metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.162D10, C10.1012- 1+4, C4⋊C4.213D10, C42⋊2C2⋊5D5, C42⋊D5⋊7C2, D10⋊Q8⋊44C2, (C4×Dic10)⋊15C2, (C4×C20).34C22, C22⋊C4.80D10, C4.Dic10⋊40C2, D10.41(C4○D4), (C2×C10).252C24, (C2×C20).604C23, Dic5⋊4D4.5C2, C23.58(C22×D5), Dic5.48(C4○D4), Dic5.Q8⋊38C2, D10.12D4.4C2, C23.D10⋊46C2, C4⋊Dic5.247C22, (C22×C10).66C23, C22.273(C23×D5), Dic5.14D4⋊46C2, C23.D5.68C22, C23.11D10⋊22C2, (C4×Dic5).238C22, (C2×Dic5).130C23, (C22×D5).236C23, C2.65(D4.10D10), D10⋊C4.140C22, C5⋊11(C22.46C24), (C2×Dic10).263C22, C10.D4.146C22, (C22×Dic5).152C22, (D5×C4⋊C4)⋊42C2, C2.99(D5×C4○D4), C4⋊C4⋊7D5⋊41C2, (C5×C42⋊2C2)⋊7C2, C10.210(C2×C4○D4), (C2×C4×D5).271C22, (C2×C4).88(C22×D5), (C5×C4⋊C4).204C22, (C2×C5⋊D4).72C22, (C5×C22⋊C4).77C22, SmallGroup(320,1380)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.162D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >
Subgroups: 678 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C22.46C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×Dic10, C42⋊D5, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊4D4, D10.12D4, Dic5.Q8, C4.Dic10, D5×C4⋊C4, C4⋊C4⋊7D5, D10⋊Q8, C5×C42⋊2C2, C42.162D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, C23×D5, D5×C4○D4, D4.10D10, C42.162D10
►On 160 points
Generators in S160
(1 127 42 70)(2 138 43 61)(3 129 44 72)(4 140 45 63)(5 131 46 74)(6 122 47 65)(7 133 48 76)(8 124 49 67)(9 135 50 78)(10 126 51 69)(11 137 52 80)(12 128 53 71)(13 139 54 62)(14 130 55 73)(15 121 56 64)(16 132 57 75)(17 123 58 66)(18 134 59 77)(19 125 60 68)(20 136 41 79)(21 119 91 145)(22 110 92 156)(23 101 93 147)(24 112 94 158)(25 103 95 149)(26 114 96 160)(27 105 97 151)(28 116 98 142)(29 107 99 153)(30 118 100 144)(31 109 81 155)(32 120 82 146)(33 111 83 157)(34 102 84 148)(35 113 85 159)(36 104 86 150)(37 115 87 141)(38 106 88 152)(39 117 89 143)(40 108 90 154)
(1 38 11 28)(2 99 12 89)(3 40 13 30)(4 81 14 91)(5 22 15 32)(6 83 16 93)(7 24 17 34)(8 85 18 95)(9 26 19 36)(10 87 20 97)(21 45 31 55)(23 47 33 57)(25 49 35 59)(27 51 37 41)(29 53 39 43)(42 88 52 98)(44 90 54 100)(46 92 56 82)(48 94 58 84)(50 96 60 86)(61 107 71 117)(62 144 72 154)(63 109 73 119)(64 146 74 156)(65 111 75 101)(66 148 76 158)(67 113 77 103)(68 150 78 160)(69 115 79 105)(70 152 80 142)(102 133 112 123)(104 135 114 125)(106 137 116 127)(108 139 118 129)(110 121 120 131)(122 157 132 147)(124 159 134 149)(126 141 136 151)(128 143 138 153)(130 145 140 155)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 84 31 94)(22 93 32 83)(23 82 33 92)(24 91 34 81)(25 100 35 90)(26 89 36 99)(27 98 37 88)(28 87 38 97)(29 96 39 86)(30 85 40 95)(41 42 51 52)(43 60 53 50)(44 49 54 59)(45 58 55 48)(46 47 56 57)(61 78 71 68)(62 67 72 77)(63 76 73 66)(64 65 74 75)(69 70 79 80)(101 156 111 146)(102 145 112 155)(103 154 113 144)(104 143 114 153)(105 152 115 142)(106 141 116 151)(107 150 117 160)(108 159 118 149)(109 148 119 158)(110 157 120 147)(121 122 131 132)(123 140 133 130)(124 129 134 139)(125 138 135 128)(126 127 136 137)
G:=sub<Sym(160)| (1,127,42,70)(2,138,43,61)(3,129,44,72)(4,140,45,63)(5,131,46,74)(6,122,47,65)(7,133,48,76)(8,124,49,67)(9,135,50,78)(10,126,51,69)(11,137,52,80)(12,128,53,71)(13,139,54,62)(14,130,55,73)(15,121,56,64)(16,132,57,75)(17,123,58,66)(18,134,59,77)(19,125,60,68)(20,136,41,79)(21,119,91,145)(22,110,92,156)(23,101,93,147)(24,112,94,158)(25,103,95,149)(26,114,96,160)(27,105,97,151)(28,116,98,142)(29,107,99,153)(30,118,100,144)(31,109,81,155)(32,120,82,146)(33,111,83,157)(34,102,84,148)(35,113,85,159)(36,104,86,150)(37,115,87,141)(38,106,88,152)(39,117,89,143)(40,108,90,154), (1,38,11,28)(2,99,12,89)(3,40,13,30)(4,81,14,91)(5,22,15,32)(6,83,16,93)(7,24,17,34)(8,85,18,95)(9,26,19,36)(10,87,20,97)(21,45,31,55)(23,47,33,57)(25,49,35,59)(27,51,37,41)(29,53,39,43)(42,88,52,98)(44,90,54,100)(46,92,56,82)(48,94,58,84)(50,96,60,86)(61,107,71,117)(62,144,72,154)(63,109,73,119)(64,146,74,156)(65,111,75,101)(66,148,76,158)(67,113,77,103)(68,150,78,160)(69,115,79,105)(70,152,80,142)(102,133,112,123)(104,135,114,125)(106,137,116,127)(108,139,118,129)(110,121,120,131)(122,157,132,147)(124,159,134,149)(126,141,136,151)(128,143,138,153)(130,145,140,155), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,84,31,94)(22,93,32,83)(23,82,33,92)(24,91,34,81)(25,100,35,90)(26,89,36,99)(27,98,37,88)(28,87,38,97)(29,96,39,86)(30,85,40,95)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57)(61,78,71,68)(62,67,72,77)(63,76,73,66)(64,65,74,75)(69,70,79,80)(101,156,111,146)(102,145,112,155)(103,154,113,144)(104,143,114,153)(105,152,115,142)(106,141,116,151)(107,150,117,160)(108,159,118,149)(109,148,119,158)(110,157,120,147)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137)>;
G:=Group( (1,127,42,70)(2,138,43,61)(3,129,44,72)(4,140,45,63)(5,131,46,74)(6,122,47,65)(7,133,48,76)(8,124,49,67)(9,135,50,78)(10,126,51,69)(11,137,52,80)(12,128,53,71)(13,139,54,62)(14,130,55,73)(15,121,56,64)(16,132,57,75)(17,123,58,66)(18,134,59,77)(19,125,60,68)(20,136,41,79)(21,119,91,145)(22,110,92,156)(23,101,93,147)(24,112,94,158)(25,103,95,149)(26,114,96,160)(27,105,97,151)(28,116,98,142)(29,107,99,153)(30,118,100,144)(31,109,81,155)(32,120,82,146)(33,111,83,157)(34,102,84,148)(35,113,85,159)(36,104,86,150)(37,115,87,141)(38,106,88,152)(39,117,89,143)(40,108,90,154), (1,38,11,28)(2,99,12,89)(3,40,13,30)(4,81,14,91)(5,22,15,32)(6,83,16,93)(7,24,17,34)(8,85,18,95)(9,26,19,36)(10,87,20,97)(21,45,31,55)(23,47,33,57)(25,49,35,59)(27,51,37,41)(29,53,39,43)(42,88,52,98)(44,90,54,100)(46,92,56,82)(48,94,58,84)(50,96,60,86)(61,107,71,117)(62,144,72,154)(63,109,73,119)(64,146,74,156)(65,111,75,101)(66,148,76,158)(67,113,77,103)(68,150,78,160)(69,115,79,105)(70,152,80,142)(102,133,112,123)(104,135,114,125)(106,137,116,127)(108,139,118,129)(110,121,120,131)(122,157,132,147)(124,159,134,149)(126,141,136,151)(128,143,138,153)(130,145,140,155), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,84,31,94)(22,93,32,83)(23,82,33,92)(24,91,34,81)(25,100,35,90)(26,89,36,99)(27,98,37,88)(28,87,38,97)(29,96,39,86)(30,85,40,95)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57)(61,78,71,68)(62,67,72,77)(63,76,73,66)(64,65,74,75)(69,70,79,80)(101,156,111,146)(102,145,112,155)(103,154,113,144)(104,143,114,153)(105,152,115,142)(106,141,116,151)(107,150,117,160)(108,159,118,149)(109,148,119,158)(110,157,120,147)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137) );
G=PermutationGroup([[(1,127,42,70),(2,138,43,61),(3,129,44,72),(4,140,45,63),(5,131,46,74),(6,122,47,65),(7,133,48,76),(8,124,49,67),(9,135,50,78),(10,126,51,69),(11,137,52,80),(12,128,53,71),(13,139,54,62),(14,130,55,73),(15,121,56,64),(16,132,57,75),(17,123,58,66),(18,134,59,77),(19,125,60,68),(20,136,41,79),(21,119,91,145),(22,110,92,156),(23,101,93,147),(24,112,94,158),(25,103,95,149),(26,114,96,160),(27,105,97,151),(28,116,98,142),(29,107,99,153),(30,118,100,144),(31,109,81,155),(32,120,82,146),(33,111,83,157),(34,102,84,148),(35,113,85,159),(36,104,86,150),(37,115,87,141),(38,106,88,152),(39,117,89,143),(40,108,90,154)], [(1,38,11,28),(2,99,12,89),(3,40,13,30),(4,81,14,91),(5,22,15,32),(6,83,16,93),(7,24,17,34),(8,85,18,95),(9,26,19,36),(10,87,20,97),(21,45,31,55),(23,47,33,57),(25,49,35,59),(27,51,37,41),(29,53,39,43),(42,88,52,98),(44,90,54,100),(46,92,56,82),(48,94,58,84),(50,96,60,86),(61,107,71,117),(62,144,72,154),(63,109,73,119),(64,146,74,156),(65,111,75,101),(66,148,76,158),(67,113,77,103),(68,150,78,160),(69,115,79,105),(70,152,80,142),(102,133,112,123),(104,135,114,125),(106,137,116,127),(108,139,118,129),(110,121,120,131),(122,157,132,147),(124,159,134,149),(126,141,136,151),(128,143,138,153),(130,145,140,155)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,84,31,94),(22,93,32,83),(23,82,33,92),(24,91,34,81),(25,100,35,90),(26,89,36,99),(27,98,37,88),(28,87,38,97),(29,96,39,86),(30,85,40,95),(41,42,51,52),(43,60,53,50),(44,49,54,59),(45,58,55,48),(46,47,56,57),(61,78,71,68),(62,67,72,77),(63,76,73,66),(64,65,74,75),(69,70,79,80),(101,156,111,146),(102,145,112,155),(103,154,113,144),(104,143,114,153),(105,152,115,142),(106,141,116,151),(107,150,117,160),(108,159,118,149),(109,148,119,158),(110,157,120,147),(121,122,131,132),(123,140,133,130),(124,129,134,139),(125,138,135,128),(126,127,136,137)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 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 | 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 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | 2- 1+4 | D5×C4○D4 | D4.10D10 |
| kernel | C42.162D10 | C4×Dic10 | C42⋊D5 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.Q8 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10⋊Q8 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | D10 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 6 | 6 | 1 | 8 | 4 |
Matrix representation of C42.162D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 23 |
| 0 | 0 | 0 | 0 | 9 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 22 |
| 0 | 0 | 0 | 0 | 40 | 29 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 22 |
| 0 | 0 | 0 | 0 | 40 | 29 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,9,0,0,0,0,23,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,12,40,0,0,0,0,22,29],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,40,0,0,0,0,22,29] >;
C42.162D10 in GAP, Magma, Sage, TeX
C_4^2._{162}D_{10} % in TeX
G:=Group("C4^2.162D10"); // GroupNames label
G:=SmallGroup(320,1380);
// by ID
G=gap.SmallGroup(320,1380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations