metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.90D10, C10.922+ 1+4, C10.482- 1+4, (C2×C20)⋊5Q8, C20⋊Q8⋊11C2, C20⋊2Q8⋊6C2, (C2×C4)⋊4Dic10, C20.78(C2×Q8), C4⋊C4.268D10, (C4×C20).7C22, C20.6Q8⋊4C2, (C2×C10).63C24, C22⋊C4.91D10, C4.Dic10⋊11C2, C4.34(C2×Dic10), C2.6(D4⋊8D10), C10.11(C22×Q8), (C2×C20).142C23, C42⋊C2.13D5, (C22×C4).187D10, C4⋊Dic5.32C22, C22.7(C2×Dic10), C22.96(C23×D5), (C2×Dic5).22C23, (C4×Dic5).76C22, C2.13(C22×Dic10), C10.D4.2C22, C23.151(C22×D5), C2.7(D4.10D10), C23.D5.92C22, (C22×C20).223C22, (C22×C10).133C23, Dic5.14D4.1C2, C5⋊2(C23.41C23), (C2×Dic10).25C22, C23.21D10.23C2, (C22×Dic5).85C22, (C2×C10).13(C2×Q8), (C2×C4⋊Dic5).45C2, (C5×C4⋊C4).304C22, (C2×C4).148(C22×D5), (C5×C42⋊C2).14C2, (C5×C22⋊C4).99C22, SmallGroup(320,1191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.90D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 638 in 206 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.41C23, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×C20, C20⋊2Q8, C20.6Q8, Dic5.14D4, C20⋊Q8, C4.Dic10, C2×C4⋊Dic5, C23.21D10, C5×C42⋊C2, C42.90D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, 2- 1+4, Dic10, C22×D5, C23.41C23, C2×Dic10, C23×D5, C22×Dic10, D4⋊8D10, D4.10D10, C42.90D10
►On 160 points
Generators in S160
(1 86 18 154)(2 82 19 160)(3 88 20 156)(4 84 16 152)(5 90 17 158)(6 129 61 139)(7 125 62 135)(8 121 63 131)(9 127 64 137)(10 123 65 133)(11 157 21 89)(12 153 22 85)(13 159 23 81)(14 155 24 87)(15 151 25 83)(26 138 34 128)(27 134 35 124)(28 140 31 130)(29 136 32 126)(30 132 33 122)(36 146 66 111)(37 142 67 117)(38 148 68 113)(39 144 69 119)(40 150 70 115)(41 107 56 99)(42 103 57 95)(43 109 58 91)(44 105 59 97)(45 101 60 93)(46 102 51 94)(47 108 52 100)(48 104 53 96)(49 110 54 92)(50 106 55 98)(71 120 79 145)(72 116 80 141)(73 112 76 147)(74 118 77 143)(75 114 78 149)
(1 50 23 45)(2 46 24 41)(3 47 25 42)(4 48 21 43)(5 49 22 44)(6 69 35 75)(7 70 31 71)(8 66 32 72)(9 67 33 73)(10 68 34 74)(11 58 16 53)(12 59 17 54)(13 60 18 55)(14 56 19 51)(15 57 20 52)(26 77 65 38)(27 78 61 39)(28 79 62 40)(29 80 63 36)(30 76 64 37)(81 101 86 106)(82 102 87 107)(83 103 88 108)(84 104 89 109)(85 105 90 110)(91 152 96 157)(92 153 97 158)(93 154 98 159)(94 155 99 160)(95 156 100 151)(111 126 116 121)(112 127 117 122)(113 128 118 123)(114 129 119 124)(115 130 120 125)(131 146 136 141)(132 147 137 142)(133 148 138 143)(134 149 139 144)(135 150 140 145)
(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 29 13 8)(2 28 14 7)(3 27 15 6)(4 26 11 10)(5 30 12 9)(16 34 21 65)(17 33 22 64)(18 32 23 63)(19 31 24 62)(20 35 25 61)(36 60 72 50)(37 59 73 49)(38 58 74 48)(39 57 75 47)(40 56 71 46)(41 79 51 70)(42 78 52 69)(43 77 53 68)(44 76 54 67)(45 80 55 66)(81 121 154 136)(82 130 155 135)(83 129 156 134)(84 128 157 133)(85 127 158 132)(86 126 159 131)(87 125 160 140)(88 124 151 139)(89 123 152 138)(90 122 153 137)(91 143 104 113)(92 142 105 112)(93 141 106 111)(94 150 107 120)(95 149 108 119)(96 148 109 118)(97 147 110 117)(98 146 101 116)(99 145 102 115)(100 144 103 114)
G:=sub<Sym(160)| (1,86,18,154)(2,82,19,160)(3,88,20,156)(4,84,16,152)(5,90,17,158)(6,129,61,139)(7,125,62,135)(8,121,63,131)(9,127,64,137)(10,123,65,133)(11,157,21,89)(12,153,22,85)(13,159,23,81)(14,155,24,87)(15,151,25,83)(26,138,34,128)(27,134,35,124)(28,140,31,130)(29,136,32,126)(30,132,33,122)(36,146,66,111)(37,142,67,117)(38,148,68,113)(39,144,69,119)(40,150,70,115)(41,107,56,99)(42,103,57,95)(43,109,58,91)(44,105,59,97)(45,101,60,93)(46,102,51,94)(47,108,52,100)(48,104,53,96)(49,110,54,92)(50,106,55,98)(71,120,79,145)(72,116,80,141)(73,112,76,147)(74,118,77,143)(75,114,78,149), (1,50,23,45)(2,46,24,41)(3,47,25,42)(4,48,21,43)(5,49,22,44)(6,69,35,75)(7,70,31,71)(8,66,32,72)(9,67,33,73)(10,68,34,74)(11,58,16,53)(12,59,17,54)(13,60,18,55)(14,56,19,51)(15,57,20,52)(26,77,65,38)(27,78,61,39)(28,79,62,40)(29,80,63,36)(30,76,64,37)(81,101,86,106)(82,102,87,107)(83,103,88,108)(84,104,89,109)(85,105,90,110)(91,152,96,157)(92,153,97,158)(93,154,98,159)(94,155,99,160)(95,156,100,151)(111,126,116,121)(112,127,117,122)(113,128,118,123)(114,129,119,124)(115,130,120,125)(131,146,136,141)(132,147,137,142)(133,148,138,143)(134,149,139,144)(135,150,140,145), (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), (1,29,13,8)(2,28,14,7)(3,27,15,6)(4,26,11,10)(5,30,12,9)(16,34,21,65)(17,33,22,64)(18,32,23,63)(19,31,24,62)(20,35,25,61)(36,60,72,50)(37,59,73,49)(38,58,74,48)(39,57,75,47)(40,56,71,46)(41,79,51,70)(42,78,52,69)(43,77,53,68)(44,76,54,67)(45,80,55,66)(81,121,154,136)(82,130,155,135)(83,129,156,134)(84,128,157,133)(85,127,158,132)(86,126,159,131)(87,125,160,140)(88,124,151,139)(89,123,152,138)(90,122,153,137)(91,143,104,113)(92,142,105,112)(93,141,106,111)(94,150,107,120)(95,149,108,119)(96,148,109,118)(97,147,110,117)(98,146,101,116)(99,145,102,115)(100,144,103,114)>;
G:=Group( (1,86,18,154)(2,82,19,160)(3,88,20,156)(4,84,16,152)(5,90,17,158)(6,129,61,139)(7,125,62,135)(8,121,63,131)(9,127,64,137)(10,123,65,133)(11,157,21,89)(12,153,22,85)(13,159,23,81)(14,155,24,87)(15,151,25,83)(26,138,34,128)(27,134,35,124)(28,140,31,130)(29,136,32,126)(30,132,33,122)(36,146,66,111)(37,142,67,117)(38,148,68,113)(39,144,69,119)(40,150,70,115)(41,107,56,99)(42,103,57,95)(43,109,58,91)(44,105,59,97)(45,101,60,93)(46,102,51,94)(47,108,52,100)(48,104,53,96)(49,110,54,92)(50,106,55,98)(71,120,79,145)(72,116,80,141)(73,112,76,147)(74,118,77,143)(75,114,78,149), (1,50,23,45)(2,46,24,41)(3,47,25,42)(4,48,21,43)(5,49,22,44)(6,69,35,75)(7,70,31,71)(8,66,32,72)(9,67,33,73)(10,68,34,74)(11,58,16,53)(12,59,17,54)(13,60,18,55)(14,56,19,51)(15,57,20,52)(26,77,65,38)(27,78,61,39)(28,79,62,40)(29,80,63,36)(30,76,64,37)(81,101,86,106)(82,102,87,107)(83,103,88,108)(84,104,89,109)(85,105,90,110)(91,152,96,157)(92,153,97,158)(93,154,98,159)(94,155,99,160)(95,156,100,151)(111,126,116,121)(112,127,117,122)(113,128,118,123)(114,129,119,124)(115,130,120,125)(131,146,136,141)(132,147,137,142)(133,148,138,143)(134,149,139,144)(135,150,140,145), (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), (1,29,13,8)(2,28,14,7)(3,27,15,6)(4,26,11,10)(5,30,12,9)(16,34,21,65)(17,33,22,64)(18,32,23,63)(19,31,24,62)(20,35,25,61)(36,60,72,50)(37,59,73,49)(38,58,74,48)(39,57,75,47)(40,56,71,46)(41,79,51,70)(42,78,52,69)(43,77,53,68)(44,76,54,67)(45,80,55,66)(81,121,154,136)(82,130,155,135)(83,129,156,134)(84,128,157,133)(85,127,158,132)(86,126,159,131)(87,125,160,140)(88,124,151,139)(89,123,152,138)(90,122,153,137)(91,143,104,113)(92,142,105,112)(93,141,106,111)(94,150,107,120)(95,149,108,119)(96,148,109,118)(97,147,110,117)(98,146,101,116)(99,145,102,115)(100,144,103,114) );
G=PermutationGroup([[(1,86,18,154),(2,82,19,160),(3,88,20,156),(4,84,16,152),(5,90,17,158),(6,129,61,139),(7,125,62,135),(8,121,63,131),(9,127,64,137),(10,123,65,133),(11,157,21,89),(12,153,22,85),(13,159,23,81),(14,155,24,87),(15,151,25,83),(26,138,34,128),(27,134,35,124),(28,140,31,130),(29,136,32,126),(30,132,33,122),(36,146,66,111),(37,142,67,117),(38,148,68,113),(39,144,69,119),(40,150,70,115),(41,107,56,99),(42,103,57,95),(43,109,58,91),(44,105,59,97),(45,101,60,93),(46,102,51,94),(47,108,52,100),(48,104,53,96),(49,110,54,92),(50,106,55,98),(71,120,79,145),(72,116,80,141),(73,112,76,147),(74,118,77,143),(75,114,78,149)], [(1,50,23,45),(2,46,24,41),(3,47,25,42),(4,48,21,43),(5,49,22,44),(6,69,35,75),(7,70,31,71),(8,66,32,72),(9,67,33,73),(10,68,34,74),(11,58,16,53),(12,59,17,54),(13,60,18,55),(14,56,19,51),(15,57,20,52),(26,77,65,38),(27,78,61,39),(28,79,62,40),(29,80,63,36),(30,76,64,37),(81,101,86,106),(82,102,87,107),(83,103,88,108),(84,104,89,109),(85,105,90,110),(91,152,96,157),(92,153,97,158),(93,154,98,159),(94,155,99,160),(95,156,100,151),(111,126,116,121),(112,127,117,122),(113,128,118,123),(114,129,119,124),(115,130,120,125),(131,146,136,141),(132,147,137,142),(133,148,138,143),(134,149,139,144),(135,150,140,145)], [(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)], [(1,29,13,8),(2,28,14,7),(3,27,15,6),(4,26,11,10),(5,30,12,9),(16,34,21,65),(17,33,22,64),(18,32,23,63),(19,31,24,62),(20,35,25,61),(36,60,72,50),(37,59,73,49),(38,58,74,48),(39,57,75,47),(40,56,71,46),(41,79,51,70),(42,78,52,69),(43,77,53,68),(44,76,54,67),(45,80,55,66),(81,121,154,136),(82,130,155,135),(83,129,156,134),(84,128,157,133),(85,127,158,132),(86,126,159,131),(87,125,160,140),(88,124,151,139),(89,123,152,138),(90,122,153,137),(91,143,104,113),(92,142,105,112),(93,141,106,111),(94,150,107,120),(95,149,108,119),(96,148,109,118),(97,147,110,117),(98,146,101,116),(99,145,102,115),(100,144,103,114)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 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 | Q8 | D5 | D10 | D10 | D10 | D10 | Dic10 | 2+ 1+4 | 2- 1+4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.90D10 | C20⋊2Q8 | C20.6Q8 | Dic5.14D4 | C20⋊Q8 | C4.Dic10 | C2×C4⋊Dic5 | C23.21D10 | C5×C42⋊C2 | C2×C20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C42.90D10 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 0 | 28 | 39 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 14 | 0 | 0 |
| 0 | 0 | 31 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 14 |
| 0 | 0 | 0 | 0 | 31 | 4 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,37,31,0,0,0,0,14,4,0,0,0,0,0,0,37,31,0,0,0,0,14,4] >;
C42.90D10 in GAP, Magma, Sage, TeX
C_4^2._{90}D_{10} % in TeX
G:=Group("C4^2.90D10"); // GroupNames label
G:=SmallGroup(320,1191);
// by ID
G=gap.SmallGroup(320,1191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,184,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations