metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.163D10, C10.1402+ 1+4, (C4×D20)⋊16C2, C4⋊D20⋊37C2, C4⋊C4.214D10, C42⋊2C2⋊6D5, C42⋊D5⋊8C2, D20⋊8C4⋊42C2, D10⋊D4⋊45C2, (C2×C20).96C23, (C4×C20).35C22, C22⋊C4.81D10, Dic5⋊4D4⋊37C2, D10.56(C4○D4), (C2×C10).253C24, D10.13D4⋊42C2, C2.65(D4⋊8D10), C23.59(C22×D5), Dic5.49(C4○D4), Dic5.Q8⋊39C2, (C2×D20).175C22, C22.D20⋊30C2, C4⋊Dic5.318C22, (C22×C10).67C23, C22.274(C23×D5), D10⋊C4.46C22, (C4×Dic5).160C22, (C2×Dic5).276C23, (C22×D5).112C23, C5⋊11(C22.47C24), C10.D4.147C22, (C22×Dic5).153C22, (D5×C4⋊C4)⋊43C2, C4⋊C4⋊D5⋊43C2, C2.100(D5×C4○D4), (C5×C42⋊2C2)⋊8C2, C10.211(C2×C4○D4), (C2×C4×D5).144C22, (C2×C4).89(C22×D5), (C5×C4⋊C4).205C22, (C2×C5⋊D4).73C22, (C5×C22⋊C4).78C22, SmallGroup(320,1381)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.163D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 918 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C22.47C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C42⋊D5, C4×D20, Dic5⋊4D4, D10⋊D4, C22.D20, Dic5.Q8, D5×C4⋊C4, D20⋊8C4, D10.13D4, C4⋊D20, C4⋊C4⋊D5, C5×C42⋊2C2, C42.163D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, C23×D5, D5×C4○D4, D4⋊8D10, C42.163D10
►On 160 points
Generators in S160
(1 102 11 112)(2 160 12 150)(3 104 13 114)(4 142 14 152)(5 106 15 116)(6 144 16 154)(7 108 17 118)(8 146 18 156)(9 110 19 120)(10 148 20 158)(21 64 31 74)(22 137 32 127)(23 66 33 76)(24 139 34 129)(25 68 35 78)(26 121 36 131)(27 70 37 80)(28 123 38 133)(29 72 39 62)(30 125 40 135)(41 140 51 130)(42 69 52 79)(43 122 53 132)(44 71 54 61)(45 124 55 134)(46 73 56 63)(47 126 57 136)(48 75 58 65)(49 128 59 138)(50 77 60 67)(81 143 91 153)(82 107 92 117)(83 145 93 155)(84 109 94 119)(85 147 95 157)(86 111 96 101)(87 149 97 159)(88 113 98 103)(89 151 99 141)(90 115 100 105)
(1 124 87 62)(2 135 88 73)(3 126 89 64)(4 137 90 75)(5 128 91 66)(6 139 92 77)(7 130 93 68)(8 121 94 79)(9 132 95 70)(10 123 96 61)(11 134 97 72)(12 125 98 63)(13 136 99 74)(14 127 100 65)(15 138 81 76)(16 129 82 67)(17 140 83 78)(18 131 84 69)(19 122 85 80)(20 133 86 71)(21 114 47 141)(22 105 48 152)(23 116 49 143)(24 107 50 154)(25 118 51 145)(26 109 52 156)(27 120 53 147)(28 111 54 158)(29 102 55 149)(30 113 56 160)(31 104 57 151)(32 115 58 142)(33 106 59 153)(34 117 60 144)(35 108 41 155)(36 119 42 146)(37 110 43 157)(38 101 44 148)(39 112 45 159)(40 103 46 150)
(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 66 11 76)(2 75 12 65)(3 64 13 74)(4 73 14 63)(5 62 15 72)(6 71 16 61)(7 80 17 70)(8 69 18 79)(9 78 19 68)(10 67 20 77)(21 141 31 151)(22 150 32 160)(23 159 33 149)(24 148 34 158)(25 157 35 147)(26 146 36 156)(27 155 37 145)(28 144 38 154)(29 153 39 143)(30 142 40 152)(41 120 51 110)(42 109 52 119)(43 118 53 108)(44 107 54 117)(45 116 55 106)(46 105 56 115)(47 114 57 104)(48 103 58 113)(49 112 59 102)(50 101 60 111)(81 134 91 124)(82 123 92 133)(83 132 93 122)(84 121 94 131)(85 130 95 140)(86 139 96 129)(87 128 97 138)(88 137 98 127)(89 126 99 136)(90 135 100 125)
G:=sub<Sym(160)| (1,102,11,112)(2,160,12,150)(3,104,13,114)(4,142,14,152)(5,106,15,116)(6,144,16,154)(7,108,17,118)(8,146,18,156)(9,110,19,120)(10,148,20,158)(21,64,31,74)(22,137,32,127)(23,66,33,76)(24,139,34,129)(25,68,35,78)(26,121,36,131)(27,70,37,80)(28,123,38,133)(29,72,39,62)(30,125,40,135)(41,140,51,130)(42,69,52,79)(43,122,53,132)(44,71,54,61)(45,124,55,134)(46,73,56,63)(47,126,57,136)(48,75,58,65)(49,128,59,138)(50,77,60,67)(81,143,91,153)(82,107,92,117)(83,145,93,155)(84,109,94,119)(85,147,95,157)(86,111,96,101)(87,149,97,159)(88,113,98,103)(89,151,99,141)(90,115,100,105), (1,124,87,62)(2,135,88,73)(3,126,89,64)(4,137,90,75)(5,128,91,66)(6,139,92,77)(7,130,93,68)(8,121,94,79)(9,132,95,70)(10,123,96,61)(11,134,97,72)(12,125,98,63)(13,136,99,74)(14,127,100,65)(15,138,81,76)(16,129,82,67)(17,140,83,78)(18,131,84,69)(19,122,85,80)(20,133,86,71)(21,114,47,141)(22,105,48,152)(23,116,49,143)(24,107,50,154)(25,118,51,145)(26,109,52,156)(27,120,53,147)(28,111,54,158)(29,102,55,149)(30,113,56,160)(31,104,57,151)(32,115,58,142)(33,106,59,153)(34,117,60,144)(35,108,41,155)(36,119,42,146)(37,110,43,157)(38,101,44,148)(39,112,45,159)(40,103,46,150), (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), (1,66,11,76)(2,75,12,65)(3,64,13,74)(4,73,14,63)(5,62,15,72)(6,71,16,61)(7,80,17,70)(8,69,18,79)(9,78,19,68)(10,67,20,77)(21,141,31,151)(22,150,32,160)(23,159,33,149)(24,148,34,158)(25,157,35,147)(26,146,36,156)(27,155,37,145)(28,144,38,154)(29,153,39,143)(30,142,40,152)(41,120,51,110)(42,109,52,119)(43,118,53,108)(44,107,54,117)(45,116,55,106)(46,105,56,115)(47,114,57,104)(48,103,58,113)(49,112,59,102)(50,101,60,111)(81,134,91,124)(82,123,92,133)(83,132,93,122)(84,121,94,131)(85,130,95,140)(86,139,96,129)(87,128,97,138)(88,137,98,127)(89,126,99,136)(90,135,100,125)>;
G:=Group( (1,102,11,112)(2,160,12,150)(3,104,13,114)(4,142,14,152)(5,106,15,116)(6,144,16,154)(7,108,17,118)(8,146,18,156)(9,110,19,120)(10,148,20,158)(21,64,31,74)(22,137,32,127)(23,66,33,76)(24,139,34,129)(25,68,35,78)(26,121,36,131)(27,70,37,80)(28,123,38,133)(29,72,39,62)(30,125,40,135)(41,140,51,130)(42,69,52,79)(43,122,53,132)(44,71,54,61)(45,124,55,134)(46,73,56,63)(47,126,57,136)(48,75,58,65)(49,128,59,138)(50,77,60,67)(81,143,91,153)(82,107,92,117)(83,145,93,155)(84,109,94,119)(85,147,95,157)(86,111,96,101)(87,149,97,159)(88,113,98,103)(89,151,99,141)(90,115,100,105), (1,124,87,62)(2,135,88,73)(3,126,89,64)(4,137,90,75)(5,128,91,66)(6,139,92,77)(7,130,93,68)(8,121,94,79)(9,132,95,70)(10,123,96,61)(11,134,97,72)(12,125,98,63)(13,136,99,74)(14,127,100,65)(15,138,81,76)(16,129,82,67)(17,140,83,78)(18,131,84,69)(19,122,85,80)(20,133,86,71)(21,114,47,141)(22,105,48,152)(23,116,49,143)(24,107,50,154)(25,118,51,145)(26,109,52,156)(27,120,53,147)(28,111,54,158)(29,102,55,149)(30,113,56,160)(31,104,57,151)(32,115,58,142)(33,106,59,153)(34,117,60,144)(35,108,41,155)(36,119,42,146)(37,110,43,157)(38,101,44,148)(39,112,45,159)(40,103,46,150), (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), (1,66,11,76)(2,75,12,65)(3,64,13,74)(4,73,14,63)(5,62,15,72)(6,71,16,61)(7,80,17,70)(8,69,18,79)(9,78,19,68)(10,67,20,77)(21,141,31,151)(22,150,32,160)(23,159,33,149)(24,148,34,158)(25,157,35,147)(26,146,36,156)(27,155,37,145)(28,144,38,154)(29,153,39,143)(30,142,40,152)(41,120,51,110)(42,109,52,119)(43,118,53,108)(44,107,54,117)(45,116,55,106)(46,105,56,115)(47,114,57,104)(48,103,58,113)(49,112,59,102)(50,101,60,111)(81,134,91,124)(82,123,92,133)(83,132,93,122)(84,121,94,131)(85,130,95,140)(86,139,96,129)(87,128,97,138)(88,137,98,127)(89,126,99,136)(90,135,100,125) );
G=PermutationGroup([[(1,102,11,112),(2,160,12,150),(3,104,13,114),(4,142,14,152),(5,106,15,116),(6,144,16,154),(7,108,17,118),(8,146,18,156),(9,110,19,120),(10,148,20,158),(21,64,31,74),(22,137,32,127),(23,66,33,76),(24,139,34,129),(25,68,35,78),(26,121,36,131),(27,70,37,80),(28,123,38,133),(29,72,39,62),(30,125,40,135),(41,140,51,130),(42,69,52,79),(43,122,53,132),(44,71,54,61),(45,124,55,134),(46,73,56,63),(47,126,57,136),(48,75,58,65),(49,128,59,138),(50,77,60,67),(81,143,91,153),(82,107,92,117),(83,145,93,155),(84,109,94,119),(85,147,95,157),(86,111,96,101),(87,149,97,159),(88,113,98,103),(89,151,99,141),(90,115,100,105)], [(1,124,87,62),(2,135,88,73),(3,126,89,64),(4,137,90,75),(5,128,91,66),(6,139,92,77),(7,130,93,68),(8,121,94,79),(9,132,95,70),(10,123,96,61),(11,134,97,72),(12,125,98,63),(13,136,99,74),(14,127,100,65),(15,138,81,76),(16,129,82,67),(17,140,83,78),(18,131,84,69),(19,122,85,80),(20,133,86,71),(21,114,47,141),(22,105,48,152),(23,116,49,143),(24,107,50,154),(25,118,51,145),(26,109,52,156),(27,120,53,147),(28,111,54,158),(29,102,55,149),(30,113,56,160),(31,104,57,151),(32,115,58,142),(33,106,59,153),(34,117,60,144),(35,108,41,155),(36,119,42,146),(37,110,43,157),(38,101,44,148),(39,112,45,159),(40,103,46,150)], [(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)], [(1,66,11,76),(2,75,12,65),(3,64,13,74),(4,73,14,63),(5,62,15,72),(6,71,16,61),(7,80,17,70),(8,69,18,79),(9,78,19,68),(10,67,20,77),(21,141,31,151),(22,150,32,160),(23,159,33,149),(24,148,34,158),(25,157,35,147),(26,146,36,156),(27,155,37,145),(28,144,38,154),(29,153,39,143),(30,142,40,152),(41,120,51,110),(42,109,52,119),(43,118,53,108),(44,107,54,117),(45,116,55,106),(46,105,56,115),(47,114,57,104),(48,103,58,113),(49,112,59,102),(50,101,60,111),(81,134,91,124),(82,123,92,133),(83,132,93,122),(84,121,94,131),(85,130,95,140),(86,139,96,129),(87,128,97,138),(88,137,98,127),(89,126,99,136),(90,135,100,125)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | 2+ 1+4 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.163D10 | C42⋊D5 | C4×D20 | Dic5⋊4D4 | D10⋊D4 | C22.D20 | Dic5.Q8 | D5×C4⋊C4 | D20⋊8C4 | D10.13D4 | C4⋊D20 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | D10 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 6 | 6 | 1 | 8 | 4 |
Matrix representation of C42.163D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 18 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 21 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 5 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 40 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,18,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,21,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,16,0,0,0,0,5,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,40,0,0,0,0,2,1] >;
C42.163D10 in GAP, Magma, Sage, TeX
C_4^2._{163}D_{10} % in TeX
G:=Group("C4^2.163D10"); // GroupNames label
G:=SmallGroup(320,1381);
// by ID
G=gap.SmallGroup(320,1381);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,184,1571,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations