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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3, D4.S3, D4.S3, C4○D12, C6×D4, C5×Dic3, C3×Dic5, C60, C6×D5, C6×D5, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, 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 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)
(1 67 114)(2 76 115 10 68 103)(3 65 116 19 69 112)(4 74 117 8 70 101)(5 63 118 17 71 110)(6 72 119)(7 61 120 15 73 108)(9 79 102 13 75 106)(11 77 104)(12 66 105 20 78 113)(14 64 107 18 80 111)(16 62 109)(21 59 98 39 41 96)(22 48 99 28 42 85)(23 57 100 37 43 94)(24 46 81 26 44 83)(25 55 82 35 45 92)(27 53 84 33 47 90)(29 51 86 31 49 88)(30 60 87 40 50 97)(32 58 89 38 52 95)(34 56 91 36 54 93)
(1 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)
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)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112), (1,67,114)(2,76,115,10,68,103)(3,65,116,19,69,112)(4,74,117,8,70,101)(5,63,118,17,71,110)(6,72,119)(7,61,120,15,73,108)(9,79,102,13,75,106)(11,77,104)(12,66,105,20,78,113)(14,64,107,18,80,111)(16,62,109)(21,59,98,39,41,96)(22,48,99,28,42,85)(23,57,100,37,43,94)(24,46,81,26,44,83)(25,55,82,35,45,92)(27,53,84,33,47,90)(29,51,86,31,49,88)(30,60,87,40,50,97)(32,58,89,38,52,95)(34,56,91,36,54,93), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)>;
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)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112), (1,67,114)(2,76,115,10,68,103)(3,65,116,19,69,112)(4,74,117,8,70,101)(5,63,118,17,71,110)(6,72,119)(7,61,120,15,73,108)(9,79,102,13,75,106)(11,77,104)(12,66,105,20,78,113)(14,64,107,18,80,111)(16,62,109)(21,59,98,39,41,96)(22,48,99,28,42,85)(23,57,100,37,43,94)(24,46,81,26,44,83)(25,55,82,35,45,92)(27,53,84,33,47,90)(29,51,86,31,49,88)(30,60,87,40,50,97)(32,58,89,38,52,95)(34,56,91,36,54,93), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111) );
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),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,82),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112)], [(1,67,114),(2,76,115,10,68,103),(3,65,116,19,69,112),(4,74,117,8,70,101),(5,63,118,17,71,110),(6,72,119),(7,61,120,15,73,108),(9,79,102,13,75,106),(11,77,104),(12,66,105,20,78,113),(14,64,107,18,80,111),(16,62,109),(21,59,98,39,41,96),(22,48,99,28,42,85),(23,57,100,37,43,94),(24,46,81,26,44,83),(25,55,82,35,45,92),(27,53,84,33,47,90),(29,51,86,31,49,88),(30,60,87,40,50,97),(32,58,89,38,52,95),(34,56,91,36,54,93)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,66),(22,65),(23,64),(24,63),(25,62),(26,61),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,74),(34,73),(35,72),(36,71),(37,70),(38,69),(39,68),(40,67),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111)]])
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