metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.141D10, C10.902- 1+4, C4.33(D4×D5), (C4×D5).12D4, C20.62(C2×D4), C4.4D4⋊9D5, C20⋊2Q8⋊30C2, D10.80(C2×D4), (C2×D4).172D10, C42⋊D5⋊20C2, (C2×C20).80C23, (C2×Q8).136D10, C22⋊C4.35D10, Dic5.91(C2×D4), C10.89(C22×D4), Dic5⋊Q8⋊24C2, C20.17D4⋊24C2, (C2×C10).219C24, (C4×C20).185C22, C4⋊Dic5.51C22, D10.12D4⋊41C2, C23.41(C22×D5), (D4×C10).154C22, (C22×C10).49C23, (Q8×C10).126C22, C22.240(C23×D5), Dic5.14D4⋊40C2, C23.D5.54C22, C5⋊4(C23.38C23), (C4×Dic5).141C22, (C2×Dic5).114C23, (C22×D5).224C23, C2.51(D4.10D10), D10⋊C4.110C22, (C2×Dic10).183C22, C10.D4.120C22, (C22×Dic5).142C22, (C2×Q8×D5)⋊10C2, C2.62(C2×D4×D5), (C5×C4.4D4)⋊11C2, (C2×C4×D5).129C22, (C2×D4⋊2D5).11C2, (C2×C4).194(C22×D5), (C2×C5⋊D4).59C22, (C5×C22⋊C4).64C22, SmallGroup(320,1347)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.141D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >
Subgroups: 926 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C23.38C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, D4⋊2D5, Q8×D5, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C20⋊2Q8, C42⋊D5, Dic5.14D4, D10.12D4, C20.17D4, Dic5⋊Q8, C5×C4.4D4, C2×D4⋊2D5, C2×Q8×D5, C42.141D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2- 1+4, C22×D5, C23.38C23, D4×D5, C23×D5, C2×D4×D5, D4.10D10, C42.141D10
►On 160 points
Generators in S160
(1 37 11 27)(2 28 12 38)(3 39 13 29)(4 30 14 40)(5 21 15 31)(6 32 16 22)(7 23 17 33)(8 34 18 24)(9 25 19 35)(10 36 20 26)(41 87 51 97)(42 98 52 88)(43 89 53 99)(44 100 54 90)(45 91 55 81)(46 82 56 92)(47 93 57 83)(48 84 58 94)(49 95 59 85)(50 86 60 96)(61 123 71 133)(62 134 72 124)(63 125 73 135)(64 136 74 126)(65 127 75 137)(66 138 76 128)(67 129 77 139)(68 140 78 130)(69 131 79 121)(70 122 80 132)(101 141 111 151)(102 152 112 142)(103 143 113 153)(104 154 114 144)(105 145 115 155)(106 156 116 146)(107 147 117 157)(108 158 118 148)(109 149 119 159)(110 160 120 150)
(1 67 42 107)(2 118 43 78)(3 69 44 109)(4 120 45 80)(5 71 46 111)(6 102 47 62)(7 73 48 113)(8 104 49 64)(9 75 50 115)(10 106 51 66)(11 77 52 117)(12 108 53 68)(13 79 54 119)(14 110 55 70)(15 61 56 101)(16 112 57 72)(17 63 58 103)(18 114 59 74)(19 65 60 105)(20 116 41 76)(21 133 82 151)(22 142 83 124)(23 135 84 153)(24 144 85 126)(25 137 86 155)(26 146 87 128)(27 139 88 157)(28 148 89 130)(29 121 90 159)(30 150 91 132)(31 123 92 141)(32 152 93 134)(33 125 94 143)(34 154 95 136)(35 127 96 145)(36 156 97 138)(37 129 98 147)(38 158 99 140)(39 131 100 149)(40 160 81 122)
(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 22 31 32)(23 40 33 30)(24 29 34 39)(25 38 35 28)(26 27 36 37)(41 42 51 52)(43 60 53 50)(44 49 54 59)(45 58 55 48)(46 47 56 57)(61 72 71 62)(63 70 73 80)(64 79 74 69)(65 68 75 78)(66 77 76 67)(81 94 91 84)(82 83 92 93)(85 90 95 100)(86 99 96 89)(87 88 97 98)(101 112 111 102)(103 110 113 120)(104 119 114 109)(105 108 115 118)(106 117 116 107)(121 136 131 126)(122 125 132 135)(123 134 133 124)(127 130 137 140)(128 139 138 129)(141 152 151 142)(143 150 153 160)(144 159 154 149)(145 148 155 158)(146 157 156 147)
G:=sub<Sym(160)| (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,87,51,97)(42,98,52,88)(43,89,53,99)(44,100,54,90)(45,91,55,81)(46,82,56,92)(47,93,57,83)(48,84,58,94)(49,95,59,85)(50,86,60,96)(61,123,71,133)(62,134,72,124)(63,125,73,135)(64,136,74,126)(65,127,75,137)(66,138,76,128)(67,129,77,139)(68,140,78,130)(69,131,79,121)(70,122,80,132)(101,141,111,151)(102,152,112,142)(103,143,113,153)(104,154,114,144)(105,145,115,155)(106,156,116,146)(107,147,117,157)(108,158,118,148)(109,149,119,159)(110,160,120,150), (1,67,42,107)(2,118,43,78)(3,69,44,109)(4,120,45,80)(5,71,46,111)(6,102,47,62)(7,73,48,113)(8,104,49,64)(9,75,50,115)(10,106,51,66)(11,77,52,117)(12,108,53,68)(13,79,54,119)(14,110,55,70)(15,61,56,101)(16,112,57,72)(17,63,58,103)(18,114,59,74)(19,65,60,105)(20,116,41,76)(21,133,82,151)(22,142,83,124)(23,135,84,153)(24,144,85,126)(25,137,86,155)(26,146,87,128)(27,139,88,157)(28,148,89,130)(29,121,90,159)(30,150,91,132)(31,123,92,141)(32,152,93,134)(33,125,94,143)(34,154,95,136)(35,127,96,145)(36,156,97,138)(37,129,98,147)(38,158,99,140)(39,131,100,149)(40,160,81,122), (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)(21,22,31,32)(23,40,33,30)(24,29,34,39)(25,38,35,28)(26,27,36,37)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,94,91,84)(82,83,92,93)(85,90,95,100)(86,99,96,89)(87,88,97,98)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107)(121,136,131,126)(122,125,132,135)(123,134,133,124)(127,130,137,140)(128,139,138,129)(141,152,151,142)(143,150,153,160)(144,159,154,149)(145,148,155,158)(146,157,156,147)>;
G:=Group( (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,87,51,97)(42,98,52,88)(43,89,53,99)(44,100,54,90)(45,91,55,81)(46,82,56,92)(47,93,57,83)(48,84,58,94)(49,95,59,85)(50,86,60,96)(61,123,71,133)(62,134,72,124)(63,125,73,135)(64,136,74,126)(65,127,75,137)(66,138,76,128)(67,129,77,139)(68,140,78,130)(69,131,79,121)(70,122,80,132)(101,141,111,151)(102,152,112,142)(103,143,113,153)(104,154,114,144)(105,145,115,155)(106,156,116,146)(107,147,117,157)(108,158,118,148)(109,149,119,159)(110,160,120,150), (1,67,42,107)(2,118,43,78)(3,69,44,109)(4,120,45,80)(5,71,46,111)(6,102,47,62)(7,73,48,113)(8,104,49,64)(9,75,50,115)(10,106,51,66)(11,77,52,117)(12,108,53,68)(13,79,54,119)(14,110,55,70)(15,61,56,101)(16,112,57,72)(17,63,58,103)(18,114,59,74)(19,65,60,105)(20,116,41,76)(21,133,82,151)(22,142,83,124)(23,135,84,153)(24,144,85,126)(25,137,86,155)(26,146,87,128)(27,139,88,157)(28,148,89,130)(29,121,90,159)(30,150,91,132)(31,123,92,141)(32,152,93,134)(33,125,94,143)(34,154,95,136)(35,127,96,145)(36,156,97,138)(37,129,98,147)(38,158,99,140)(39,131,100,149)(40,160,81,122), (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)(21,22,31,32)(23,40,33,30)(24,29,34,39)(25,38,35,28)(26,27,36,37)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,94,91,84)(82,83,92,93)(85,90,95,100)(86,99,96,89)(87,88,97,98)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107)(121,136,131,126)(122,125,132,135)(123,134,133,124)(127,130,137,140)(128,139,138,129)(141,152,151,142)(143,150,153,160)(144,159,154,149)(145,148,155,158)(146,157,156,147) );
G=PermutationGroup([[(1,37,11,27),(2,28,12,38),(3,39,13,29),(4,30,14,40),(5,21,15,31),(6,32,16,22),(7,23,17,33),(8,34,18,24),(9,25,19,35),(10,36,20,26),(41,87,51,97),(42,98,52,88),(43,89,53,99),(44,100,54,90),(45,91,55,81),(46,82,56,92),(47,93,57,83),(48,84,58,94),(49,95,59,85),(50,86,60,96),(61,123,71,133),(62,134,72,124),(63,125,73,135),(64,136,74,126),(65,127,75,137),(66,138,76,128),(67,129,77,139),(68,140,78,130),(69,131,79,121),(70,122,80,132),(101,141,111,151),(102,152,112,142),(103,143,113,153),(104,154,114,144),(105,145,115,155),(106,156,116,146),(107,147,117,157),(108,158,118,148),(109,149,119,159),(110,160,120,150)], [(1,67,42,107),(2,118,43,78),(3,69,44,109),(4,120,45,80),(5,71,46,111),(6,102,47,62),(7,73,48,113),(8,104,49,64),(9,75,50,115),(10,106,51,66),(11,77,52,117),(12,108,53,68),(13,79,54,119),(14,110,55,70),(15,61,56,101),(16,112,57,72),(17,63,58,103),(18,114,59,74),(19,65,60,105),(20,116,41,76),(21,133,82,151),(22,142,83,124),(23,135,84,153),(24,144,85,126),(25,137,86,155),(26,146,87,128),(27,139,88,157),(28,148,89,130),(29,121,90,159),(30,150,91,132),(31,123,92,141),(32,152,93,134),(33,125,94,143),(34,154,95,136),(35,127,96,145),(36,156,97,138),(37,129,98,147),(38,158,99,140),(39,131,100,149),(40,160,81,122)], [(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),(21,22,31,32),(23,40,33,30),(24,29,34,39),(25,38,35,28),(26,27,36,37),(41,42,51,52),(43,60,53,50),(44,49,54,59),(45,58,55,48),(46,47,56,57),(61,72,71,62),(63,70,73,80),(64,79,74,69),(65,68,75,78),(66,77,76,67),(81,94,91,84),(82,83,92,93),(85,90,95,100),(86,99,96,89),(87,88,97,98),(101,112,111,102),(103,110,113,120),(104,119,114,109),(105,108,115,118),(106,117,116,107),(121,136,131,126),(122,125,132,135),(123,134,133,124),(127,130,137,140),(128,139,138,129),(141,152,151,142),(143,150,153,160),(144,159,154,149),(145,148,155,158),(146,157,156,147)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 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 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 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 | D4 | D5 | D10 | D10 | D10 | D10 | 2- 1+4 | D4×D5 | D4.10D10 |
| kernel | C42.141D10 | C20⋊2Q8 | C42⋊D5 | Dic5.14D4 | D10.12D4 | C20.17D4 | Dic5⋊Q8 | C5×C4.4D4 | C2×D4⋊2D5 | C2×Q8×D5 | C4×D5 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C42.141D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 34 | 0 |
| 0 | 0 | 0 | 14 | 0 | 34 |
| 0 | 0 | 34 | 0 | 27 | 0 |
| 0 | 0 | 0 | 34 | 0 | 27 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 32 | 17 | 0 | 0 | 0 | 0 |
| 17 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 9 | 24 | 0 | 0 | 0 | 0 |
| 24 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 40 | 34 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,34,0,0,0,0,14,0,34,0,0,34,0,27,0,0,0,0,34,0,27],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[32,17,0,0,0,0,17,9,0,0,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,7,34,0,0,0,0,7,40,0,0],[9,24,0,0,0,0,24,32,0,0,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,7,40,0,0,0,0,7,34,0,0] >;
C42.141D10 in GAP, Magma, Sage, TeX
C_4^2._{141}D_{10} % in TeX
G:=Group("C4^2.141D10"); // GroupNames label
G:=SmallGroup(320,1347);
// by ID
G=gap.SmallGroup(320,1347);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,192,12550]);
// 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,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations