metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.9D6, D60⋊4C22, Dic6⋊10D10, C60.15C23, C3⋊C8⋊8D10, (D4×D5)⋊3S3, D4⋊D15⋊5C2, C3⋊D40⋊3C2, D4.S3⋊4D5, (C5×D4).7D6, (C4×D5).8D6, C3⋊6(D40⋊C2), (C6×D5).63D4, D4.11(S3×D5), C6.144(D4×D5), C12.28D10⋊1C2, C15⋊17(C8⋊C22), C15⋊3C8⋊8C22, (C3×D4).22D10, C30.D4⋊3C2, C30.177(C2×D4), C5⋊3(D12⋊6C22), C20.32D6⋊4C2, C20.15(C22×S3), (C3×Dic5).14D4, (C5×Dic6)⋊4C22, (D5×C12).7C22, (C3×D20).5C22, (D4×C15).9C22, C12.15(C22×D5), D10.29(C3⋊D4), Dic5.23(C3⋊D4), (C3×D4×D5)⋊3C2, C4.15(C2×S3×D5), (C5×C3⋊C8)⋊8C22, (C5×D4.S3)⋊5C2, C2.26(D5×C3⋊D4), C10.47(C2×C3⋊D4), SmallGroup(480,567)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.9D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a8b, dbd=a3b, dcd=a10c-1 >
Subgroups: 812 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3, C6, C6 [×3], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4 [×2], C22×C6, C3×D5 [×2], D15, C30, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], D4.S3, D4.S3, C4○D12, C6×D4, C5×Dic3, C3×Dic5, C60, C6×D5, C6×D5 [×3], D30, C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, D12⋊6C22, C5×C3⋊C8, C15⋊3C8, D30.C2, C3⋊D20, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, D60, D4×C15, D5×C2×C6, D40⋊C2, C20.32D6, C3⋊D40, C30.D4, C5×D4.S3, D4⋊D15, C12.28D10, C3×D4×D5, D20.9D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D12⋊6C22, C2×S3×D5, D40⋊C2, D5×C3⋊D4, D20.9D6
►On 120 points
Generators in S120
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)
(1 43 112)(2 52 113 10 44 101)(3 41 114 19 45 110)(4 50 115 8 46 119)(5 59 116 17 47 108)(6 48 117)(7 57 118 15 49 106)(9 55 120 13 51 104)(11 53 102)(12 42 103 20 54 111)(14 60 105 18 56 109)(16 58 107)(21 64 87 35 70 81)(22 73 88 24 71 90)(23 62 89 33 72 99)(25 80 91 31 74 97)(26 69 92 40 75 86)(27 78 93 29 76 95)(28 67 94 38 77 84)(30 65 96 36 79 82)(32 63 98 34 61 100)(37 68 83 39 66 85)
(1 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 84)(17 83)(18 82)(19 81)(20 100)(21 104)(22 103)(23 102)(24 101)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 64)(42 63)(43 62)(44 61)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)
G:=sub<Sym(120)| (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120), (1,43,112)(2,52,113,10,44,101)(3,41,114,19,45,110)(4,50,115,8,46,119)(5,59,116,17,47,108)(6,48,117)(7,57,118,15,49,106)(9,55,120,13,51,104)(11,53,102)(12,42,103,20,54,111)(14,60,105,18,56,109)(16,58,107)(21,64,87,35,70,81)(22,73,88,24,71,90)(23,62,89,33,72,99)(25,80,91,31,74,97)(26,69,92,40,75,86)(27,78,93,29,76,95)(28,67,94,38,77,84)(30,65,96,36,79,82)(32,63,98,34,61,100)(37,68,83,39,66,85), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,84)(17,83)(18,82)(19,81)(20,100)(21,104)(22,103)(23,102)(24,101)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,64)(42,63)(43,62)(44,61)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120), (1,43,112)(2,52,113,10,44,101)(3,41,114,19,45,110)(4,50,115,8,46,119)(5,59,116,17,47,108)(6,48,117)(7,57,118,15,49,106)(9,55,120,13,51,104)(11,53,102)(12,42,103,20,54,111)(14,60,105,18,56,109)(16,58,107)(21,64,87,35,70,81)(22,73,88,24,71,90)(23,62,89,33,72,99)(25,80,91,31,74,97)(26,69,92,40,75,86)(27,78,93,29,76,95)(28,67,94,38,77,84)(30,65,96,36,79,82)(32,63,98,34,61,100)(37,68,83,39,66,85), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,84)(17,83)(18,82)(19,81)(20,100)(21,104)(22,103)(23,102)(24,101)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,64)(42,63)(43,62)(44,61)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,80),(74,79),(75,78),(76,77),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(118,120)], [(1,43,112),(2,52,113,10,44,101),(3,41,114,19,45,110),(4,50,115,8,46,119),(5,59,116,17,47,108),(6,48,117),(7,57,118,15,49,106),(9,55,120,13,51,104),(11,53,102),(12,42,103,20,54,111),(14,60,105,18,56,109),(16,58,107),(21,64,87,35,70,81),(22,73,88,24,71,90),(23,62,89,33,72,99),(25,80,91,31,74,97),(26,69,92,40,75,86),(27,78,93,29,76,95),(28,67,94,38,77,84),(30,65,96,36,79,82),(32,63,98,34,61,100),(37,68,83,39,66,85)], [(1,99),(2,98),(3,97),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,90),(11,89),(12,88),(13,87),(14,86),(15,85),(16,84),(17,83),(18,82),(19,81),(20,100),(21,104),(22,103),(23,102),(24,101),(25,120),(26,119),(27,118),(28,117),(29,116),(30,115),(31,114),(32,113),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,105),(41,64),(42,63),(43,62),(44,61),(45,80),(46,79),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65)])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 20 | 60 | 2 | 2 | 10 | 12 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 12 | 60 | 2 | 2 | 8 | 8 | 4 | 20 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C8⋊C22 | S3×D5 | D4×D5 | D12⋊6C22 | C2×S3×D5 | D40⋊C2 | D5×C3⋊D4 | D20.9D6 |
| kernel | D20.9D6 | C20.32D6 | C3⋊D40 | C30.D4 | C5×D4.S3 | D4⋊D15 | C12.28D10 | C3×D4×D5 | D4×D5 | C3×Dic5 | C6×D5 | D4.S3 | C4×D5 | D20 | C5×D4 | C3⋊C8 | Dic6 | C3×D4 | Dic5 | D10 | C15 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.9D6 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 51 | 188 |
| 0 | 0 | 0 | 0 | 52 | 240 |
| 0 | 0 | 124 | 125 | 51 | 240 |
| 0 | 0 | 234 | 233 | 1 | 189 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 52 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 117 | 118 | 240 | 240 |
| 0 | 0 | 7 | 59 | 0 | 1 |
| 240 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 52 | 0 | 0 |
| 0 | 0 | 240 | 52 | 0 | 0 |
| 0 | 0 | 189 | 0 | 51 | 52 |
| 0 | 0 | 240 | 52 | 191 | 190 |
| 112 | 221 | 0 | 0 | 0 | 0 |
| 109 | 129 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 33 | 73 | 0 |
| 0 | 0 | 135 | 208 | 108 | 168 |
| 0 | 0 | 0 | 0 | 212 | 33 |
| 0 | 0 | 0 | 0 | 106 | 29 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,124,234,0,0,1,0,125,233,0,0,51,52,51,1,0,0,188,240,240,189],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,117,7,0,0,52,1,118,59,0,0,0,0,240,0,0,0,0,0,240,1],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,189,240,189,240,0,0,52,52,0,52,0,0,0,0,51,191,0,0,0,0,52,190],[112,109,0,0,0,0,221,129,0,0,0,0,0,0,33,135,0,0,0,0,33,208,0,0,0,0,73,108,212,106,0,0,0,168,33,29] >;
D20.9D6 in GAP, Magma, Sage, TeX
D_{20}._9D_6 % in TeX
G:=Group("D20.9D6"); // GroupNames label
G:=SmallGroup(480,567);
// by ID
G=gap.SmallGroup(480,567);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,135,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=a^10*c^-1>;
// generators/relations