metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.37C42, M4(2)⋊5Dic5, C40.81(C2×C4), C40⋊8C4⋊27C2, C4.7(C4×Dic5), (C8×Dic5)⋊30C2, (C2×C8).275D10, (C5×M4(2))⋊9C4, C4⋊Dic5.33C4, C8.12(C2×Dic5), C23.28(C4×D5), C10.55(C8○D4), C5⋊9(C8○2M4(2)), (C2×C10).29C42, C10.46(C2×C42), C23.D5.18C4, C22.7(C4×Dic5), C20.236(C22×C4), (C2×C40).233C22, (C2×C20).865C23, (C22×C4).347D10, (C10×M4(2)).6C2, (C2×M4(2)).19D5, C4.35(C22×Dic5), C2.5(D20.2C4), (C22×C20).181C22, (C4×Dic5).316C22, C23.21D10.18C2, C4.115(C2×C4×D5), (C2×C5⋊2C8)⋊11C4, (C2×C4).83(C4×D5), C2.14(C2×C4×Dic5), C22.63(C2×C4×D5), C5⋊2C8.46(C2×C4), (C2×C20).274(C2×C4), (C2×C4).47(C2×Dic5), (C22×C5⋊2C8).11C2, (C2×C4).807(C22×D5), (C2×C10).236(C22×C4), (C22×C10).133(C2×C4), (C2×C5⋊2C8).357C22, (C2×Dic5).112(C2×C4), SmallGroup(320,749)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.37C42
G = < a,b,c | a20=1, b4=c4=a10, bab-1=a9, cac-1=a11, bc=cb >
Subgroups: 286 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C8○2M4(2), C2×C5⋊2C8, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C22×C20, C8×Dic5, C40⋊8C4, C22×C5⋊2C8, C23.21D10, C10×M4(2), C20.37C42
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, D20.2C4, C2×C4×Dic5, C20.37C42
►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)
(1 100 6 85 11 90 16 95)(2 89 7 94 12 99 17 84)(3 98 8 83 13 88 18 93)(4 87 9 92 14 97 19 82)(5 96 10 81 15 86 20 91)(21 118 36 113 31 108 26 103)(22 107 37 102 32 117 27 112)(23 116 38 111 33 106 28 101)(24 105 39 120 34 115 29 110)(25 114 40 109 35 104 30 119)(41 71 46 76 51 61 56 66)(42 80 47 65 52 70 57 75)(43 69 48 74 53 79 58 64)(44 78 49 63 54 68 59 73)(45 67 50 72 55 77 60 62)(121 143 136 158 131 153 126 148)(122 152 137 147 132 142 127 157)(123 141 138 156 133 151 128 146)(124 150 139 145 134 160 129 155)(125 159 140 154 135 149 130 144)
(1 34 67 150 11 24 77 160)(2 25 68 141 12 35 78 151)(3 36 69 152 13 26 79 142)(4 27 70 143 14 37 80 153)(5 38 71 154 15 28 61 144)(6 29 72 145 16 39 62 155)(7 40 73 156 17 30 63 146)(8 31 74 147 18 21 64 157)(9 22 75 158 19 32 65 148)(10 33 76 149 20 23 66 159)(41 140 81 106 51 130 91 116)(42 131 82 117 52 121 92 107)(43 122 83 108 53 132 93 118)(44 133 84 119 54 123 94 109)(45 124 85 110 55 134 95 120)(46 135 86 101 56 125 96 111)(47 126 87 112 57 136 97 102)(48 137 88 103 58 127 98 113)(49 128 89 114 59 138 99 104)(50 139 90 105 60 129 100 115)
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), (1,100,6,85,11,90,16,95)(2,89,7,94,12,99,17,84)(3,98,8,83,13,88,18,93)(4,87,9,92,14,97,19,82)(5,96,10,81,15,86,20,91)(21,118,36,113,31,108,26,103)(22,107,37,102,32,117,27,112)(23,116,38,111,33,106,28,101)(24,105,39,120,34,115,29,110)(25,114,40,109,35,104,30,119)(41,71,46,76,51,61,56,66)(42,80,47,65,52,70,57,75)(43,69,48,74,53,79,58,64)(44,78,49,63,54,68,59,73)(45,67,50,72,55,77,60,62)(121,143,136,158,131,153,126,148)(122,152,137,147,132,142,127,157)(123,141,138,156,133,151,128,146)(124,150,139,145,134,160,129,155)(125,159,140,154,135,149,130,144), (1,34,67,150,11,24,77,160)(2,25,68,141,12,35,78,151)(3,36,69,152,13,26,79,142)(4,27,70,143,14,37,80,153)(5,38,71,154,15,28,61,144)(6,29,72,145,16,39,62,155)(7,40,73,156,17,30,63,146)(8,31,74,147,18,21,64,157)(9,22,75,158,19,32,65,148)(10,33,76,149,20,23,66,159)(41,140,81,106,51,130,91,116)(42,131,82,117,52,121,92,107)(43,122,83,108,53,132,93,118)(44,133,84,119,54,123,94,109)(45,124,85,110,55,134,95,120)(46,135,86,101,56,125,96,111)(47,126,87,112,57,136,97,102)(48,137,88,103,58,127,98,113)(49,128,89,114,59,138,99,104)(50,139,90,105,60,129,100,115)>;
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), (1,100,6,85,11,90,16,95)(2,89,7,94,12,99,17,84)(3,98,8,83,13,88,18,93)(4,87,9,92,14,97,19,82)(5,96,10,81,15,86,20,91)(21,118,36,113,31,108,26,103)(22,107,37,102,32,117,27,112)(23,116,38,111,33,106,28,101)(24,105,39,120,34,115,29,110)(25,114,40,109,35,104,30,119)(41,71,46,76,51,61,56,66)(42,80,47,65,52,70,57,75)(43,69,48,74,53,79,58,64)(44,78,49,63,54,68,59,73)(45,67,50,72,55,77,60,62)(121,143,136,158,131,153,126,148)(122,152,137,147,132,142,127,157)(123,141,138,156,133,151,128,146)(124,150,139,145,134,160,129,155)(125,159,140,154,135,149,130,144), (1,34,67,150,11,24,77,160)(2,25,68,141,12,35,78,151)(3,36,69,152,13,26,79,142)(4,27,70,143,14,37,80,153)(5,38,71,154,15,28,61,144)(6,29,72,145,16,39,62,155)(7,40,73,156,17,30,63,146)(8,31,74,147,18,21,64,157)(9,22,75,158,19,32,65,148)(10,33,76,149,20,23,66,159)(41,140,81,106,51,130,91,116)(42,131,82,117,52,121,92,107)(43,122,83,108,53,132,93,118)(44,133,84,119,54,123,94,109)(45,124,85,110,55,134,95,120)(46,135,86,101,56,125,96,111)(47,126,87,112,57,136,97,102)(48,137,88,103,58,127,98,113)(49,128,89,114,59,138,99,104)(50,139,90,105,60,129,100,115) );
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)], [(1,100,6,85,11,90,16,95),(2,89,7,94,12,99,17,84),(3,98,8,83,13,88,18,93),(4,87,9,92,14,97,19,82),(5,96,10,81,15,86,20,91),(21,118,36,113,31,108,26,103),(22,107,37,102,32,117,27,112),(23,116,38,111,33,106,28,101),(24,105,39,120,34,115,29,110),(25,114,40,109,35,104,30,119),(41,71,46,76,51,61,56,66),(42,80,47,65,52,70,57,75),(43,69,48,74,53,79,58,64),(44,78,49,63,54,68,59,73),(45,67,50,72,55,77,60,62),(121,143,136,158,131,153,126,148),(122,152,137,147,132,142,127,157),(123,141,138,156,133,151,128,146),(124,150,139,145,134,160,129,155),(125,159,140,154,135,149,130,144)], [(1,34,67,150,11,24,77,160),(2,25,68,141,12,35,78,151),(3,36,69,152,13,26,79,142),(4,27,70,143,14,37,80,153),(5,38,71,154,15,28,61,144),(6,29,72,145,16,39,62,155),(7,40,73,156,17,30,63,146),(8,31,74,147,18,21,64,157),(9,22,75,158,19,32,65,148),(10,33,76,149,20,23,66,159),(41,140,81,106,51,130,91,116),(42,131,82,117,52,121,92,107),(43,122,83,108,53,132,93,118),(44,133,84,119,54,123,94,109),(45,124,85,110,55,134,95,120),(46,135,86,101,56,125,96,111),(47,126,87,112,57,136,97,102),(48,137,88,103,58,127,98,113),(49,128,89,114,59,138,99,104),(50,139,90,105,60,129,100,115)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.2C4 |
| kernel | C20.37C42 | C8×Dic5 | C40⋊8C4 | C22×C5⋊2C8 | C23.21D10 | C10×M4(2) | C2×C5⋊2C8 | C4⋊Dic5 | C23.D5 | C5×M4(2) | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 8 | 2 | 4 | 8 | 2 | 8 | 12 | 4 | 8 |
Matrix representation of C20.37C42 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 35 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 15 | 2 |
| 0 | 0 | 0 | 10 | 26 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,35,40,0,0,0,1,0],[32,0,0,0,0,0,14,0,0,0,0,0,14,0,0,0,0,0,15,10,0,0,0,2,26],[40,0,0,0,0,0,0,32,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,32] >;
C20.37C42 in GAP, Magma, Sage, TeX
C_{20}._{37}C_4^2 % in TeX
G:=Group("C20.37C4^2"); // GroupNames label
G:=SmallGroup(320,749);
// by ID
G=gap.SmallGroup(320,749);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,136,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^9,c*a*c^-1=a^11,b*c=c*b>;
// generators/relations