)(18,54,144,101)(19,75,145,82)(20,56,146,103)(21,77,147,84)(22,58,148,105)(23,79,149,86)(24,60,150,107)(25,41,151,88)(26,62,152,109)(27,43,153,90)(28,64,154,111)(29,45,155,92)(30,66,156,113)(31,47,157,94)(32,68,158,115)(33,49,159,96)(34,70,160,117)(35,51,121,98)(36,72,122,119)(37,53,123,100)(38,74,124,81)(39,55,125,102)(40,76,126,83), (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), (1,10,21,30)(2,29,22,9)(3,8,23,28)(4,27,24,7)(5,6,25,26)(11,40,31,20)(12,19,32,39)(13,38,33,18)(14,17,34,37)(15,36,35,16)(41,42,61,62)(43,80,63,60)(44,59,64,79)(45,78,65,58)(46,57,66,77)(47,76,67,56)(48,55,68,75)(49,74,69,54)(50,53,70,73)(51,72,71,52)(81,116,101,96)(82,95,102,115)(83,114,103,94)(84,93,104,113)(85,112,105,92)(86,91,106,111)(87,110,107,90)(88,89,108,109)(97,100,117,120)(98,119,118,99)(121,142,141,122)(123,140,143,160)(124,159,144,139)(125,138,145,158)(126,157,146,137)(127,136,147,156)(128,155,148,135)(129,134,149,154)(130,153,150,133)(131,132,151,152)>;
G:=Group( (1,57,127,104)(2,78,128,85)(3,59,129,106)(4,80,130,87)(5,61,131,108)(6,42,132,89)(7,63,133,110)(8,44,134,91)(9,65,135,112)(10,46,136,93)(11,67,137,114)(12,48,138,95)(13,69,139,116)(14,50,140,97)(15,71,141,118)(16,52,142,99)(17,73,143,120)(18,54,144,101)(19,75,145,82)(20,56,146,103)(21,77,147,84)(22,58,148,105)(23,79,149,86)(24,60,150,107)(25,41,151,88)(26,62,152,109)(27,43,153,90)(28,64,154,111)(29,45,155,92)(30,66,156,113)(31,47,157,94)(32,68,158,115)(33,49,159,96)(34,70,160,117)(35,51,121,98)(36,72,122,119)(37,53,123,100)(38,74,124,81)(39,55,125,102)(40,76,126,83), (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), (1,10,21,30)(2,29,22,9)(3,8,23,28)(4,27,24,7)(5,6,25,26)(11,40,31,20)(12,19,32,39)(13,38,33,18)(14,17,34,37)(15,36,35,16)(41,42,61,62)(43,80,63,60)(44,59,64,79)(45,78,65,58)(46,57,66,77)(47,76,67,56)(48,55,68,75)(49,74,69,54)(50,53,70,73)(51,72,71,52)(81,116,101,96)(82,95,102,115)(83,114,103,94)(84,93,104,113)(85,112,105,92)(86,91,106,111)(87,110,107,90)(88,89,108,109)(97,100,117,120)(98,119,118,99)(121,142,141,122)(123,140,143,160)(124,159,144,139)(125,138,145,158)(126,157,146,137)(127,136,147,156)(128,155,148,135)(129,134,149,154)(130,153,150,133)(131,132,151,152) );
G=PermutationGroup([[(1,57,127,104),(2,78,128,85),(3,59,129,106),(4,80,130,87),(5,61,131,108),(6,42,132,89),(7,63,133,110),(8,44,134,91),(9,65,135,112),(10,46,136,93),(11,67,137,114),(12,48,138,95),(13,69,139,116),(14,50,140,97),(15,71,141,118),(16,52,142,99),(17,73,143,120),(18,54,144,101),(19,75,145,82),(20,56,146,103),(21,77,147,84),(22,58,148,105),(23,79,149,86),(24,60,150,107),(25,41,151,88),(26,62,152,109),(27,43,153,90),(28,64,154,111),(29,45,155,92),(30,66,156,113),(31,47,157,94),(32,68,158,115),(33,49,159,96),(34,70,160,117),(35,51,121,98),(36,72,122,119),(37,53,123,100),(38,74,124,81),(39,55,125,102),(40,76,126,83)], [(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)], [(1,10,21,30),(2,29,22,9),(3,8,23,28),(4,27,24,7),(5,6,25,26),(11,40,31,20),(12,19,32,39),(13,38,33,18),(14,17,34,37),(15,36,35,16),(41,42,61,62),(43,80,63,60),(44,59,64,79),(45,78,65,58),(46,57,66,77),(47,76,67,56),(48,55,68,75),(49,74,69,54),(50,53,70,73),(51,72,71,52),(81,116,101,96),(82,95,102,115),(83,114,103,94),(84,93,104,113),(85,112,105,92),(86,91,106,111),(87,110,107,90),(88,89,108,109),(97,100,117,120),(98,119,118,99),(121,142,141,122),(123,140,143,160),(124,159,144,139),(125,138,145,158),(126,157,146,137),(127,136,147,156),(128,155,148,135),(129,134,149,154),(130,153,150,133),(131,132,151,152)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4L | 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 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C4○D20 | C8⋊C22 | C8.C22 | C8⋊D10 | C8.D10 |
| kernel | C42.16D10 | C20.44D4 | C40⋊5C4 | D20⋊5C4 | C5×C8⋊C4 | C4×Dic10 | C4×D20 | C2×C40⋊C2 | C40⋊C2 | C2×C20 | C8⋊C4 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C42.16D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 4 | 39 | 0 |
| 0 | 0 | 37 | 31 | 0 | 39 |
| 0 | 0 | 11 | 16 | 23 | 37 |
| 0 | 0 | 25 | 22 | 4 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 34 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 10 | 5 | 12 |
| 0 | 0 | 31 | 29 | 29 | 3 |
| 0 | 0 | 13 | 17 | 24 | 31 |
| 0 | 0 | 24 | 17 | 10 | 12 |
| 6 | 34 | 0 | 0 | 0 | 0 |
| 5 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 39 | 12 | 5 |
| 0 | 0 | 21 | 15 | 3 | 29 |
| 0 | 0 | 7 | 20 | 31 | 24 |
| 0 | 0 | 12 | 34 | 12 | 10 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,18,37,11,25,0,0,4,31,16,22,0,0,39,0,23,4,0,0,0,39,37,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[0,6,0,0,0,0,34,35,0,0,0,0,0,0,17,31,13,24,0,0,10,29,17,17,0,0,5,29,24,10,0,0,12,3,31,12],[6,5,0,0,0,0,34,35,0,0,0,0,0,0,26,21,7,12,0,0,39,15,20,34,0,0,12,3,31,12,0,0,5,29,24,10] >;
C42.16D10 in GAP, Magma, Sage, TeX
C_4^2._{16}D_{10} % in TeX
G:=Group("C4^2.16D10"); // GroupNames label
G:=SmallGroup(320,337);
// by ID
G=gap.SmallGroup(320,337);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations