metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.161D10, C10.1002- 1+4, C10.1392+ 1+4, (C4×D20)⋊15C2, C4⋊C4.118D10, C42⋊2C2⋊4D5, C20.6Q8⋊9C2, D10⋊Q8⋊43C2, D10⋊2Q8⋊41C2, D10⋊D4.4C2, (C4×C20).33C22, C22⋊C4.79D10, C4.Dic10⋊39C2, Dic5⋊4D4⋊36C2, D10.20(C4○D4), (C2×C20).194C23, (C2×C10).251C24, C4⋊Dic5.54C22, D10.12D4⋊51C2, D10.13D4⋊41C2, C2.64(D4⋊8D10), C23.57(C22×D5), (C2×D20).235C22, C22.D20⋊29C2, (C22×C10).65C23, C22.272(C23×D5), Dic5.14D4⋊45C2, C23.D5.67C22, C5⋊9(C22.33C24), (C2×Dic5).275C23, (C4×Dic5).159C22, C10.D4.56C22, (C22×D5).235C23, C2.64(D4.10D10), D10⋊C4.114C22, (C2×Dic10).190C22, (C22×Dic5).151C22, (D5×C4⋊C4)⋊41C2, C2.98(D5×C4○D4), C4⋊C4⋊D5⋊42C2, (C5×C42⋊2C2)⋊6C2, C10.209(C2×C4○D4), (C2×C4×D5).270C22, (C5×C4⋊C4).203C22, (C2×C4).209(C22×D5), (C2×C5⋊D4).71C22, (C5×C22⋊C4).76C22, SmallGroup(320,1379)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.161D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 798 in 218 conjugacy classes, 93 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, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C22.33C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C20.6Q8, C4×D20, Dic5.14D4, Dic5⋊4D4, D10.12D4, D10⋊D4, C22.D20, C4.Dic10, D5×C4⋊C4, D10.13D4, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42.161D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.33C24, C23×D5, D5×C4○D4, D4⋊8D10, D4.10D10, C42.161D10
►On 160 points
Generators in S160
(1 60 69 147)(2 51 70 158)(3 42 71 149)(4 53 72 160)(5 44 73 151)(6 55 74 142)(7 46 75 153)(8 57 76 144)(9 48 77 155)(10 59 78 146)(11 50 79 157)(12 41 80 148)(13 52 61 159)(14 43 62 150)(15 54 63 141)(16 45 64 152)(17 56 65 143)(18 47 66 154)(19 58 67 145)(20 49 68 156)(21 110 95 132)(22 101 96 123)(23 112 97 134)(24 103 98 125)(25 114 99 136)(26 105 100 127)(27 116 81 138)(28 107 82 129)(29 118 83 140)(30 109 84 131)(31 120 85 122)(32 111 86 133)(33 102 87 124)(34 113 88 135)(35 104 89 126)(36 115 90 137)(37 106 91 128)(38 117 92 139)(39 108 93 130)(40 119 94 121)
(1 124 11 134)(2 113 12 103)(3 126 13 136)(4 115 14 105)(5 128 15 138)(6 117 16 107)(7 130 17 140)(8 119 18 109)(9 132 19 122)(10 101 20 111)(21 58 31 48)(22 156 32 146)(23 60 33 50)(24 158 34 148)(25 42 35 52)(26 160 36 150)(27 44 37 54)(28 142 38 152)(29 46 39 56)(30 144 40 154)(41 98 51 88)(43 100 53 90)(45 82 55 92)(47 84 57 94)(49 86 59 96)(61 114 71 104)(62 127 72 137)(63 116 73 106)(64 129 74 139)(65 118 75 108)(66 131 76 121)(67 120 77 110)(68 133 78 123)(69 102 79 112)(70 135 80 125)(81 151 91 141)(83 153 93 143)(85 155 95 145)(87 157 97 147)(89 159 99 149)
(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)
(1 123 11 133)(2 132 12 122)(3 121 13 131)(4 130 14 140)(5 139 15 129)(6 128 16 138)(7 137 17 127)(8 126 18 136)(9 135 19 125)(10 124 20 134)(21 51 31 41)(22 60 32 50)(23 49 33 59)(24 58 34 48)(25 47 35 57)(26 56 36 46)(27 45 37 55)(28 54 38 44)(29 43 39 53)(30 52 40 42)(61 109 71 119)(62 118 72 108)(63 107 73 117)(64 116 74 106)(65 105 75 115)(66 114 76 104)(67 103 77 113)(68 112 78 102)(69 101 79 111)(70 110 80 120)(81 152 91 142)(82 141 92 151)(83 150 93 160)(84 159 94 149)(85 148 95 158)(86 157 96 147)(87 146 97 156)(88 155 98 145)(89 144 99 154)(90 153 100 143)
G:=sub<Sym(160)| (1,60,69,147)(2,51,70,158)(3,42,71,149)(4,53,72,160)(5,44,73,151)(6,55,74,142)(7,46,75,153)(8,57,76,144)(9,48,77,155)(10,59,78,146)(11,50,79,157)(12,41,80,148)(13,52,61,159)(14,43,62,150)(15,54,63,141)(16,45,64,152)(17,56,65,143)(18,47,66,154)(19,58,67,145)(20,49,68,156)(21,110,95,132)(22,101,96,123)(23,112,97,134)(24,103,98,125)(25,114,99,136)(26,105,100,127)(27,116,81,138)(28,107,82,129)(29,118,83,140)(30,109,84,131)(31,120,85,122)(32,111,86,133)(33,102,87,124)(34,113,88,135)(35,104,89,126)(36,115,90,137)(37,106,91,128)(38,117,92,139)(39,108,93,130)(40,119,94,121), (1,124,11,134)(2,113,12,103)(3,126,13,136)(4,115,14,105)(5,128,15,138)(6,117,16,107)(7,130,17,140)(8,119,18,109)(9,132,19,122)(10,101,20,111)(21,58,31,48)(22,156,32,146)(23,60,33,50)(24,158,34,148)(25,42,35,52)(26,160,36,150)(27,44,37,54)(28,142,38,152)(29,46,39,56)(30,144,40,154)(41,98,51,88)(43,100,53,90)(45,82,55,92)(47,84,57,94)(49,86,59,96)(61,114,71,104)(62,127,72,137)(63,116,73,106)(64,129,74,139)(65,118,75,108)(66,131,76,121)(67,120,77,110)(68,133,78,123)(69,102,79,112)(70,135,80,125)(81,151,91,141)(83,153,93,143)(85,155,95,145)(87,157,97,147)(89,159,99,149), (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), (1,123,11,133)(2,132,12,122)(3,121,13,131)(4,130,14,140)(5,139,15,129)(6,128,16,138)(7,137,17,127)(8,126,18,136)(9,135,19,125)(10,124,20,134)(21,51,31,41)(22,60,32,50)(23,49,33,59)(24,58,34,48)(25,47,35,57)(26,56,36,46)(27,45,37,55)(28,54,38,44)(29,43,39,53)(30,52,40,42)(61,109,71,119)(62,118,72,108)(63,107,73,117)(64,116,74,106)(65,105,75,115)(66,114,76,104)(67,103,77,113)(68,112,78,102)(69,101,79,111)(70,110,80,120)(81,152,91,142)(82,141,92,151)(83,150,93,160)(84,159,94,149)(85,148,95,158)(86,157,96,147)(87,146,97,156)(88,155,98,145)(89,144,99,154)(90,153,100,143)>;
G:=Group( (1,60,69,147)(2,51,70,158)(3,42,71,149)(4,53,72,160)(5,44,73,151)(6,55,74,142)(7,46,75,153)(8,57,76,144)(9,48,77,155)(10,59,78,146)(11,50,79,157)(12,41,80,148)(13,52,61,159)(14,43,62,150)(15,54,63,141)(16,45,64,152)(17,56,65,143)(18,47,66,154)(19,58,67,145)(20,49,68,156)(21,110,95,132)(22,101,96,123)(23,112,97,134)(24,103,98,125)(25,114,99,136)(26,105,100,127)(27,116,81,138)(28,107,82,129)(29,118,83,140)(30,109,84,131)(31,120,85,122)(32,111,86,133)(33,102,87,124)(34,113,88,135)(35,104,89,126)(36,115,90,137)(37,106,91,128)(38,117,92,139)(39,108,93,130)(40,119,94,121), (1,124,11,134)(2,113,12,103)(3,126,13,136)(4,115,14,105)(5,128,15,138)(6,117,16,107)(7,130,17,140)(8,119,18,109)(9,132,19,122)(10,101,20,111)(21,58,31,48)(22,156,32,146)(23,60,33,50)(24,158,34,148)(25,42,35,52)(26,160,36,150)(27,44,37,54)(28,142,38,152)(29,46,39,56)(30,144,40,154)(41,98,51,88)(43,100,53,90)(45,82,55,92)(47,84,57,94)(49,86,59,96)(61,114,71,104)(62,127,72,137)(63,116,73,106)(64,129,74,139)(65,118,75,108)(66,131,76,121)(67,120,77,110)(68,133,78,123)(69,102,79,112)(70,135,80,125)(81,151,91,141)(83,153,93,143)(85,155,95,145)(87,157,97,147)(89,159,99,149), (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), (1,123,11,133)(2,132,12,122)(3,121,13,131)(4,130,14,140)(5,139,15,129)(6,128,16,138)(7,137,17,127)(8,126,18,136)(9,135,19,125)(10,124,20,134)(21,51,31,41)(22,60,32,50)(23,49,33,59)(24,58,34,48)(25,47,35,57)(26,56,36,46)(27,45,37,55)(28,54,38,44)(29,43,39,53)(30,52,40,42)(61,109,71,119)(62,118,72,108)(63,107,73,117)(64,116,74,106)(65,105,75,115)(66,114,76,104)(67,103,77,113)(68,112,78,102)(69,101,79,111)(70,110,80,120)(81,152,91,142)(82,141,92,151)(83,150,93,160)(84,159,94,149)(85,148,95,158)(86,157,96,147)(87,146,97,156)(88,155,98,145)(89,144,99,154)(90,153,100,143) );
G=PermutationGroup([[(1,60,69,147),(2,51,70,158),(3,42,71,149),(4,53,72,160),(5,44,73,151),(6,55,74,142),(7,46,75,153),(8,57,76,144),(9,48,77,155),(10,59,78,146),(11,50,79,157),(12,41,80,148),(13,52,61,159),(14,43,62,150),(15,54,63,141),(16,45,64,152),(17,56,65,143),(18,47,66,154),(19,58,67,145),(20,49,68,156),(21,110,95,132),(22,101,96,123),(23,112,97,134),(24,103,98,125),(25,114,99,136),(26,105,100,127),(27,116,81,138),(28,107,82,129),(29,118,83,140),(30,109,84,131),(31,120,85,122),(32,111,86,133),(33,102,87,124),(34,113,88,135),(35,104,89,126),(36,115,90,137),(37,106,91,128),(38,117,92,139),(39,108,93,130),(40,119,94,121)], [(1,124,11,134),(2,113,12,103),(3,126,13,136),(4,115,14,105),(5,128,15,138),(6,117,16,107),(7,130,17,140),(8,119,18,109),(9,132,19,122),(10,101,20,111),(21,58,31,48),(22,156,32,146),(23,60,33,50),(24,158,34,148),(25,42,35,52),(26,160,36,150),(27,44,37,54),(28,142,38,152),(29,46,39,56),(30,144,40,154),(41,98,51,88),(43,100,53,90),(45,82,55,92),(47,84,57,94),(49,86,59,96),(61,114,71,104),(62,127,72,137),(63,116,73,106),(64,129,74,139),(65,118,75,108),(66,131,76,121),(67,120,77,110),(68,133,78,123),(69,102,79,112),(70,135,80,125),(81,151,91,141),(83,153,93,143),(85,155,95,145),(87,157,97,147),(89,159,99,149)], [(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)], [(1,123,11,133),(2,132,12,122),(3,121,13,131),(4,130,14,140),(5,139,15,129),(6,128,16,138),(7,137,17,127),(8,126,18,136),(9,135,19,125),(10,124,20,134),(21,51,31,41),(22,60,32,50),(23,49,33,59),(24,58,34,48),(25,47,35,57),(26,56,36,46),(27,45,37,55),(28,54,38,44),(29,43,39,53),(30,52,40,42),(61,109,71,119),(62,118,72,108),(63,107,73,117),(64,116,74,106),(65,105,75,115),(66,114,76,104),(67,103,77,113),(68,112,78,102),(69,101,79,111),(70,110,80,120),(81,152,91,142),(82,141,92,151),(83,150,93,160),(84,159,94,149),(85,148,95,158),(86,157,96,147),(87,146,97,156),(88,155,98,145),(89,144,99,154),(90,153,100,143)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D5×C4○D4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.161D10 | C20.6Q8 | C4×D20 | Dic5.14D4 | Dic5⋊4D4 | D10.12D4 | D10⋊D4 | C22.D20 | C4.Dic10 | D5×C4⋊C4 | D10.13D4 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | D10 | C42 | C22⋊C4 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 6 | 6 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.161D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 39 | 0 |
| 0 | 0 | 0 | 1 | 0 | 39 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 40 | 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 |
| 33 | 2 | 0 | 0 | 0 | 0 |
| 30 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 13 | 13 |
| 0 | 0 | 26 | 33 | 28 | 4 |
| 0 | 0 | 1 | 1 | 26 | 26 |
| 0 | 0 | 40 | 35 | 15 | 8 |
| 33 | 2 | 0 | 0 | 0 | 0 |
| 30 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 30 | 15 | 26 |
| 0 | 0 | 35 | 30 | 3 | 26 |
| 0 | 0 | 39 | 2 | 30 | 11 |
| 0 | 0 | 16 | 2 | 6 | 11 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,39,0,40,0,0,0,0,39,0,40],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[33,30,0,0,0,0,2,8,0,0,0,0,0,0,15,26,1,40,0,0,15,33,1,35,0,0,13,28,26,15,0,0,13,4,26,8],[33,30,0,0,0,0,2,8,0,0,0,0,0,0,11,35,39,16,0,0,30,30,2,2,0,0,15,3,30,6,0,0,26,26,11,11] >;
C42.161D10 in GAP, Magma, Sage, TeX
C_4^2._{161}D_{10} % in TeX
G:=Group("C4^2.161D10"); // GroupNames label
G:=SmallGroup(320,1379);
// by ID
G=gap.SmallGroup(320,1379);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,80,12550]);
// 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=a^2*b,d*c*d^-1=c^9>;
// generators/relations