metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.116D10, C10.1052+ 1+4, (C4×D4)⋊24D5, (D4×C20)⋊26C2, (C4×D20)⋊33C2, C20⋊D4⋊10C2, C4⋊D20⋊16C2, C20⋊7D4⋊12C2, C20⋊2D4⋊10C2, C4⋊C4.287D10, D10⋊D4⋊11C2, (C2×D4).223D10, C4.46(C4○D20), C42⋊D5⋊15C2, C4.Dic10⋊16C2, D10.12D4⋊9C2, C20.113(C4○D4), (C2×C10).106C24, (C4×C20).160C22, (C2×C20).164C23, C22⋊C4.118D10, (C22×C4).214D10, C2.24(D4⋊6D10), C2.18(D4⋊8D10), (D4×C10).265C22, (C2×D20).268C22, C23.23D10⋊4C2, C4⋊Dic5.364C22, (C22×C20).83C22, (C4×Dic5).86C22, (C2×Dic5).47C23, C10.D4.7C22, (C22×D5).40C23, C22.131(C23×D5), C23.103(C22×D5), C23.D5.16C22, D10⋊C4.88C22, (C22×C10).176C23, C5⋊2(C22.34C24), C10.48(C2×C4○D4), C2.55(C2×C4○D20), (C2×C4×D5).254C22, (C5×C4⋊C4).334C22, (C2×C4).581(C22×D5), (C2×C5⋊D4).19C22, (C5×C22⋊C4).129C22, SmallGroup(320,1234)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.116D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, dbd-1=a2b-1, dcd-1=a2c-1 >
Subgroups: 958 in 240 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C22.D4, C42.C2, C4⋊1D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22.34C24, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, C42⋊D5, C4×D20, D10.12D4, D10⋊D4, C4.Dic10, C4⋊D20, C23.23D10, C20⋊7D4, C20⋊2D4, C20⋊D4, D4×C20, C42.116D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.34C24, C4○D20, C23×D5, C2×C4○D20, D4⋊6D10, D4⋊8D10, C42.116D10
►On 160 points
Generators in S160
(1 110 63 118)(2 144 64 84)(3 102 65 120)(4 146 66 86)(5 104 67 112)(6 148 68 88)(7 106 69 114)(8 150 70 90)(9 108 61 116)(10 142 62 82)(11 43 97 123)(12 151 98 139)(13 45 99 125)(14 153 100 131)(15 47 91 127)(16 155 92 133)(17 49 93 129)(18 157 94 135)(19 41 95 121)(20 159 96 137)(21 156 76 134)(22 50 77 130)(23 158 78 136)(24 42 79 122)(25 160 80 138)(26 44 71 124)(27 152 72 140)(28 46 73 126)(29 154 74 132)(30 48 75 128)(31 119 52 101)(32 85 53 145)(33 111 54 103)(34 87 55 147)(35 113 56 105)(36 89 57 149)(37 115 58 107)(38 81 59 141)(39 117 60 109)(40 83 51 143)
(1 100 40 28)(2 91 31 29)(3 92 32 30)(4 93 33 21)(5 94 34 22)(6 95 35 23)(7 96 36 24)(8 97 37 25)(9 98 38 26)(10 99 39 27)(11 58 80 70)(12 59 71 61)(13 60 72 62)(14 51 73 63)(15 52 74 64)(16 53 75 65)(17 54 76 66)(18 55 77 67)(19 56 78 68)(20 57 79 69)(41 105 136 88)(42 106 137 89)(43 107 138 90)(44 108 139 81)(45 109 140 82)(46 110 131 83)(47 101 132 84)(48 102 133 85)(49 103 134 86)(50 104 135 87)(111 156 146 129)(112 157 147 130)(113 158 148 121)(114 159 149 122)(115 160 150 123)(116 151 141 124)(117 152 142 125)(118 153 143 126)(119 154 144 127)(120 155 145 128)
(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)
(1 125 63 45)(2 44 64 124)(3 123 65 43)(4 42 66 122)(5 121 67 41)(6 50 68 130)(7 129 69 49)(8 48 70 128)(9 127 61 47)(10 46 62 126)(11 85 97 145)(12 144 98 84)(13 83 99 143)(14 142 100 82)(15 81 91 141)(16 150 92 90)(17 89 93 149)(18 148 94 88)(19 87 95 147)(20 146 96 86)(21 114 76 106)(22 105 77 113)(23 112 78 104)(24 103 79 111)(25 120 80 102)(26 101 71 119)(27 118 72 110)(28 109 73 117)(29 116 74 108)(30 107 75 115)(31 139 52 151)(32 160 53 138)(33 137 54 159)(34 158 55 136)(35 135 56 157)(36 156 57 134)(37 133 58 155)(38 154 59 132)(39 131 60 153)(40 152 51 140)
G:=sub<Sym(160)| (1,110,63,118)(2,144,64,84)(3,102,65,120)(4,146,66,86)(5,104,67,112)(6,148,68,88)(7,106,69,114)(8,150,70,90)(9,108,61,116)(10,142,62,82)(11,43,97,123)(12,151,98,139)(13,45,99,125)(14,153,100,131)(15,47,91,127)(16,155,92,133)(17,49,93,129)(18,157,94,135)(19,41,95,121)(20,159,96,137)(21,156,76,134)(22,50,77,130)(23,158,78,136)(24,42,79,122)(25,160,80,138)(26,44,71,124)(27,152,72,140)(28,46,73,126)(29,154,74,132)(30,48,75,128)(31,119,52,101)(32,85,53,145)(33,111,54,103)(34,87,55,147)(35,113,56,105)(36,89,57,149)(37,115,58,107)(38,81,59,141)(39,117,60,109)(40,83,51,143), (1,100,40,28)(2,91,31,29)(3,92,32,30)(4,93,33,21)(5,94,34,22)(6,95,35,23)(7,96,36,24)(8,97,37,25)(9,98,38,26)(10,99,39,27)(11,58,80,70)(12,59,71,61)(13,60,72,62)(14,51,73,63)(15,52,74,64)(16,53,75,65)(17,54,76,66)(18,55,77,67)(19,56,78,68)(20,57,79,69)(41,105,136,88)(42,106,137,89)(43,107,138,90)(44,108,139,81)(45,109,140,82)(46,110,131,83)(47,101,132,84)(48,102,133,85)(49,103,134,86)(50,104,135,87)(111,156,146,129)(112,157,147,130)(113,158,148,121)(114,159,149,122)(115,160,150,123)(116,151,141,124)(117,152,142,125)(118,153,143,126)(119,154,144,127)(120,155,145,128), (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), (1,125,63,45)(2,44,64,124)(3,123,65,43)(4,42,66,122)(5,121,67,41)(6,50,68,130)(7,129,69,49)(8,48,70,128)(9,127,61,47)(10,46,62,126)(11,85,97,145)(12,144,98,84)(13,83,99,143)(14,142,100,82)(15,81,91,141)(16,150,92,90)(17,89,93,149)(18,148,94,88)(19,87,95,147)(20,146,96,86)(21,114,76,106)(22,105,77,113)(23,112,78,104)(24,103,79,111)(25,120,80,102)(26,101,71,119)(27,118,72,110)(28,109,73,117)(29,116,74,108)(30,107,75,115)(31,139,52,151)(32,160,53,138)(33,137,54,159)(34,158,55,136)(35,135,56,157)(36,156,57,134)(37,133,58,155)(38,154,59,132)(39,131,60,153)(40,152,51,140)>;
G:=Group( (1,110,63,118)(2,144,64,84)(3,102,65,120)(4,146,66,86)(5,104,67,112)(6,148,68,88)(7,106,69,114)(8,150,70,90)(9,108,61,116)(10,142,62,82)(11,43,97,123)(12,151,98,139)(13,45,99,125)(14,153,100,131)(15,47,91,127)(16,155,92,133)(17,49,93,129)(18,157,94,135)(19,41,95,121)(20,159,96,137)(21,156,76,134)(22,50,77,130)(23,158,78,136)(24,42,79,122)(25,160,80,138)(26,44,71,124)(27,152,72,140)(28,46,73,126)(29,154,74,132)(30,48,75,128)(31,119,52,101)(32,85,53,145)(33,111,54,103)(34,87,55,147)(35,113,56,105)(36,89,57,149)(37,115,58,107)(38,81,59,141)(39,117,60,109)(40,83,51,143), (1,100,40,28)(2,91,31,29)(3,92,32,30)(4,93,33,21)(5,94,34,22)(6,95,35,23)(7,96,36,24)(8,97,37,25)(9,98,38,26)(10,99,39,27)(11,58,80,70)(12,59,71,61)(13,60,72,62)(14,51,73,63)(15,52,74,64)(16,53,75,65)(17,54,76,66)(18,55,77,67)(19,56,78,68)(20,57,79,69)(41,105,136,88)(42,106,137,89)(43,107,138,90)(44,108,139,81)(45,109,140,82)(46,110,131,83)(47,101,132,84)(48,102,133,85)(49,103,134,86)(50,104,135,87)(111,156,146,129)(112,157,147,130)(113,158,148,121)(114,159,149,122)(115,160,150,123)(116,151,141,124)(117,152,142,125)(118,153,143,126)(119,154,144,127)(120,155,145,128), (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), (1,125,63,45)(2,44,64,124)(3,123,65,43)(4,42,66,122)(5,121,67,41)(6,50,68,130)(7,129,69,49)(8,48,70,128)(9,127,61,47)(10,46,62,126)(11,85,97,145)(12,144,98,84)(13,83,99,143)(14,142,100,82)(15,81,91,141)(16,150,92,90)(17,89,93,149)(18,148,94,88)(19,87,95,147)(20,146,96,86)(21,114,76,106)(22,105,77,113)(23,112,78,104)(24,103,79,111)(25,120,80,102)(26,101,71,119)(27,118,72,110)(28,109,73,117)(29,116,74,108)(30,107,75,115)(31,139,52,151)(32,160,53,138)(33,137,54,159)(34,158,55,136)(35,135,56,157)(36,156,57,134)(37,133,58,155)(38,154,59,132)(39,131,60,153)(40,152,51,140) );
G=PermutationGroup([[(1,110,63,118),(2,144,64,84),(3,102,65,120),(4,146,66,86),(5,104,67,112),(6,148,68,88),(7,106,69,114),(8,150,70,90),(9,108,61,116),(10,142,62,82),(11,43,97,123),(12,151,98,139),(13,45,99,125),(14,153,100,131),(15,47,91,127),(16,155,92,133),(17,49,93,129),(18,157,94,135),(19,41,95,121),(20,159,96,137),(21,156,76,134),(22,50,77,130),(23,158,78,136),(24,42,79,122),(25,160,80,138),(26,44,71,124),(27,152,72,140),(28,46,73,126),(29,154,74,132),(30,48,75,128),(31,119,52,101),(32,85,53,145),(33,111,54,103),(34,87,55,147),(35,113,56,105),(36,89,57,149),(37,115,58,107),(38,81,59,141),(39,117,60,109),(40,83,51,143)], [(1,100,40,28),(2,91,31,29),(3,92,32,30),(4,93,33,21),(5,94,34,22),(6,95,35,23),(7,96,36,24),(8,97,37,25),(9,98,38,26),(10,99,39,27),(11,58,80,70),(12,59,71,61),(13,60,72,62),(14,51,73,63),(15,52,74,64),(16,53,75,65),(17,54,76,66),(18,55,77,67),(19,56,78,68),(20,57,79,69),(41,105,136,88),(42,106,137,89),(43,107,138,90),(44,108,139,81),(45,109,140,82),(46,110,131,83),(47,101,132,84),(48,102,133,85),(49,103,134,86),(50,104,135,87),(111,156,146,129),(112,157,147,130),(113,158,148,121),(114,159,149,122),(115,160,150,123),(116,151,141,124),(117,152,142,125),(118,153,143,126),(119,154,144,127),(120,155,145,128)], [(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)], [(1,125,63,45),(2,44,64,124),(3,123,65,43),(4,42,66,122),(5,121,67,41),(6,50,68,130),(7,129,69,49),(8,48,70,128),(9,127,61,47),(10,46,62,126),(11,85,97,145),(12,144,98,84),(13,83,99,143),(14,142,100,82),(15,81,91,141),(16,150,92,90),(17,89,93,149),(18,148,94,88),(19,87,95,147),(20,146,96,86),(21,114,76,106),(22,105,77,113),(23,112,78,104),(24,103,79,111),(25,120,80,102),(26,101,71,119),(27,118,72,110),(28,109,73,117),(29,116,74,108),(30,107,75,115),(31,139,52,151),(32,160,53,138),(33,137,54,159),(34,158,55,136),(35,135,56,157),(36,156,57,134),(37,133,58,155),(38,154,59,132),(39,131,60,153),(40,152,51,140)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4⋊6D10 | D4⋊8D10 |
| kernel | C42.116D10 | C42⋊D5 | C4×D20 | D10.12D4 | D10⋊D4 | C4.Dic10 | C4⋊D20 | C23.23D10 | C20⋊7D4 | C20⋊2D4 | C20⋊D4 | D4×C20 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 4 | 4 |
Matrix representation of C42.116D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 9 | 0 |
| 0 | 0 | 0 | 30 | 0 | 9 |
| 0 | 0 | 32 | 0 | 11 | 0 |
| 0 | 0 | 0 | 32 | 0 | 11 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 17 | 39 | 0 | 0 | 0 | 0 |
| 21 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 17 | 39 | 0 | 0 | 0 | 0 |
| 22 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 11 | 33 | 23 |
| 0 | 0 | 27 | 27 | 8 | 8 |
| 0 | 0 | 8 | 18 | 27 | 30 |
| 0 | 0 | 33 | 33 | 14 | 14 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,0,32,0,0,0,0,30,0,32,0,0,9,0,11,0,0,0,0,9,0,11],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[17,21,0,0,0,0,39,24,0,0,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,40,34,0,0,0,0,7,7,0,0],[17,22,0,0,0,0,39,24,0,0,0,0,0,0,14,27,8,33,0,0,11,27,18,33,0,0,33,8,27,14,0,0,23,8,30,14] >;
C42.116D10 in GAP, Magma, Sage, TeX
C_4^2._{116}D_{10} % in TeX
G:=Group("C4^2.116D10"); // GroupNames label
G:=SmallGroup(320,1234);
// by ID
G=gap.SmallGroup(320,1234);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations