metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.35C42, C42.186D10, C4.5(C4×Dic5), C4.Dic5⋊15C4, C4⋊C4.10Dic5, C10.59(C8○D4), C5⋊7(C8○2M4(2)), C10.41(C2×C42), (C2×C10).27C42, C22⋊C4.6Dic5, C22.5(C4×Dic5), C20.195(C22×C4), (C2×C20).845C23, (C4×C20).231C22, C42⋊C2.14D5, C42.D5⋊21C2, (C22×C4).336D10, C2.1(D4.Dic5), C23.14(C2×Dic5), (C22×C20).147C22, C22.21(C22×Dic5), (C2×C5⋊2C8)⋊9C4, C4.110(C2×C4×D5), (C4×C5⋊2C8)⋊25C2, (C5×C4⋊C4).20C4, (C2×C4).80(C4×D5), C2.10(C2×C4×Dic5), C5⋊2C8.45(C2×C4), (C2×C20).261(C2×C4), (C5×C22⋊C4).15C4, (C22×C5⋊2C8).6C2, (C2×C4).42(C2×Dic5), (C5×C42⋊C2).6C2, (C2×C4).787(C22×D5), (C2×C4.Dic5).17C2, (C2×C10).283(C22×C4), (C22×C10).124(C2×C4), (C2×C5⋊2C8).355C22, SmallGroup(320,624)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.35C42
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a9, cac-1=a11, bc=cb >
Subgroups: 254 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8, C2×C20, C2×C20, C22×C10, C8○2M4(2), C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C4×C5⋊2C8, C42.D5, C22×C5⋊2C8, C2×C4.Dic5, C5×C42⋊C2, C20.35C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, Dic5, D10, C2×C42, C8○D4, C4×D5, C2×Dic5, C22×D5, C8○2M4(2), C4×Dic5, C2×C4×D5, C22×Dic5, C2×C4×Dic5, D4.Dic5, C20.35C42
►On 160 points
Generators in S160
(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)
(1 117 6 102 11 107 16 112)(2 106 7 111 12 116 17 101)(3 115 8 120 13 105 18 110)(4 104 9 109 14 114 19 119)(5 113 10 118 15 103 20 108)(21 65 26 70 31 75 36 80)(22 74 27 79 32 64 37 69)(23 63 28 68 33 73 38 78)(24 72 29 77 34 62 39 67)(25 61 30 66 35 71 40 76)(41 97 56 92 51 87 46 82)(42 86 57 81 52 96 47 91)(43 95 58 90 53 85 48 100)(44 84 59 99 54 94 49 89)(45 93 60 88 55 83 50 98)(121 156 136 151 131 146 126 141)(122 145 137 160 132 155 127 150)(123 154 138 149 133 144 128 159)(124 143 139 158 134 153 129 148)(125 152 140 147 135 142 130 157)
(1 131 77 54)(2 122 78 45)(3 133 79 56)(4 124 80 47)(5 135 61 58)(6 126 62 49)(7 137 63 60)(8 128 64 51)(9 139 65 42)(10 130 66 53)(11 121 67 44)(12 132 68 55)(13 123 69 46)(14 134 70 57)(15 125 71 48)(16 136 72 59)(17 127 73 50)(18 138 74 41)(19 129 75 52)(20 140 76 43)(21 91 104 143)(22 82 105 154)(23 93 106 145)(24 84 107 156)(25 95 108 147)(26 86 109 158)(27 97 110 149)(28 88 111 160)(29 99 112 151)(30 90 113 142)(31 81 114 153)(32 92 115 144)(33 83 116 155)(34 94 117 146)(35 85 118 157)(36 96 119 148)(37 87 120 159)(38 98 101 150)(39 89 102 141)(40 100 103 152)
G:=sub<Sym(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), (1,117,6,102,11,107,16,112)(2,106,7,111,12,116,17,101)(3,115,8,120,13,105,18,110)(4,104,9,109,14,114,19,119)(5,113,10,118,15,103,20,108)(21,65,26,70,31,75,36,80)(22,74,27,79,32,64,37,69)(23,63,28,68,33,73,38,78)(24,72,29,77,34,62,39,67)(25,61,30,66,35,71,40,76)(41,97,56,92,51,87,46,82)(42,86,57,81,52,96,47,91)(43,95,58,90,53,85,48,100)(44,84,59,99,54,94,49,89)(45,93,60,88,55,83,50,98)(121,156,136,151,131,146,126,141)(122,145,137,160,132,155,127,150)(123,154,138,149,133,144,128,159)(124,143,139,158,134,153,129,148)(125,152,140,147,135,142,130,157), (1,131,77,54)(2,122,78,45)(3,133,79,56)(4,124,80,47)(5,135,61,58)(6,126,62,49)(7,137,63,60)(8,128,64,51)(9,139,65,42)(10,130,66,53)(11,121,67,44)(12,132,68,55)(13,123,69,46)(14,134,70,57)(15,125,71,48)(16,136,72,59)(17,127,73,50)(18,138,74,41)(19,129,75,52)(20,140,76,43)(21,91,104,143)(22,82,105,154)(23,93,106,145)(24,84,107,156)(25,95,108,147)(26,86,109,158)(27,97,110,149)(28,88,111,160)(29,99,112,151)(30,90,113,142)(31,81,114,153)(32,92,115,144)(33,83,116,155)(34,94,117,146)(35,85,118,157)(36,96,119,148)(37,87,120,159)(38,98,101,150)(39,89,102,141)(40,100,103,152)>;
G:=Group( (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), (1,117,6,102,11,107,16,112)(2,106,7,111,12,116,17,101)(3,115,8,120,13,105,18,110)(4,104,9,109,14,114,19,119)(5,113,10,118,15,103,20,108)(21,65,26,70,31,75,36,80)(22,74,27,79,32,64,37,69)(23,63,28,68,33,73,38,78)(24,72,29,77,34,62,39,67)(25,61,30,66,35,71,40,76)(41,97,56,92,51,87,46,82)(42,86,57,81,52,96,47,91)(43,95,58,90,53,85,48,100)(44,84,59,99,54,94,49,89)(45,93,60,88,55,83,50,98)(121,156,136,151,131,146,126,141)(122,145,137,160,132,155,127,150)(123,154,138,149,133,144,128,159)(124,143,139,158,134,153,129,148)(125,152,140,147,135,142,130,157), (1,131,77,54)(2,122,78,45)(3,133,79,56)(4,124,80,47)(5,135,61,58)(6,126,62,49)(7,137,63,60)(8,128,64,51)(9,139,65,42)(10,130,66,53)(11,121,67,44)(12,132,68,55)(13,123,69,46)(14,134,70,57)(15,125,71,48)(16,136,72,59)(17,127,73,50)(18,138,74,41)(19,129,75,52)(20,140,76,43)(21,91,104,143)(22,82,105,154)(23,93,106,145)(24,84,107,156)(25,95,108,147)(26,86,109,158)(27,97,110,149)(28,88,111,160)(29,99,112,151)(30,90,113,142)(31,81,114,153)(32,92,115,144)(33,83,116,155)(34,94,117,146)(35,85,118,157)(36,96,119,148)(37,87,120,159)(38,98,101,150)(39,89,102,141)(40,100,103,152) );
G=PermutationGroup([[(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)], [(1,117,6,102,11,107,16,112),(2,106,7,111,12,116,17,101),(3,115,8,120,13,105,18,110),(4,104,9,109,14,114,19,119),(5,113,10,118,15,103,20,108),(21,65,26,70,31,75,36,80),(22,74,27,79,32,64,37,69),(23,63,28,68,33,73,38,78),(24,72,29,77,34,62,39,67),(25,61,30,66,35,71,40,76),(41,97,56,92,51,87,46,82),(42,86,57,81,52,96,47,91),(43,95,58,90,53,85,48,100),(44,84,59,99,54,94,49,89),(45,93,60,88,55,83,50,98),(121,156,136,151,131,146,126,141),(122,145,137,160,132,155,127,150),(123,154,138,149,133,144,128,159),(124,143,139,158,134,153,129,148),(125,152,140,147,135,142,130,157)], [(1,131,77,54),(2,122,78,45),(3,133,79,56),(4,124,80,47),(5,135,61,58),(6,126,62,49),(7,137,63,60),(8,128,64,51),(9,139,65,42),(10,130,66,53),(11,121,67,44),(12,132,68,55),(13,123,69,46),(14,134,70,57),(15,125,71,48),(16,136,72,59),(17,127,73,50),(18,138,74,41),(19,129,75,52),(20,140,76,43),(21,91,104,143),(22,82,105,154),(23,93,106,145),(24,84,107,156),(25,95,108,147),(26,86,109,158),(27,97,110,149),(28,88,111,160),(29,99,112,151),(30,90,113,142),(31,81,114,153),(32,92,115,144),(33,83,116,155),(34,94,117,146),(35,85,118,157),(36,96,119,148),(37,87,120,159),(38,98,101,150),(39,89,102,141),(40,100,103,152)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8T | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | Dic5 | Dic5 | D10 | C8○D4 | C4×D5 | D4.Dic5 |
| kernel | C20.35C42 | C4×C5⋊2C8 | C42.D5 | C22×C5⋊2C8 | C2×C4.Dic5 | C5×C42⋊C2 | C2×C5⋊2C8 | C4.Dic5 | C5×C22⋊C4 | C5×C4⋊C4 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C10 | C2×C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 8 | 4 | 4 | 2 | 4 | 4 | 4 | 2 | 8 | 16 | 8 |
Matrix representation of C20.35C42 ►in GL4(𝔽41) generated by
| 7 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 18 | 9 |
| 12 | 16 | 0 | 0 |
| 14 | 29 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 9 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,32,18,0,0,0,9],[12,14,0,0,16,29,0,0,0,0,27,0,0,0,0,27],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,9,32] >;
C20.35C42 in GAP, Magma, Sage, TeX
C_{20}._{35}C_4^2 % in TeX
G:=Group("C20.35C4^2"); // GroupNames label
G:=SmallGroup(320,624);
// by ID
G=gap.SmallGroup(320,624);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^9,c*a*c^-1=a^11,b*c=c*b>;
// generators/relations