metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.50D8, D4⋊3Dic10, C42.44D10, (C5×D4)⋊3Q8, (C4×D4).2D5, C5⋊4(D4⋊Q8), (D4×C20).2C2, C20⋊3C8⋊17C2, C10.50(C2×D8), (C2×C20).57D4, C20.25(C2×Q8), C4.23(D4⋊D5), C4⋊C4.238D10, C20⋊2Q8⋊14C2, C4.9(C2×Dic10), (C2×D4).185D10, C20.45(C4○D4), C4.59(C4○D20), C10.D8⋊30C2, (C4×C20).78C22, D4⋊Dic5.7C2, (C2×C20).332C23, C10.61(C22⋊Q8), C2.7(D4.9D10), (D4×C10).227C22, C4⋊Dic5.136C22, C2.12(C20.48D4), C10.107(C8.C22), C2.6(C2×D4⋊D5), (C2×C10).463(C2×D4), (C2×C4).243(C5⋊D4), (C5×C4⋊C4).269C22, (C2×C5⋊2C8).89C22, (C2×C4).432(C22×D5), C22.146(C2×C5⋊D4), SmallGroup(320,634)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.50D8
G = < a,b,c | a20=b8=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a10b-1 >
Subgroups: 358 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4⋊Q8, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C20⋊3C8, C10.D8, D4⋊Dic5, C20⋊2Q8, D4×C20, C20.50D8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×D8, C8.C22, Dic10, C5⋊D4, C22×D5, D4⋊Q8, D4⋊D5, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, C2×D4⋊D5, D4.9D10, C20.50D8
►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 22 49 154 87 116 65 127)(2 21 50 153 88 115 66 126)(3 40 51 152 89 114 67 125)(4 39 52 151 90 113 68 124)(5 38 53 150 91 112 69 123)(6 37 54 149 92 111 70 122)(7 36 55 148 93 110 71 121)(8 35 56 147 94 109 72 140)(9 34 57 146 95 108 73 139)(10 33 58 145 96 107 74 138)(11 32 59 144 97 106 75 137)(12 31 60 143 98 105 76 136)(13 30 41 142 99 104 77 135)(14 29 42 141 100 103 78 134)(15 28 43 160 81 102 79 133)(16 27 44 159 82 101 80 132)(17 26 45 158 83 120 61 131)(18 25 46 157 84 119 62 130)(19 24 47 156 85 118 63 129)(20 23 48 155 86 117 64 128)
(1 137 11 127)(2 136 12 126)(3 135 13 125)(4 134 14 124)(5 133 15 123)(6 132 16 122)(7 131 17 121)(8 130 18 140)(9 129 19 139)(10 128 20 138)(21 66 31 76)(22 65 32 75)(23 64 33 74)(24 63 34 73)(25 62 35 72)(26 61 36 71)(27 80 37 70)(28 79 38 69)(29 78 39 68)(30 77 40 67)(41 114 51 104)(42 113 52 103)(43 112 53 102)(44 111 54 101)(45 110 55 120)(46 109 56 119)(47 108 57 118)(48 107 58 117)(49 106 59 116)(50 105 60 115)(81 150 91 160)(82 149 92 159)(83 148 93 158)(84 147 94 157)(85 146 95 156)(86 145 96 155)(87 144 97 154)(88 143 98 153)(89 142 99 152)(90 141 100 151)
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), (1,22,49,154,87,116,65,127)(2,21,50,153,88,115,66,126)(3,40,51,152,89,114,67,125)(4,39,52,151,90,113,68,124)(5,38,53,150,91,112,69,123)(6,37,54,149,92,111,70,122)(7,36,55,148,93,110,71,121)(8,35,56,147,94,109,72,140)(9,34,57,146,95,108,73,139)(10,33,58,145,96,107,74,138)(11,32,59,144,97,106,75,137)(12,31,60,143,98,105,76,136)(13,30,41,142,99,104,77,135)(14,29,42,141,100,103,78,134)(15,28,43,160,81,102,79,133)(16,27,44,159,82,101,80,132)(17,26,45,158,83,120,61,131)(18,25,46,157,84,119,62,130)(19,24,47,156,85,118,63,129)(20,23,48,155,86,117,64,128), (1,137,11,127)(2,136,12,126)(3,135,13,125)(4,134,14,124)(5,133,15,123)(6,132,16,122)(7,131,17,121)(8,130,18,140)(9,129,19,139)(10,128,20,138)(21,66,31,76)(22,65,32,75)(23,64,33,74)(24,63,34,73)(25,62,35,72)(26,61,36,71)(27,80,37,70)(28,79,38,69)(29,78,39,68)(30,77,40,67)(41,114,51,104)(42,113,52,103)(43,112,53,102)(44,111,54,101)(45,110,55,120)(46,109,56,119)(47,108,57,118)(48,107,58,117)(49,106,59,116)(50,105,60,115)(81,150,91,160)(82,149,92,159)(83,148,93,158)(84,147,94,157)(85,146,95,156)(86,145,96,155)(87,144,97,154)(88,143,98,153)(89,142,99,152)(90,141,100,151)>;
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), (1,22,49,154,87,116,65,127)(2,21,50,153,88,115,66,126)(3,40,51,152,89,114,67,125)(4,39,52,151,90,113,68,124)(5,38,53,150,91,112,69,123)(6,37,54,149,92,111,70,122)(7,36,55,148,93,110,71,121)(8,35,56,147,94,109,72,140)(9,34,57,146,95,108,73,139)(10,33,58,145,96,107,74,138)(11,32,59,144,97,106,75,137)(12,31,60,143,98,105,76,136)(13,30,41,142,99,104,77,135)(14,29,42,141,100,103,78,134)(15,28,43,160,81,102,79,133)(16,27,44,159,82,101,80,132)(17,26,45,158,83,120,61,131)(18,25,46,157,84,119,62,130)(19,24,47,156,85,118,63,129)(20,23,48,155,86,117,64,128), (1,137,11,127)(2,136,12,126)(3,135,13,125)(4,134,14,124)(5,133,15,123)(6,132,16,122)(7,131,17,121)(8,130,18,140)(9,129,19,139)(10,128,20,138)(21,66,31,76)(22,65,32,75)(23,64,33,74)(24,63,34,73)(25,62,35,72)(26,61,36,71)(27,80,37,70)(28,79,38,69)(29,78,39,68)(30,77,40,67)(41,114,51,104)(42,113,52,103)(43,112,53,102)(44,111,54,101)(45,110,55,120)(46,109,56,119)(47,108,57,118)(48,107,58,117)(49,106,59,116)(50,105,60,115)(81,150,91,160)(82,149,92,159)(83,148,93,158)(84,147,94,157)(85,146,95,156)(86,145,96,155)(87,144,97,154)(88,143,98,153)(89,142,99,152)(90,141,100,151) );
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)], [(1,22,49,154,87,116,65,127),(2,21,50,153,88,115,66,126),(3,40,51,152,89,114,67,125),(4,39,52,151,90,113,68,124),(5,38,53,150,91,112,69,123),(6,37,54,149,92,111,70,122),(7,36,55,148,93,110,71,121),(8,35,56,147,94,109,72,140),(9,34,57,146,95,108,73,139),(10,33,58,145,96,107,74,138),(11,32,59,144,97,106,75,137),(12,31,60,143,98,105,76,136),(13,30,41,142,99,104,77,135),(14,29,42,141,100,103,78,134),(15,28,43,160,81,102,79,133),(16,27,44,159,82,101,80,132),(17,26,45,158,83,120,61,131),(18,25,46,157,84,119,62,130),(19,24,47,156,85,118,63,129),(20,23,48,155,86,117,64,128)], [(1,137,11,127),(2,136,12,126),(3,135,13,125),(4,134,14,124),(5,133,15,123),(6,132,16,122),(7,131,17,121),(8,130,18,140),(9,129,19,139),(10,128,20,138),(21,66,31,76),(22,65,32,75),(23,64,33,74),(24,63,34,73),(25,62,35,72),(26,61,36,71),(27,80,37,70),(28,79,38,69),(29,78,39,68),(30,77,40,67),(41,114,51,104),(42,113,52,103),(43,112,53,102),(44,111,54,101),(45,110,55,120),(46,109,56,119),(47,108,57,118),(48,107,58,117),(49,106,59,116),(50,105,60,115),(81,150,91,160),(82,149,92,159),(83,148,93,158),(84,147,94,157),(85,146,95,156),(86,145,96,155),(87,144,97,154),(88,143,98,153),(89,142,99,152),(90,141,100,151)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | D8 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | Dic10 | C4○D20 | C8.C22 | D4⋊D5 | D4.9D10 |
| kernel | C20.50D8 | C20⋊3C8 | C10.D8 | D4⋊Dic5 | C20⋊2Q8 | D4×C20 | C2×C20 | C5×D4 | C4×D4 | C20 | C20 | C42 | C4⋊C4 | C2×D4 | C2×C4 | D4 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of C20.50D8 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 25 | 30 |
| 0 | 0 | 27 | 39 |
| 0 | 17 | 0 | 0 |
| 12 | 17 | 0 | 0 |
| 0 | 0 | 0 | 22 |
| 0 | 0 | 13 | 0 |
| 0 | 17 | 0 | 0 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 |
| 0 | 0 | 13 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,25,27,0,0,30,39],[0,12,0,0,17,17,0,0,0,0,0,13,0,0,22,0],[0,29,0,0,17,0,0,0,0,0,0,13,0,0,22,0] >;
C20.50D8 in GAP, Magma, Sage, TeX
C_{20}._{50}D_8 % in TeX
G:=Group("C20.50D8"); // GroupNames label
G:=SmallGroup(320,634);
// by ID
G=gap.SmallGroup(320,634);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations