metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊6D6, D12.26D10, C60.26C23, D60.9C22, C3⋊C8⋊11D10, Q8⋊5(S3×D5), (C5×Q8)⋊5D6, (D5×D12)⋊3C2, C3⋊D40⋊5C2, C15⋊D8⋊5C2, (C3×Q8)⋊8D10, (C4×D5).9D6, C5⋊3(D4⋊D6), C3⋊7(D40⋊C2), Q8⋊2S3⋊1D5, Q8⋊2D5⋊4S3, (C6×D5).12D4, C6.147(D4×D5), C15⋊22(C8⋊C22), Q8⋊2D15⋊2C2, C30.188(C2×D4), (C3×D20)⋊6C22, (Q8×C15)⋊2C22, C15⋊3C8⋊12C22, C20.32D6⋊5C2, C20.26(C22×S3), (C3×Dic5).70D4, (C5×D12).9C22, C12.26(C22×D5), D10.19(C3⋊D4), (D5×C12).10C22, Dic5.33(C3⋊D4), C4.26(C2×S3×D5), (C5×C3⋊C8)⋊12C22, C2.29(D5×C3⋊D4), (C5×Q8⋊2S3)⋊2C2, (C3×Q8⋊2D5)⋊1C2, C10.50(C2×C3⋊D4), SmallGroup(480,578)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a3b, dcd=c-1 >
Subgroups: 924 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, C23, D5, C10, C10, 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, D12, D12, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3, Q8⋊2S3, Q8⋊2S3, C2×D12, C3×C4○D4, C3×Dic5, C60, C60, S3×D5, C6×D5, C6×D5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, D4⋊D6, C5×C3⋊C8, C15⋊3C8, C5⋊D12, D5×C12, D5×C12, C3×D20, C3×D20, C5×D12, D60, Q8×C15, C2×S3×D5, D40⋊C2, C20.32D6, C15⋊D8, C3⋊D40, C5×Q8⋊2S3, Q8⋊2D15, D5×D12, C3×Q8⋊2D5, D20⋊D6
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, D4⋊D6, C2×S3×D5, D40⋊C2, D5×C3⋊D4, D20⋊D6
►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 61)(2 80)(3 79)(4 78)(5 77)(6 76)(7 75)(8 74)(9 73)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 103)(22 102)(23 101)(24 120)(25 119)(26 118)(27 117)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 104)(41 85)(42 84)(43 83)(44 82)(45 81)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)
(1 83 109)(2 92 110 10 84 118)(3 81 111 19 85 107)(4 90 112 8 86 116)(5 99 113 17 87 105)(6 88 114)(7 97 115 15 89 103)(9 95 117 13 91 101)(11 93 119)(12 82 120 20 94 108)(14 100 102 18 96 106)(16 98 104)(21 77 49 31 67 59)(22 66 50 40 68 48)(23 75 51 29 69 57)(24 64 52 38 70 46)(25 73 53 27 71 55)(26 62 54 36 72 44)(28 80 56 34 74 42)(30 78 58 32 76 60)(33 65 41 39 79 47)(35 63 43 37 61 45)
(1 104)(2 103)(3 102)(4 101)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 80)(39 79)(40 78)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)
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), (1,61)(2,80)(3,79)(4,78)(5,77)(6,76)(7,75)(8,74)(9,73)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,103)(22,102)(23,101)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86), (1,83,109)(2,92,110,10,84,118)(3,81,111,19,85,107)(4,90,112,8,86,116)(5,99,113,17,87,105)(6,88,114)(7,97,115,15,89,103)(9,95,117,13,91,101)(11,93,119)(12,82,120,20,94,108)(14,100,102,18,96,106)(16,98,104)(21,77,49,31,67,59)(22,66,50,40,68,48)(23,75,51,29,69,57)(24,64,52,38,70,46)(25,73,53,27,71,55)(26,62,54,36,72,44)(28,80,56,34,74,42)(30,78,58,32,76,60)(33,65,41,39,79,47)(35,63,43,37,61,45), (1,104)(2,103)(3,102)(4,101)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,61)(38,80)(39,79)(40,78)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)>;
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), (1,61)(2,80)(3,79)(4,78)(5,77)(6,76)(7,75)(8,74)(9,73)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,103)(22,102)(23,101)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86), (1,83,109)(2,92,110,10,84,118)(3,81,111,19,85,107)(4,90,112,8,86,116)(5,99,113,17,87,105)(6,88,114)(7,97,115,15,89,103)(9,95,117,13,91,101)(11,93,119)(12,82,120,20,94,108)(14,100,102,18,96,106)(16,98,104)(21,77,49,31,67,59)(22,66,50,40,68,48)(23,75,51,29,69,57)(24,64,52,38,70,46)(25,73,53,27,71,55)(26,62,54,36,72,44)(28,80,56,34,74,42)(30,78,58,32,76,60)(33,65,41,39,79,47)(35,63,43,37,61,45), (1,104)(2,103)(3,102)(4,101)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,61)(38,80)(39,79)(40,78)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91) );
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)], [(1,61),(2,80),(3,79),(4,78),(5,77),(6,76),(7,75),(8,74),(9,73),(10,72),(11,71),(12,70),(13,69),(14,68),(15,67),(16,66),(17,65),(18,64),(19,63),(20,62),(21,103),(22,102),(23,101),(24,120),(25,119),(26,118),(27,117),(28,116),(29,115),(30,114),(31,113),(32,112),(33,111),(34,110),(35,109),(36,108),(37,107),(38,106),(39,105),(40,104),(41,85),(42,84),(43,83),(44,82),(45,81),(46,100),(47,99),(48,98),(49,97),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86)], [(1,83,109),(2,92,110,10,84,118),(3,81,111,19,85,107),(4,90,112,8,86,116),(5,99,113,17,87,105),(6,88,114),(7,97,115,15,89,103),(9,95,117,13,91,101),(11,93,119),(12,82,120,20,94,108),(14,100,102,18,96,106),(16,98,104),(21,77,49,31,67,59),(22,66,50,40,68,48),(23,75,51,29,69,57),(24,64,52,38,70,46),(25,73,53,27,71,55),(26,62,54,36,72,44),(28,80,56,34,74,42),(30,78,58,32,76,60),(33,65,41,39,79,47),(35,63,43,37,61,45)], [(1,104),(2,103),(3,102),(4,101),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,77),(22,76),(23,75),(24,74),(25,73),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62),(37,61),(38,80),(39,79),(40,78),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 10 | 12 | 20 | 60 | 2 | 2 | 4 | 10 | 2 | 2 | 2 | 20 | 20 | 20 | 12 | 60 | 2 | 2 | 24 | 24 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 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 | D4⋊D6 | C2×S3×D5 | D40⋊C2 | D5×C3⋊D4 | D20⋊D6 |
| kernel | D20⋊D6 | C20.32D6 | C15⋊D8 | C3⋊D40 | C5×Q8⋊2S3 | Q8⋊2D15 | D5×D12 | C3×Q8⋊2D5 | Q8⋊2D5 | C3×Dic5 | C6×D5 | Q8⋊2S3 | C4×D5 | D20 | C5×Q8 | C3⋊C8 | D12 | C3×Q8 | Dic5 | D10 | C15 | Q8 | 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⋊D6 ►in GL8(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 50 | 190 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 192 | 0 | 0 |
| 0 | 0 | 0 | 0 | 123 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 118 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 191 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 171 | 140 | 0 | 0 | 0 | 0 |
| 0 | 0 | 101 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 192 |
| 0 | 0 | 0 | 0 | 118 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 123 | 240 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 51 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 51 | 190 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 118 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 118 | 0 | 0 | 240 |
| 190 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 190 | 51 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 192 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 118 | 1 | 0 | 240 |
G:=sub<GL(8,GF(241))| [240,50,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,123,118,0,0,0,0,0,192,240,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0],[1,191,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,171,101,0,0,0,0,0,0,140,70,0,0,0,0,0,0,0,0,240,118,123,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,192,240,1,1],[51,51,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,1,0,118,118,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[190,190,0,0,0,0,0,0,240,51,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,118,0,0,0,0,192,240,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240] >;
D20⋊D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes D_6 % in TeX
G:=Group("D20:D6"); // GroupNames label
G:=SmallGroup(480,578);
// by ID
G=gap.SmallGroup(480,578);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,100,346,185,80,1356,18822]);
// 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^18*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations