metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.143D10, C10.1272+ 1+4, (C4×D20)⋊46C2, (D4×Dic5)⋊31C2, (Q8×Dic5)⋊20C2, C4.4D4⋊14D5, (C4×Dic10)⋊46C2, (C2×D4).176D10, C20⋊D4.11C2, (C2×Q8).139D10, C22⋊C4.36D10, Dic5⋊4D4⋊34C2, C20.126(C4○D4), C20.23D4⋊23C2, C4.16(D4⋊2D5), (C2×C20).505C23, (C4×C20).188C22, (C2×C10).225C24, D10.12D4⋊46C2, C2.51(D4⋊8D10), C23.47(C22×D5), Dic5.65(C4○D4), Dic5.5D4⋊41C2, (C2×D20).274C22, (D4×C10).158C22, C22.D20⋊26C2, C4⋊Dic5.235C22, (C22×C10).55C23, (Q8×C10).129C22, (C22×D5).97C23, C22.246(C23×D5), C23.D5.58C22, D10⋊C4.37C22, C5⋊4(C22.53C24), (C4×Dic5).143C22, (C2×Dic5).266C23, (C2×Dic10).258C22, C10.D4.142C22, (C22×Dic5).145C22, C2.81(D5×C4○D4), C10.192(C2×C4○D4), C2.57(C2×D4⋊2D5), (C5×C4.4D4)⋊17C2, (C2×C4×D5).267C22, (C2×C4).198(C22×D5), (C2×C5⋊D4).63C22, (C5×C22⋊C4).67C22, SmallGroup(320,1353)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.143D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 854 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×D4, C4×Q8, C22.D4, C4.4D4, C4.4D4, C4⋊1D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C22.53C24, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic5⋊4D4, D10.12D4, Dic5.5D4, C22.D20, D4×Dic5, C20⋊D4, Q8×Dic5, C20.23D4, C5×C4.4D4, C42.143D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.53C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, D5×C4○D4, D4⋊8D10, C42.143D10
►On 160 points
Generators in S160
(1 110 63 112)(2 144 64 88)(3 102 65 114)(4 146 66 90)(5 104 67 116)(6 148 68 82)(7 106 69 118)(8 150 70 84)(9 108 61 120)(10 142 62 86)(11 45 99 129)(12 153 100 135)(13 47 91 121)(14 155 92 137)(15 49 93 123)(16 157 94 139)(17 41 95 125)(18 159 96 131)(19 43 97 127)(20 151 98 133)(21 152 72 134)(22 46 73 130)(23 154 74 136)(24 48 75 122)(25 156 76 138)(26 50 77 124)(27 158 78 140)(28 42 79 126)(29 160 80 132)(30 44 71 128)(31 113 56 101)(32 89 57 145)(33 115 58 103)(34 81 59 147)(35 117 60 105)(36 83 51 149)(37 119 52 107)(38 85 53 141)(39 111 54 109)(40 87 55 143)
(1 12 55 22)(2 91 56 74)(3 14 57 24)(4 93 58 76)(5 16 59 26)(6 95 60 78)(7 18 51 28)(8 97 52 80)(9 20 53 30)(10 99 54 72)(11 39 21 62)(13 31 23 64)(15 33 25 66)(17 35 27 68)(19 37 29 70)(32 75 65 92)(34 77 67 94)(36 79 69 96)(38 71 61 98)(40 73 63 100)(41 117 158 82)(42 106 159 149)(43 119 160 84)(44 108 151 141)(45 111 152 86)(46 110 153 143)(47 113 154 88)(48 102 155 145)(49 115 156 90)(50 104 157 147)(81 124 116 139)(83 126 118 131)(85 128 120 133)(87 130 112 135)(89 122 114 137)(101 136 144 121)(103 138 146 123)(105 140 148 125)(107 132 150 127)(109 134 142 129)
(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 63 62)(2 61 64 9)(3 8 65 70)(4 69 66 7)(5 6 67 68)(11 22 99 73)(12 72 100 21)(13 30 91 71)(14 80 92 29)(15 28 93 79)(16 78 94 27)(17 26 95 77)(18 76 96 25)(19 24 97 75)(20 74 98 23)(31 53 56 38)(32 37 57 52)(33 51 58 36)(34 35 59 60)(39 55 54 40)(41 157 125 139)(42 138 126 156)(43 155 127 137)(44 136 128 154)(45 153 129 135)(46 134 130 152)(47 151 121 133)(48 132 122 160)(49 159 123 131)(50 140 124 158)(81 82 147 148)(83 90 149 146)(84 145 150 89)(85 88 141 144)(86 143 142 87)(101 120 113 108)(102 107 114 119)(103 118 115 106)(104 105 116 117)(109 112 111 110)
G:=sub<Sym(160)| (1,110,63,112)(2,144,64,88)(3,102,65,114)(4,146,66,90)(5,104,67,116)(6,148,68,82)(7,106,69,118)(8,150,70,84)(9,108,61,120)(10,142,62,86)(11,45,99,129)(12,153,100,135)(13,47,91,121)(14,155,92,137)(15,49,93,123)(16,157,94,139)(17,41,95,125)(18,159,96,131)(19,43,97,127)(20,151,98,133)(21,152,72,134)(22,46,73,130)(23,154,74,136)(24,48,75,122)(25,156,76,138)(26,50,77,124)(27,158,78,140)(28,42,79,126)(29,160,80,132)(30,44,71,128)(31,113,56,101)(32,89,57,145)(33,115,58,103)(34,81,59,147)(35,117,60,105)(36,83,51,149)(37,119,52,107)(38,85,53,141)(39,111,54,109)(40,87,55,143), (1,12,55,22)(2,91,56,74)(3,14,57,24)(4,93,58,76)(5,16,59,26)(6,95,60,78)(7,18,51,28)(8,97,52,80)(9,20,53,30)(10,99,54,72)(11,39,21,62)(13,31,23,64)(15,33,25,66)(17,35,27,68)(19,37,29,70)(32,75,65,92)(34,77,67,94)(36,79,69,96)(38,71,61,98)(40,73,63,100)(41,117,158,82)(42,106,159,149)(43,119,160,84)(44,108,151,141)(45,111,152,86)(46,110,153,143)(47,113,154,88)(48,102,155,145)(49,115,156,90)(50,104,157,147)(81,124,116,139)(83,126,118,131)(85,128,120,133)(87,130,112,135)(89,122,114,137)(101,136,144,121)(103,138,146,123)(105,140,148,125)(107,132,150,127)(109,134,142,129), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,22,99,73)(12,72,100,21)(13,30,91,71)(14,80,92,29)(15,28,93,79)(16,78,94,27)(17,26,95,77)(18,76,96,25)(19,24,97,75)(20,74,98,23)(31,53,56,38)(32,37,57,52)(33,51,58,36)(34,35,59,60)(39,55,54,40)(41,157,125,139)(42,138,126,156)(43,155,127,137)(44,136,128,154)(45,153,129,135)(46,134,130,152)(47,151,121,133)(48,132,122,160)(49,159,123,131)(50,140,124,158)(81,82,147,148)(83,90,149,146)(84,145,150,89)(85,88,141,144)(86,143,142,87)(101,120,113,108)(102,107,114,119)(103,118,115,106)(104,105,116,117)(109,112,111,110)>;
G:=Group( (1,110,63,112)(2,144,64,88)(3,102,65,114)(4,146,66,90)(5,104,67,116)(6,148,68,82)(7,106,69,118)(8,150,70,84)(9,108,61,120)(10,142,62,86)(11,45,99,129)(12,153,100,135)(13,47,91,121)(14,155,92,137)(15,49,93,123)(16,157,94,139)(17,41,95,125)(18,159,96,131)(19,43,97,127)(20,151,98,133)(21,152,72,134)(22,46,73,130)(23,154,74,136)(24,48,75,122)(25,156,76,138)(26,50,77,124)(27,158,78,140)(28,42,79,126)(29,160,80,132)(30,44,71,128)(31,113,56,101)(32,89,57,145)(33,115,58,103)(34,81,59,147)(35,117,60,105)(36,83,51,149)(37,119,52,107)(38,85,53,141)(39,111,54,109)(40,87,55,143), (1,12,55,22)(2,91,56,74)(3,14,57,24)(4,93,58,76)(5,16,59,26)(6,95,60,78)(7,18,51,28)(8,97,52,80)(9,20,53,30)(10,99,54,72)(11,39,21,62)(13,31,23,64)(15,33,25,66)(17,35,27,68)(19,37,29,70)(32,75,65,92)(34,77,67,94)(36,79,69,96)(38,71,61,98)(40,73,63,100)(41,117,158,82)(42,106,159,149)(43,119,160,84)(44,108,151,141)(45,111,152,86)(46,110,153,143)(47,113,154,88)(48,102,155,145)(49,115,156,90)(50,104,157,147)(81,124,116,139)(83,126,118,131)(85,128,120,133)(87,130,112,135)(89,122,114,137)(101,136,144,121)(103,138,146,123)(105,140,148,125)(107,132,150,127)(109,134,142,129), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,22,99,73)(12,72,100,21)(13,30,91,71)(14,80,92,29)(15,28,93,79)(16,78,94,27)(17,26,95,77)(18,76,96,25)(19,24,97,75)(20,74,98,23)(31,53,56,38)(32,37,57,52)(33,51,58,36)(34,35,59,60)(39,55,54,40)(41,157,125,139)(42,138,126,156)(43,155,127,137)(44,136,128,154)(45,153,129,135)(46,134,130,152)(47,151,121,133)(48,132,122,160)(49,159,123,131)(50,140,124,158)(81,82,147,148)(83,90,149,146)(84,145,150,89)(85,88,141,144)(86,143,142,87)(101,120,113,108)(102,107,114,119)(103,118,115,106)(104,105,116,117)(109,112,111,110) );
G=PermutationGroup([[(1,110,63,112),(2,144,64,88),(3,102,65,114),(4,146,66,90),(5,104,67,116),(6,148,68,82),(7,106,69,118),(8,150,70,84),(9,108,61,120),(10,142,62,86),(11,45,99,129),(12,153,100,135),(13,47,91,121),(14,155,92,137),(15,49,93,123),(16,157,94,139),(17,41,95,125),(18,159,96,131),(19,43,97,127),(20,151,98,133),(21,152,72,134),(22,46,73,130),(23,154,74,136),(24,48,75,122),(25,156,76,138),(26,50,77,124),(27,158,78,140),(28,42,79,126),(29,160,80,132),(30,44,71,128),(31,113,56,101),(32,89,57,145),(33,115,58,103),(34,81,59,147),(35,117,60,105),(36,83,51,149),(37,119,52,107),(38,85,53,141),(39,111,54,109),(40,87,55,143)], [(1,12,55,22),(2,91,56,74),(3,14,57,24),(4,93,58,76),(5,16,59,26),(6,95,60,78),(7,18,51,28),(8,97,52,80),(9,20,53,30),(10,99,54,72),(11,39,21,62),(13,31,23,64),(15,33,25,66),(17,35,27,68),(19,37,29,70),(32,75,65,92),(34,77,67,94),(36,79,69,96),(38,71,61,98),(40,73,63,100),(41,117,158,82),(42,106,159,149),(43,119,160,84),(44,108,151,141),(45,111,152,86),(46,110,153,143),(47,113,154,88),(48,102,155,145),(49,115,156,90),(50,104,157,147),(81,124,116,139),(83,126,118,131),(85,128,120,133),(87,130,112,135),(89,122,114,137),(101,136,144,121),(103,138,146,123),(105,140,148,125),(107,132,150,127),(109,134,142,129)], [(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)], [(1,10,63,62),(2,61,64,9),(3,8,65,70),(4,69,66,7),(5,6,67,68),(11,22,99,73),(12,72,100,21),(13,30,91,71),(14,80,92,29),(15,28,93,79),(16,78,94,27),(17,26,95,77),(18,76,96,25),(19,24,97,75),(20,74,98,23),(31,53,56,38),(32,37,57,52),(33,51,58,36),(34,35,59,60),(39,55,54,40),(41,157,125,139),(42,138,126,156),(43,155,127,137),(44,136,128,154),(45,153,129,135),(46,134,130,152),(47,151,121,133),(48,132,122,160),(49,159,123,131),(50,140,124,158),(81,82,147,148),(83,90,149,146),(84,145,150,89),(85,88,141,144),(86,143,142,87),(101,120,113,108),(102,107,114,119),(103,118,115,106),(104,105,116,117),(109,112,111,110)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊2D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.143D10 | C4×Dic10 | C4×D20 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C22.D20 | D4×Dic5 | C20⋊D4 | Q8×Dic5 | C20.23D4 | C5×C4.4D4 | C4.4D4 | Dic5 | C20 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C42.143D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 16 | 32 |
| 32 | 16 | 0 | 0 | 0 | 0 |
| 36 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 16 | 32 |
| 40 | 20 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 8 |
| 0 | 0 | 0 | 0 | 15 | 39 |
| 40 | 20 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 33 |
| 0 | 0 | 0 | 0 | 16 | 2 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[32,36,0,0,0,0,16,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[40,0,0,0,0,0,20,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,2,15,0,0,0,0,8,39],[40,4,0,0,0,0,20,1,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,39,16,0,0,0,0,33,2] >;
C42.143D10 in GAP, Magma, Sage, TeX
C_4^2._{143}D_{10} % in TeX
G:=Group("C4^2.143D10"); // GroupNames label
G:=SmallGroup(320,1353);
// by ID
G=gap.SmallGroup(320,1353);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,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,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations