metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D20, C40⋊10D4, C5⋊3(C8⋊D4), C2.D8⋊11D5, C4⋊C4.51D10, (C2×C8).66D10, C4.54(C2×D20), D10⋊2Q8⋊7C2, C20.134(C2×D4), D20⋊6C4⋊23C2, C20.41(C4○D4), C10.Q16⋊20C2, C4⋊D20.10C2, (C2×Dic5).58D4, (C22×D5).37D4, C22.232(D4×D5), C2.24(D8⋊D5), C2.20(C4⋊D20), C10.47(C4⋊D4), C10.43(C8⋊C22), (C2×C20).302C23, (C2×C40).144C22, C4.10(Q8⋊2D5), (C2×D20).88C22, C2.23(Q16⋊D5), C10.71(C8.C22), (C2×Dic10).94C22, (C5×C2.D8)⋊8C2, (C2×C8⋊D5)⋊6C2, (C2×C40⋊C2)⋊22C2, (C2×C4×D5).45C22, (C2×C10).307(C2×D4), (C5×C4⋊C4).95C22, (C2×C5⋊2C8).72C22, (C2×C4).405(C22×D5), SmallGroup(320,513)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊3D20
G = < a,b,c | a8=b20=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >
Subgroups: 598 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C8⋊D4, C8⋊D5, C40⋊C2, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D20⋊6C4, C10.Q16, C5×C2.D8, C4⋊D20, D10⋊2Q8, C2×C8⋊D5, C2×C40⋊C2, C8⋊3D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, D20, C22×D5, C8⋊D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D8⋊D5, Q16⋊D5, C8⋊3D20
►On 160 points
Generators in S160
(1 104 68 154 48 90 24 123)(2 124 25 91 49 155 69 105)(3 106 70 156 50 92 26 125)(4 126 27 93 51 157 71 107)(5 108 72 158 52 94 28 127)(6 128 29 95 53 159 73 109)(7 110 74 160 54 96 30 129)(8 130 31 97 55 141 75 111)(9 112 76 142 56 98 32 131)(10 132 33 99 57 143 77 113)(11 114 78 144 58 100 34 133)(12 134 35 81 59 145 79 115)(13 116 80 146 60 82 36 135)(14 136 37 83 41 147 61 117)(15 118 62 148 42 84 38 137)(16 138 39 85 43 149 63 119)(17 120 64 150 44 86 40 139)(18 140 21 87 45 151 65 101)(19 102 66 152 46 88 22 121)(20 122 23 89 47 153 67 103)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 65)(22 64)(23 63)(24 62)(25 61)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 72)(35 71)(36 70)(37 69)(38 68)(39 67)(40 66)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(81 126)(82 125)(83 124)(84 123)(85 122)(86 121)(87 140)(88 139)(89 138)(90 137)(91 136)(92 135)(93 134)(94 133)(95 132)(96 131)(97 130)(98 129)(99 128)(100 127)(101 151)(102 150)(103 149)(104 148)(105 147)(106 146)(107 145)(108 144)(109 143)(110 142)(111 141)(112 160)(113 159)(114 158)(115 157)(116 156)(117 155)(118 154)(119 153)(120 152)
G:=sub<Sym(160)| (1,104,68,154,48,90,24,123)(2,124,25,91,49,155,69,105)(3,106,70,156,50,92,26,125)(4,126,27,93,51,157,71,107)(5,108,72,158,52,94,28,127)(6,128,29,95,53,159,73,109)(7,110,74,160,54,96,30,129)(8,130,31,97,55,141,75,111)(9,112,76,142,56,98,32,131)(10,132,33,99,57,143,77,113)(11,114,78,144,58,100,34,133)(12,134,35,81,59,145,79,115)(13,116,80,146,60,82,36,135)(14,136,37,83,41,147,61,117)(15,118,62,148,42,84,38,137)(16,138,39,85,43,149,63,119)(17,120,64,150,44,86,40,139)(18,140,21,87,45,151,65,101)(19,102,66,152,46,88,22,121)(20,122,23,89,47,153,67,103), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,65)(22,64)(23,63)(24,62)(25,61)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,140)(88,139)(89,138)(90,137)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,130)(98,129)(99,128)(100,127)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,160)(113,159)(114,158)(115,157)(116,156)(117,155)(118,154)(119,153)(120,152)>;
G:=Group( (1,104,68,154,48,90,24,123)(2,124,25,91,49,155,69,105)(3,106,70,156,50,92,26,125)(4,126,27,93,51,157,71,107)(5,108,72,158,52,94,28,127)(6,128,29,95,53,159,73,109)(7,110,74,160,54,96,30,129)(8,130,31,97,55,141,75,111)(9,112,76,142,56,98,32,131)(10,132,33,99,57,143,77,113)(11,114,78,144,58,100,34,133)(12,134,35,81,59,145,79,115)(13,116,80,146,60,82,36,135)(14,136,37,83,41,147,61,117)(15,118,62,148,42,84,38,137)(16,138,39,85,43,149,63,119)(17,120,64,150,44,86,40,139)(18,140,21,87,45,151,65,101)(19,102,66,152,46,88,22,121)(20,122,23,89,47,153,67,103), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,65)(22,64)(23,63)(24,62)(25,61)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,140)(88,139)(89,138)(90,137)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,130)(98,129)(99,128)(100,127)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,160)(113,159)(114,158)(115,157)(116,156)(117,155)(118,154)(119,153)(120,152) );
G=PermutationGroup([[(1,104,68,154,48,90,24,123),(2,124,25,91,49,155,69,105),(3,106,70,156,50,92,26,125),(4,126,27,93,51,157,71,107),(5,108,72,158,52,94,28,127),(6,128,29,95,53,159,73,109),(7,110,74,160,54,96,30,129),(8,130,31,97,55,141,75,111),(9,112,76,142,56,98,32,131),(10,132,33,99,57,143,77,113),(11,114,78,144,58,100,34,133),(12,134,35,81,59,145,79,115),(13,116,80,146,60,82,36,135),(14,136,37,83,41,147,61,117),(15,118,62,148,42,84,38,137),(16,138,39,85,43,149,63,119),(17,120,64,150,44,86,40,139),(18,140,21,87,45,151,65,101),(19,102,66,152,46,88,22,121),(20,122,23,89,47,153,67,103)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,65),(22,64),(23,63),(24,62),(25,61),(26,80),(27,79),(28,78),(29,77),(30,76),(31,75),(32,74),(33,73),(34,72),(35,71),(36,70),(37,69),(38,68),(39,67),(40,66),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(81,126),(82,125),(83,124),(84,123),(85,122),(86,121),(87,140),(88,139),(89,138),(90,137),(91,136),(92,135),(93,134),(94,133),(95,132),(96,131),(97,130),(98,129),(99,128),(100,127),(101,151),(102,150),(103,149),(104,148),(105,147),(106,146),(107,145),(108,144),(109,143),(110,142),(111,141),(112,160),(113,159),(114,158),(115,157),(116,156),(117,155),(118,154),(119,153),(120,152)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 40 | 2 | 2 | 8 | 8 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C8⋊C22 | C8.C22 | Q8⋊2D5 | D4×D5 | D8⋊D5 | Q16⋊D5 |
| kernel | C8⋊3D20 | D20⋊6C4 | C10.Q16 | C5×C2.D8 | C4⋊D20 | D10⋊2Q8 | C2×C8⋊D5 | C2×C40⋊C2 | C40 | C2×Dic5 | C22×D5 | C2.D8 | C20 | C4⋊C4 | C2×C8 | C8 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C8⋊3D20 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 15 | 26 |
| 0 | 0 | 0 | 0 | 32 | 18 | 24 | 0 |
| 0 | 0 | 0 | 0 | 28 | 21 | 3 | 20 |
| 0 | 0 | 0 | 0 | 0 | 38 | 21 | 20 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 18 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 18 | 40 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,32,28,0,0,0,0,0,1,18,21,38,0,0,0,0,15,24,3,21,0,0,0,0,26,0,20,20],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,1,0,18,18,0,0,0,0,0,0,0,40,0,0,0,0,9,40,40,40,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,23,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40] >;
C8⋊3D20 in GAP, Magma, Sage, TeX
C_8\rtimes_3D_{20} % in TeX
G:=Group("C8:3D20"); // GroupNames label
G:=SmallGroup(320,513);
// by ID
G=gap.SmallGroup(320,513);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,219,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations