metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊29D6, D12⋊29D10, Dic10⋊26D6, Dic6⋊26D10, D60⋊41C22, C30.23C24, D30.9C23, C15⋊32+ 1+4, C60.115C23, (C4×D5)⋊2D6, C5⋊D4⋊9D6, C4○D20⋊9S3, C4○D12⋊9D5, (C4×S3)⋊2D10, (C2×C20)⋊10D6, C5⋊2(D4○D12), C3⋊D4⋊9D10, (S3×D20)⋊12C2, (C2×D60)⋊19C2, (D5×D12)⋊12C2, (C2×C12)⋊10D10, D10⋊D6⋊1C2, C3⋊2(D4⋊8D10), (C2×C60)⋊11C22, C3⋊D20⋊2C22, C5⋊D12⋊2C22, D60⋊C2⋊12C2, C12.28D10⋊12C2, D6.9(C22×D5), (C6×D5).9C23, C6.23(C23×D5), (S3×C20)⋊11C22, (D5×C12)⋊11C22, (C3×D20)⋊26C22, (C5×D12)⋊26C22, (S3×C10).9C23, C10.23(S3×C23), D30.C2⋊1C22, D10.9(C22×S3), (C2×C30).242C23, C20.132(C22×S3), (C5×Dic6)⋊23C22, C12.130(C22×D5), (C3×Dic10)⋊23C22, (C22×D15)⋊10C22, Dic3.11(C22×D5), (C3×Dic5).11C23, (C5×Dic3).11C23, Dic5.11(C22×S3), (C2×C4)⋊6(S3×D5), C4.89(C2×S3×D5), (C2×S3×D5)⋊3C22, (C3×C4○D20)⋊8C2, (C5×C4○D12)⋊8C2, C2.26(C22×S3×D5), C22.19(C2×S3×D5), (C5×C3⋊D4)⋊11C22, (C3×C5⋊D4)⋊11C22, (C2×C6).13(C22×D5), (C2×C10).13(C22×S3), SmallGroup(480,1095)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊29D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a10b, dbd=a8b, dcd=c-1 >
Subgroups: 2060 in 332 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C30, 2+ 1+4, Dic10, C4×D5, C4×D5, D20, D20, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C2×D12, C4○D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, C60, S3×D5, C6×D5, S3×C10, D30, D30, C2×C30, C2×D20, C4○D20, C4○D20, D4×D5, Q8⋊2D5, C5×C4○D4, D4○D12, D30.C2, C3⋊D20, C5⋊D12, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, D60, C2×C60, C2×S3×D5, C22×D15, D4⋊8D10, D60⋊C2, C12.28D10, D5×D12, S3×D20, D10⋊D6, C3×C4○D20, C5×C4○D12, C2×D60, D20⋊29D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2+ 1+4, C22×D5, S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5, D4⋊8D10, C22×S3×D5, D20⋊29D6
►On 120 points
(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 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 109)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)
(1 31 76 113 92 55)(2 32 77 114 93 56)(3 33 78 115 94 57)(4 34 79 116 95 58)(5 35 80 117 96 59)(6 36 61 118 97 60)(7 37 62 119 98 41)(8 38 63 120 99 42)(9 39 64 101 100 43)(10 40 65 102 81 44)(11 21 66 103 82 45)(12 22 67 104 83 46)(13 23 68 105 84 47)(14 24 69 106 85 48)(15 25 70 107 86 49)(16 26 71 108 87 50)(17 27 72 109 88 51)(18 28 73 110 89 52)(19 29 74 111 90 53)(20 30 75 112 91 54)
(1 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 80)(18 79)(19 78)(20 77)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 108)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)
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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116), (1,31,76,113,92,55)(2,32,77,114,93,56)(3,33,78,115,94,57)(4,34,79,116,95,58)(5,35,80,117,96,59)(6,36,61,118,97,60)(7,37,62,119,98,41)(8,38,63,120,99,42)(9,39,64,101,100,43)(10,40,65,102,81,44)(11,21,66,103,82,45)(12,22,67,104,83,46)(13,23,68,105,84,47)(14,24,69,106,85,48)(15,25,70,107,86,49)(16,26,71,108,87,50)(17,27,72,109,88,51)(18,28,73,110,89,52)(19,29,74,111,90,53)(20,30,75,112,91,54), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,80)(18,79)(19,78)(20,77)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)>;
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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116), (1,31,76,113,92,55)(2,32,77,114,93,56)(3,33,78,115,94,57)(4,34,79,116,95,58)(5,35,80,117,96,59)(6,36,61,118,97,60)(7,37,62,119,98,41)(8,38,63,120,99,42)(9,39,64,101,100,43)(10,40,65,102,81,44)(11,21,66,103,82,45)(12,22,67,104,83,46)(13,23,68,105,84,47)(14,24,69,106,85,48)(15,25,70,107,86,49)(16,26,71,108,87,50)(17,27,72,109,88,51)(18,28,73,110,89,52)(19,29,74,111,90,53)(20,30,75,112,91,54), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,80)(18,79)(19,78)(20,77)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93) );
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),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,65),(62,64),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100),(101,109),(102,108),(103,107),(104,106),(110,120),(111,119),(112,118),(113,117),(114,116)], [(1,31,76,113,92,55),(2,32,77,114,93,56),(3,33,78,115,94,57),(4,34,79,116,95,58),(5,35,80,117,96,59),(6,36,61,118,97,60),(7,37,62,119,98,41),(8,38,63,120,99,42),(9,39,64,101,100,43),(10,40,65,102,81,44),(11,21,66,103,82,45),(12,22,67,104,83,46),(13,23,68,105,84,47),(14,24,69,106,85,48),(15,25,70,107,86,49),(16,26,71,108,87,50),(17,27,72,109,88,51),(18,28,73,110,89,52),(19,29,74,111,90,53),(20,30,75,112,91,54)], [(1,76),(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,80),(18,79),(19,78),(20,77),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,107),(42,106),(43,105),(44,104),(45,103),(46,102),(47,101),(48,120),(49,119),(50,118),(51,117),(52,116),(53,115),(54,114),(55,113),(56,112),(57,111),(58,110),(59,109),(60,108),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D10 | 2+ 1+4 | S3×D5 | D4○D12 | C2×S3×D5 | C2×S3×D5 | D4⋊8D10 | D20⋊29D6 |
| kernel | D20⋊29D6 | D60⋊C2 | C12.28D10 | D5×D12 | S3×D20 | D10⋊D6 | C3×C4○D20 | C5×C4○D12 | C2×D60 | C4○D20 | C4○D12 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of D20⋊29D6 ►in GL8(𝔽61)
| 43 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 0 | 7 | 32 |
| 0 | 0 | 0 | 0 | 3 | 10 | 29 | 2 |
| 1 | 43 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 27 | 0 | 60 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 19 | 44 | 44 |
| 0 | 0 | 0 | 0 | 3 | 28 | 0 | 43 |
| 0 | 0 | 0 | 0 | 30 | 19 | 9 | 42 |
| 0 | 0 | 0 | 0 | 46 | 6 | 48 | 20 |
| 43 | 18 | 18 | 43 | 0 | 0 | 0 | 0 |
| 60 | 18 | 1 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 37 | 60 | 0 |
| 0 | 0 | 0 | 0 | 31 | 9 | 44 | 1 |
G:=sub<GL(8,GF(61))| [43,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,2,51,3,0,0,0,0,32,34,0,10,0,0,0,0,0,0,7,29,0,0,0,0,0,0,32,2],[1,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,1,19,27,27,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0,60,0,0,0,0,0,0,60,0,60,0,0,0,0,0,0,0,0,4,3,30,46,0,0,0,0,19,28,19,6,0,0,0,0,44,0,9,48,0,0,0,0,44,43,42,20],[43,60,0,0,0,0,0,0,18,18,0,0,0,0,0,0,18,1,18,1,0,0,0,0,43,43,43,43,0,0,0,0,0,0,0,0,43,43,42,31,0,0,0,0,1,18,37,9,0,0,0,0,0,0,60,44,0,0,0,0,0,0,0,1] >;
D20⋊29D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{29}D_6 % in TeX
G:=Group("D20:29D6"); // GroupNames label
G:=SmallGroup(480,1095);
// by ID
G=gap.SmallGroup(480,1095);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^10*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations