metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.243D10, (C4×C8)⋊4D5, (C4×C40)⋊4C2, (C2×C8).284D10, C20.8Q8⋊1C2, D10⋊1C8.1C2, C10.30(C8○D4), D10⋊C4.12C4, C20.243(C4○D4), C4.127(C4○D20), (C4×C20).341C22, (C2×C40).344C22, (C2×C20).808C23, C10.D4.12C4, C42⋊D5.10C2, C42.D5⋊16C2, C2.8(D20.3C4), C2.11(C42⋊D5), C5⋊3(C42.7C22), C10.27(C42⋊C2), (C4×Dic5).199C22, (C2×C4).89(C4×D5), C22.96(C2×C4×D5), (C2×C20).379(C2×C4), (C2×C4×D5).227C22, (C2×Dic5).14(C2×C4), (C22×D5).15(C2×C4), (C2×C4).750(C22×D5), (C2×C10).164(C22×C4), (C2×C5⋊2C8).192C22, SmallGroup(320,317)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.243D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=a2c9 >
Subgroups: 302 in 96 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C42.7C22, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C42.D5, C20.8Q8, D10⋊1C8, C4×C40, C42⋊D5, C42.243D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C8○D4, C4×D5, C22×D5, C42.7C22, C2×C4×D5, C4○D20, C42⋊D5, D20.3C4, C42.243D10
►On 160 points
Generators in S160
(1 96 128 60)(2 97 129 61)(3 98 130 62)(4 99 131 63)(5 100 132 64)(6 101 133 65)(7 102 134 66)(8 103 135 67)(9 104 136 68)(10 105 137 69)(11 106 138 70)(12 107 139 71)(13 108 140 72)(14 109 141 73)(15 110 142 74)(16 111 143 75)(17 112 144 76)(18 113 145 77)(19 114 146 78)(20 115 147 79)(21 116 148 80)(22 117 149 41)(23 118 150 42)(24 119 151 43)(25 120 152 44)(26 81 153 45)(27 82 154 46)(28 83 155 47)(29 84 156 48)(30 85 157 49)(31 86 158 50)(32 87 159 51)(33 88 160 52)(34 89 121 53)(35 90 122 54)(36 91 123 55)(37 92 124 56)(38 93 125 57)(39 94 126 58)(40 95 127 59)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 10 138 147 21 30 158 127)(2 146 139 29 22 126 159 9)(3 28 140 125 23 8 160 145)(4 124 141 7 24 144 121 27)(5 6 142 143 25 26 122 123)(11 20 148 157 31 40 128 137)(12 156 149 39 32 136 129 19)(13 38 150 135 33 18 130 155)(14 134 151 17 34 154 131 37)(15 16 152 153 35 36 132 133)(41 114 87 48 61 94 107 68)(42 47 88 93 62 67 108 113)(43 92 89 66 63 112 109 46)(44 65 90 111 64 45 110 91)(49 106 95 80 69 86 115 60)(50 79 96 85 70 59 116 105)(51 84 97 58 71 104 117 78)(52 57 98 103 72 77 118 83)(53 102 99 76 73 82 119 56)(54 75 100 81 74 55 120 101)
G:=sub<Sym(160)| (1,96,128,60)(2,97,129,61)(3,98,130,62)(4,99,131,63)(5,100,132,64)(6,101,133,65)(7,102,134,66)(8,103,135,67)(9,104,136,68)(10,105,137,69)(11,106,138,70)(12,107,139,71)(13,108,140,72)(14,109,141,73)(15,110,142,74)(16,111,143,75)(17,112,144,76)(18,113,145,77)(19,114,146,78)(20,115,147,79)(21,116,148,80)(22,117,149,41)(23,118,150,42)(24,119,151,43)(25,120,152,44)(26,81,153,45)(27,82,154,46)(28,83,155,47)(29,84,156,48)(30,85,157,49)(31,86,158,50)(32,87,159,51)(33,88,160,52)(34,89,121,53)(35,90,122,54)(36,91,123,55)(37,92,124,56)(38,93,125,57)(39,94,126,58)(40,95,127,59), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,138,147,21,30,158,127)(2,146,139,29,22,126,159,9)(3,28,140,125,23,8,160,145)(4,124,141,7,24,144,121,27)(5,6,142,143,25,26,122,123)(11,20,148,157,31,40,128,137)(12,156,149,39,32,136,129,19)(13,38,150,135,33,18,130,155)(14,134,151,17,34,154,131,37)(15,16,152,153,35,36,132,133)(41,114,87,48,61,94,107,68)(42,47,88,93,62,67,108,113)(43,92,89,66,63,112,109,46)(44,65,90,111,64,45,110,91)(49,106,95,80,69,86,115,60)(50,79,96,85,70,59,116,105)(51,84,97,58,71,104,117,78)(52,57,98,103,72,77,118,83)(53,102,99,76,73,82,119,56)(54,75,100,81,74,55,120,101)>;
G:=Group( (1,96,128,60)(2,97,129,61)(3,98,130,62)(4,99,131,63)(5,100,132,64)(6,101,133,65)(7,102,134,66)(8,103,135,67)(9,104,136,68)(10,105,137,69)(11,106,138,70)(12,107,139,71)(13,108,140,72)(14,109,141,73)(15,110,142,74)(16,111,143,75)(17,112,144,76)(18,113,145,77)(19,114,146,78)(20,115,147,79)(21,116,148,80)(22,117,149,41)(23,118,150,42)(24,119,151,43)(25,120,152,44)(26,81,153,45)(27,82,154,46)(28,83,155,47)(29,84,156,48)(30,85,157,49)(31,86,158,50)(32,87,159,51)(33,88,160,52)(34,89,121,53)(35,90,122,54)(36,91,123,55)(37,92,124,56)(38,93,125,57)(39,94,126,58)(40,95,127,59), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,138,147,21,30,158,127)(2,146,139,29,22,126,159,9)(3,28,140,125,23,8,160,145)(4,124,141,7,24,144,121,27)(5,6,142,143,25,26,122,123)(11,20,148,157,31,40,128,137)(12,156,149,39,32,136,129,19)(13,38,150,135,33,18,130,155)(14,134,151,17,34,154,131,37)(15,16,152,153,35,36,132,133)(41,114,87,48,61,94,107,68)(42,47,88,93,62,67,108,113)(43,92,89,66,63,112,109,46)(44,65,90,111,64,45,110,91)(49,106,95,80,69,86,115,60)(50,79,96,85,70,59,116,105)(51,84,97,58,71,104,117,78)(52,57,98,103,72,77,118,83)(53,102,99,76,73,82,119,56)(54,75,100,81,74,55,120,101) );
G=PermutationGroup([[(1,96,128,60),(2,97,129,61),(3,98,130,62),(4,99,131,63),(5,100,132,64),(6,101,133,65),(7,102,134,66),(8,103,135,67),(9,104,136,68),(10,105,137,69),(11,106,138,70),(12,107,139,71),(13,108,140,72),(14,109,141,73),(15,110,142,74),(16,111,143,75),(17,112,144,76),(18,113,145,77),(19,114,146,78),(20,115,147,79),(21,116,148,80),(22,117,149,41),(23,118,150,42),(24,119,151,43),(25,120,152,44),(26,81,153,45),(27,82,154,46),(28,83,155,47),(29,84,156,48),(30,85,157,49),(31,86,158,50),(32,87,159,51),(33,88,160,52),(34,89,121,53),(35,90,122,54),(36,91,123,55),(37,92,124,56),(38,93,125,57),(39,94,126,58),(40,95,127,59)], [(1,11,21,31),(2,12,22,32),(3,13,23,33),(4,14,24,34),(5,15,25,35),(6,16,26,36),(7,17,27,37),(8,18,28,38),(9,19,29,39),(10,20,30,40),(41,51,61,71),(42,52,62,72),(43,53,63,73),(44,54,64,74),(45,55,65,75),(46,56,66,76),(47,57,67,77),(48,58,68,78),(49,59,69,79),(50,60,70,80),(81,91,101,111),(82,92,102,112),(83,93,103,113),(84,94,104,114),(85,95,105,115),(86,96,106,116),(87,97,107,117),(88,98,108,118),(89,99,109,119),(90,100,110,120),(121,131,141,151),(122,132,142,152),(123,133,143,153),(124,134,144,154),(125,135,145,155),(126,136,146,156),(127,137,147,157),(128,138,148,158),(129,139,149,159),(130,140,150,160)], [(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)], [(1,10,138,147,21,30,158,127),(2,146,139,29,22,126,159,9),(3,28,140,125,23,8,160,145),(4,124,141,7,24,144,121,27),(5,6,142,143,25,26,122,123),(11,20,148,157,31,40,128,137),(12,156,149,39,32,136,129,19),(13,38,150,135,33,18,130,155),(14,134,151,17,34,154,131,37),(15,16,152,153,35,36,132,133),(41,114,87,48,61,94,107,68),(42,47,88,93,62,67,108,113),(43,92,89,66,63,112,109,46),(44,65,90,111,64,45,110,91),(49,106,95,80,69,86,115,60),(50,79,96,85,70,59,116,105),(51,84,97,58,71,104,117,78),(52,57,98,103,72,77,118,83),(53,102,99,76,73,82,119,56),(54,75,100,81,74,55,120,101)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D5 | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C4○D20 | D20.3C4 |
| kernel | C42.243D10 | C42.D5 | C20.8Q8 | D10⋊1C8 | C4×C40 | C42⋊D5 | C10.D4 | D10⋊C4 | C4×C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 2 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of C42.243D10 ►in GL4(𝔽41) generated by
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 26 |
| 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 16 | 25 | 0 | 0 |
| 16 | 2 | 0 | 0 |
| 0 | 0 | 3 | 5 |
| 0 | 0 | 0 | 38 |
| 25 | 16 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 38 | 36 |
| 0 | 0 | 20 | 3 |
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,32,0,0,0,26,9],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[16,16,0,0,25,2,0,0,0,0,3,0,0,0,5,38],[25,2,0,0,16,16,0,0,0,0,38,20,0,0,36,3] >;
C42.243D10 in GAP, Magma, Sage, TeX
C_4^2._{243}D_{10} % in TeX
G:=Group("C4^2.243D10"); // GroupNames label
G:=SmallGroup(320,317);
// by ID
G=gap.SmallGroup(320,317);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,422,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=a^2*b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations