metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊29D6, D12⋊29D10, Dic6⋊26D10, Dic10⋊26D6, 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, C5⋊D12⋊2C22, C3⋊D20⋊2C22, D60⋊C2⋊12C2, C12.28D10⋊12C2, D6.9(C22×D5), (C6×D5).9C23, C6.23(C23×D5), (S3×C20)⋊11C22, (C5×D12)⋊26C22, (C3×D20)⋊26C22, (D5×C12)⋊11C22, (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, (C5×C4○D12)⋊8C2, (C3×C4○D20)⋊8C2, C22.19(C2×S3×D5), C2.26(C22×S3×D5), (C3×C5⋊D4)⋊11C22, (C5×C3⋊D4)⋊11C22, (C2×C6).13(C22×D5), (C2×C10).13(C22×S3), SmallGroup(480,1095)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2060 in 332 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], C5, S3 [×6], C6, C6 [×3], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×10], C2×C6, C2×C6 [×2], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×10], C2×C10, C2×C10 [×2], Dic6, C4×S3 [×2], C4×S3 [×4], D12, D12 [×8], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×6], C5×S3 [×2], C3×D5 [×2], D15 [×4], C30, C30, 2+ (1+4), Dic10, C4×D5 [×2], C4×D5 [×4], D20, D20 [×8], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5 [×6], C2×D12 [×3], C4○D12, C4○D12 [×2], S3×D4 [×6], Q8⋊3S3 [×2], C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30 [×4], D30 [×2], C2×C30, C2×D20 [×3], C4○D20, C4○D20 [×2], D4×D5 [×6], Q8⋊2D5 [×2], C5×C4○D4, D4○D12, D30.C2 [×4], C3⋊D20 [×4], C5⋊D12 [×4], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], D60 [×4], C2×C60, C2×S3×D5 [×4], C22×D15 [×2], D4⋊8D10, D60⋊C2 [×2], C12.28D10 [×2], D5×D12 [×2], S3×D20 [×2], D10⋊D6 [×4], C3×C4○D20, C5×C4○D12, C2×D60, D20⋊29D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5 [×3], D4⋊8D10, C22×S3×D5, D20⋊29D6
Generators and relations
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a10b, dbd=a8b, dcd=c-1 >
►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)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 111)(102 110)(103 109)(104 108)(105 107)(112 120)(113 119)(114 118)(115 117)
(1 43 119 29 77 90)(2 44 120 30 78 91)(3 45 101 31 79 92)(4 46 102 32 80 93)(5 47 103 33 61 94)(6 48 104 34 62 95)(7 49 105 35 63 96)(8 50 106 36 64 97)(9 51 107 37 65 98)(10 52 108 38 66 99)(11 53 109 39 67 100)(12 54 110 40 68 81)(13 55 111 21 69 82)(14 56 112 22 70 83)(15 57 113 23 71 84)(16 58 114 24 72 85)(17 59 115 25 73 86)(18 60 116 26 74 87)(19 41 117 27 75 88)(20 42 118 28 76 89)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 120)(21 98)(22 97)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 90)(30 89)(31 88)(32 87)(33 86)(34 85)(35 84)(36 83)(37 82)(38 81)(39 100)(40 99)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)
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)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,111)(102,110)(103,109)(104,108)(105,107)(112,120)(113,119)(114,118)(115,117), (1,43,119,29,77,90)(2,44,120,30,78,91)(3,45,101,31,79,92)(4,46,102,32,80,93)(5,47,103,33,61,94)(6,48,104,34,62,95)(7,49,105,35,63,96)(8,50,106,36,64,97)(9,51,107,37,65,98)(10,52,108,38,66,99)(11,53,109,39,67,100)(12,54,110,40,68,81)(13,55,111,21,69,82)(14,56,112,22,70,83)(15,57,113,23,71,84)(16,58,114,24,72,85)(17,59,115,25,73,86)(18,60,116,26,74,87)(19,41,117,27,75,88)(20,42,118,28,76,89), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,120)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,100)(40,99)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)>;
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)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,111)(102,110)(103,109)(104,108)(105,107)(112,120)(113,119)(114,118)(115,117), (1,43,119,29,77,90)(2,44,120,30,78,91)(3,45,101,31,79,92)(4,46,102,32,80,93)(5,47,103,33,61,94)(6,48,104,34,62,95)(7,49,105,35,63,96)(8,50,106,36,64,97)(9,51,107,37,65,98)(10,52,108,38,66,99)(11,53,109,39,67,100)(12,54,110,40,68,81)(13,55,111,21,69,82)(14,56,112,22,70,83)(15,57,113,23,71,84)(16,58,114,24,72,85)(17,59,115,25,73,86)(18,60,116,26,74,87)(19,41,117,27,75,88)(20,42,118,28,76,89), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,120)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,100)(40,99)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78) );
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),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(101,111),(102,110),(103,109),(104,108),(105,107),(112,120),(113,119),(114,118),(115,117)], [(1,43,119,29,77,90),(2,44,120,30,78,91),(3,45,101,31,79,92),(4,46,102,32,80,93),(5,47,103,33,61,94),(6,48,104,34,62,95),(7,49,105,35,63,96),(8,50,106,36,64,97),(9,51,107,37,65,98),(10,52,108,38,66,99),(11,53,109,39,67,100),(12,54,110,40,68,81),(13,55,111,21,69,82),(14,56,112,22,70,83),(15,57,113,23,71,84),(16,58,114,24,72,85),(17,59,115,25,73,86),(18,60,116,26,74,87),(19,41,117,27,75,88),(20,42,118,28,76,89)], [(1,119),(2,118),(3,117),(4,116),(5,115),(6,114),(7,113),(8,112),(9,111),(10,110),(11,109),(12,108),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,120),(21,98),(22,97),(23,96),(24,95),(25,94),(26,93),(27,92),(28,91),(29,90),(30,89),(31,88),(32,87),(33,86),(34,85),(35,84),(36,83),(37,82),(38,81),(39,100),(40,99),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78)])
Matrix representation ►G ⊆ 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] >;
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 |
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