metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊5D20, C40⋊23D4, C20⋊7SD16, C42.260D10, (C4×C8)⋊12D5, (C4×C40)⋊17C2, C5⋊1(C8⋊5D4), C4⋊1(C40⋊C2), C20⋊2Q8⋊1C2, C4.30(C2×D20), (C2×C4).79D20, (C2×C20).376D4, C20⋊4D4.2C2, C20.273(C2×D4), (C2×C8).317D10, C10.4(C2×SD16), C10.3(C4⋊1D4), C2.5(C20⋊4D4), (C2×D20).3C22, C22.89(C2×D20), (C2×C40).389C22, (C4×C20).306C22, (C2×C20).722C23, (C2×Dic10).4C22, (C2×C40⋊C2)⋊7C2, C2.7(C2×C40⋊C2), (C2×C10).105(C2×D4), (C2×C4).665(C22×D5), SmallGroup(320,320)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊5D20
G = < a,b,c | a8=b20=c2=1, ab=ba, cac=a3, cbc=b-1 >
Subgroups: 782 in 142 conjugacy classes, 55 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C40, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C8⋊5D4, C40⋊C2, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C2×D20, C2×D20, C4×C40, C20⋊2Q8, C20⋊4D4, C2×C40⋊C2, C8⋊5D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C4⋊1D4, C2×SD16, D20, C22×D5, C8⋊5D4, C40⋊C2, C2×D20, C20⋊4D4, C2×C40⋊C2, C8⋊5D20
►On 160 points
Generators in S160
(1 100 48 140 112 28 150 75)(2 81 49 121 113 29 151 76)(3 82 50 122 114 30 152 77)(4 83 51 123 115 31 153 78)(5 84 52 124 116 32 154 79)(6 85 53 125 117 33 155 80)(7 86 54 126 118 34 156 61)(8 87 55 127 119 35 157 62)(9 88 56 128 120 36 158 63)(10 89 57 129 101 37 159 64)(11 90 58 130 102 38 160 65)(12 91 59 131 103 39 141 66)(13 92 60 132 104 40 142 67)(14 93 41 133 105 21 143 68)(15 94 42 134 106 22 144 69)(16 95 43 135 107 23 145 70)(17 96 44 136 108 24 146 71)(18 97 45 137 109 25 147 72)(19 98 46 138 110 26 148 73)(20 99 47 139 111 27 149 74)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 61)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 160)(58 159)(59 158)(60 157)(81 138)(82 137)(83 136)(84 135)(85 134)(86 133)(87 132)(88 131)(89 130)(90 129)(91 128)(92 127)(93 126)(94 125)(95 124)(96 123)(97 122)(98 121)(99 140)(100 139)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
G:=sub<Sym(160)| (1,100,48,140,112,28,150,75)(2,81,49,121,113,29,151,76)(3,82,50,122,114,30,152,77)(4,83,51,123,115,31,153,78)(5,84,52,124,116,32,154,79)(6,85,53,125,117,33,155,80)(7,86,54,126,118,34,156,61)(8,87,55,127,119,35,157,62)(9,88,56,128,120,36,158,63)(10,89,57,129,101,37,159,64)(11,90,58,130,102,38,160,65)(12,91,59,131,103,39,141,66)(13,92,60,132,104,40,142,67)(14,93,41,133,105,21,143,68)(15,94,42,134,106,22,144,69)(16,95,43,135,107,23,145,70)(17,96,44,136,108,24,146,71)(18,97,45,137,109,25,147,72)(19,98,46,138,110,26,148,73)(20,99,47,139,111,27,149,74), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,61)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,160)(58,159)(59,158)(60,157)(81,138)(82,137)(83,136)(84,135)(85,134)(86,133)(87,132)(88,131)(89,130)(90,129)(91,128)(92,127)(93,126)(94,125)(95,124)(96,123)(97,122)(98,121)(99,140)(100,139)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)>;
G:=Group( (1,100,48,140,112,28,150,75)(2,81,49,121,113,29,151,76)(3,82,50,122,114,30,152,77)(4,83,51,123,115,31,153,78)(5,84,52,124,116,32,154,79)(6,85,53,125,117,33,155,80)(7,86,54,126,118,34,156,61)(8,87,55,127,119,35,157,62)(9,88,56,128,120,36,158,63)(10,89,57,129,101,37,159,64)(11,90,58,130,102,38,160,65)(12,91,59,131,103,39,141,66)(13,92,60,132,104,40,142,67)(14,93,41,133,105,21,143,68)(15,94,42,134,106,22,144,69)(16,95,43,135,107,23,145,70)(17,96,44,136,108,24,146,71)(18,97,45,137,109,25,147,72)(19,98,46,138,110,26,148,73)(20,99,47,139,111,27,149,74), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,61)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,160)(58,159)(59,158)(60,157)(81,138)(82,137)(83,136)(84,135)(85,134)(86,133)(87,132)(88,131)(89,130)(90,129)(91,128)(92,127)(93,126)(94,125)(95,124)(96,123)(97,122)(98,121)(99,140)(100,139)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112) );
G=PermutationGroup([[(1,100,48,140,112,28,150,75),(2,81,49,121,113,29,151,76),(3,82,50,122,114,30,152,77),(4,83,51,123,115,31,153,78),(5,84,52,124,116,32,154,79),(6,85,53,125,117,33,155,80),(7,86,54,126,118,34,156,61),(8,87,55,127,119,35,157,62),(9,88,56,128,120,36,158,63),(10,89,57,129,101,37,159,64),(11,90,58,130,102,38,160,65),(12,91,59,131,103,39,141,66),(13,92,60,132,104,40,142,67),(14,93,41,133,105,21,143,68),(15,94,42,134,106,22,144,69),(16,95,43,135,107,23,145,70),(17,96,44,136,108,24,146,71),(18,97,45,137,109,25,147,72),(19,98,46,138,110,26,148,73),(20,99,47,139,111,27,149,74)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,61),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,156),(42,155),(43,154),(44,153),(45,152),(46,151),(47,150),(48,149),(49,148),(50,147),(51,146),(52,145),(53,144),(54,143),(55,142),(56,141),(57,160),(58,159),(59,158),(60,157),(81,138),(82,137),(83,136),(84,135),(85,134),(86,133),(87,132),(88,131),(89,130),(90,129),(91,128),(92,127),(93,126),(94,125),(95,124),(96,123),(97,122),(98,121),(99,140),(100,139),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | ··· | 2 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 |
| kernel | C8⋊5D20 | C4×C40 | C20⋊2Q8 | C20⋊4D4 | C2×C40⋊C2 | C40 | C2×C20 | C4×C8 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 4 | 16 | 8 | 32 |
Matrix representation of C8⋊5D20 ►in GL4(𝔽41) generated by
| 39 | 34 | 0 | 0 |
| 27 | 13 | 0 | 0 |
| 0 | 0 | 14 | 10 |
| 0 | 0 | 2 | 16 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 27 | 2 |
| 0 | 0 | 25 | 11 |
| 40 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 |
| 0 | 0 | 30 | 2 |
| 0 | 0 | 22 | 11 |
G:=sub<GL(4,GF(41))| [39,27,0,0,34,13,0,0,0,0,14,2,0,0,10,16],[40,0,0,0,0,40,0,0,0,0,27,25,0,0,2,11],[40,8,0,0,0,1,0,0,0,0,30,22,0,0,2,11] >;
C8⋊5D20 in GAP, Magma, Sage, TeX
C_8\rtimes_5D_{20} % in TeX
G:=Group("C8:5D20"); // GroupNames label
G:=SmallGroup(320,320);
// by ID
G=gap.SmallGroup(320,320);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations