metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊13D6, D12⋊13D10, C30.29C24, C60.53C23, C15⋊42+ (1+4), Dic30⋊9C22, D30.14C23, Dic15.16C23, (D4×D5)⋊4S3, (C4×D5)⋊3D6, (S3×D4)⋊4D5, C5⋊D4⋊5D6, D4⋊10(S3×D5), (C5×D4)⋊13D6, (C4×S3)⋊3D10, C3⋊D4⋊5D10, (C3×D4)⋊13D10, C20⋊D6⋊5C2, C5⋊2(D4⋊6D6), (C22×D5)⋊6D6, C15⋊Q8⋊12C22, D12⋊5D5⋊5C2, D4⋊2D15⋊7C2, D20⋊5S3⋊5C2, C3⋊2(D4⋊6D10), (S3×C20)⋊4C22, (C4×D15)⋊4C22, (C3×D20)⋊9C22, (C22×S3)⋊5D10, (D5×C12)⋊4C22, (C5×D12)⋊9C22, C15⋊7D4⋊5C22, (C2×C30).5C23, C6.29(C23×D5), D6.D10⋊2C2, C30.C23⋊5C2, (D4×C15)⋊11C22, C5⋊D12⋊14C22, C3⋊D20⋊14C22, C15⋊D4⋊14C22, C10.29(S3×C23), C20.53(C22×S3), (S3×Dic5)⋊2C22, (D5×Dic3)⋊2C22, (C6×D5).45C23, D6.26(C22×D5), C12.53(C22×D5), (S3×C10).14C23, D10.14(C22×S3), (C2×Dic15)⋊18C22, Dic3.15(C22×D5), Dic5.15(C22×S3), (C3×Dic5).14C23, (C5×Dic3).16C23, (C5×S3×D4)⋊6C2, (C3×D4×D5)⋊6C2, C4.53(C2×S3×D5), (D5×C3⋊D4)⋊4C2, (S3×C5⋊D4)⋊4C2, (C2×S3×D5)⋊4C22, (D5×C2×C6)⋊8C22, C22.5(C2×S3×D5), (S3×C2×C10)⋊8C22, (C2×C15⋊D4)⋊20C2, C2.32(C22×S3×D5), (C3×C5⋊D4)⋊5C22, (C5×C3⋊D4)⋊5C22, (C2×C6).5(C22×D5), (C2×C10).5(C22×S3), SmallGroup(480,1101)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1740 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×5], C22 [×2], C22 [×13], C5, S3 [×4], C6, C6 [×5], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], D5 [×4], C10, C10 [×5], Dic3, Dic3 [×3], C12, C12, D6, D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×5], C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5 [×3], C20, C20, D10, D10 [×2], D10 [×5], C2×C10 [×2], C2×C10 [×5], Dic6 [×2], C4×S3, C4×S3 [×3], D12, D12, C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12, C3×D4, C3×D4 [×3], C22×S3 [×2], C22×S3 [×2], C22×C6 [×2], C5×S3 [×3], C3×D5 [×3], D15, C30, C30 [×2], 2+ (1+4), Dic10 [×2], C4×D5, C4×D5 [×3], D20, D20, C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4, C5×D4 [×3], C22×D5 [×2], C22×D5 [×2], C22×C10 [×2], C4○D12 [×2], S3×D4, S3×D4 [×3], D4⋊2S3 [×4], C2×C3⋊D4 [×4], C6×D4, C5×Dic3, C3×Dic5, Dic15, Dic15 [×2], C60, S3×D5 [×2], C6×D5, C6×D5 [×2], C6×D5 [×2], S3×C10, S3×C10 [×2], S3×C10 [×2], D30, C2×C30 [×2], C4○D20 [×2], D4×D5, D4×D5 [×3], D4⋊2D5 [×4], C2×C5⋊D4 [×4], D4×C10, D4⋊6D6, D5×Dic3 [×2], S3×Dic5 [×2], C15⋊D4, C15⋊D4 [×6], C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4 [×2], S3×C20, C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15, C2×Dic15 [×2], C15⋊7D4 [×2], D4×C15, C2×S3×D5 [×2], D5×C2×C6 [×2], S3×C2×C10 [×2], D4⋊6D10, D20⋊5S3, D6.D10, D12⋊5D5, C20⋊D6, C30.C23 [×2], C2×C15⋊D4 [×2], D5×C3⋊D4 [×2], S3×C5⋊D4 [×2], C3×D4×D5, C5×S3×D4, D4⋊2D15, D20⋊13D6
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⋊6D6, C2×S3×D5 [×3], D4⋊6D10, C22×S3×D5, D20⋊13D6
Generators and relations
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd=a10b, 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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)
(1 22 64 6 37 69)(2 33 65 17 38 80)(3 24 66 8 39 71)(4 35 67 19 40 62)(5 26 68 10 21 73)(7 28 70 12 23 75)(9 30 72 14 25 77)(11 32 74 16 27 79)(13 34 76 18 29 61)(15 36 78 20 31 63)(41 89 108 46 84 113)(42 100 109 57 85 104)(43 91 110 48 86 115)(44 82 111 59 87 106)(45 93 112 50 88 117)(47 95 114 52 90 119)(49 97 116 54 92 101)(51 99 118 56 94 103)(53 81 120 58 96 105)(55 83 102 60 98 107)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 101)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 81)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117), (1,22,64,6,37,69)(2,33,65,17,38,80)(3,24,66,8,39,71)(4,35,67,19,40,62)(5,26,68,10,21,73)(7,28,70,12,23,75)(9,30,72,14,25,77)(11,32,74,16,27,79)(13,34,76,18,29,61)(15,36,78,20,31,63)(41,89,108,46,84,113)(42,100,109,57,85,104)(43,91,110,48,86,115)(44,82,111,59,87,106)(45,93,112,50,88,117)(47,95,114,52,90,119)(49,97,116,54,92,101)(51,99,118,56,94,103)(53,81,120,58,96,105)(55,83,102,60,98,107), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,101)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,81)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117), (1,22,64,6,37,69)(2,33,65,17,38,80)(3,24,66,8,39,71)(4,35,67,19,40,62)(5,26,68,10,21,73)(7,28,70,12,23,75)(9,30,72,14,25,77)(11,32,74,16,27,79)(13,34,76,18,29,61)(15,36,78,20,31,63)(41,89,108,46,84,113)(42,100,109,57,85,104)(43,91,110,48,86,115)(44,82,111,59,87,106)(45,93,112,50,88,117)(47,95,114,52,90,119)(49,97,116,54,92,101)(51,99,118,56,94,103)(53,81,120,58,96,105)(55,83,102,60,98,107), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,101)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,81)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,120),(114,119),(115,118),(116,117)], [(1,22,64,6,37,69),(2,33,65,17,38,80),(3,24,66,8,39,71),(4,35,67,19,40,62),(5,26,68,10,21,73),(7,28,70,12,23,75),(9,30,72,14,25,77),(11,32,74,16,27,79),(13,34,76,18,29,61),(15,36,78,20,31,63),(41,89,108,46,84,113),(42,100,109,57,85,104),(43,91,110,48,86,115),(44,82,111,59,87,106),(45,93,112,50,88,117),(47,95,114,52,90,119),(49,97,116,54,92,101),(51,99,118,56,94,103),(53,81,120,58,96,105),(55,83,102,60,98,107)], [(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,118),(18,119),(19,120),(20,101),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,81),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 43 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 57 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 44 | 0 | 0 |
| 0 | 0 | 17 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 44 |
| 0 | 0 | 0 | 0 | 17 | 31 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,1,43,0,0,0,1,0,0,0,0,60,18,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0],[47,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,15,0,0,0,0,57,0,0,0,0,0,0,0,30,17,0,0,0,0,44,31,0,0,0,0,0,0,30,17,0,0,0,0,44,31] >;
57 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 | 6E | 6F | 6G | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| 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 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 10 | 10 | 10 | 30 | 2 | 2 | 6 | 10 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 20 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | 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⋊6D6 | C2×S3×D5 | C2×S3×D5 | D4⋊6D10 | D20⋊13D6 |
| kernel | D20⋊13D6 | D20⋊5S3 | D6.D10 | D12⋊5D5 | C20⋊D6 | C30.C23 | C2×C15⋊D4 | D5×C3⋊D4 | S3×C5⋊D4 | C3×D4×D5 | C5×S3×D4 | D4⋊2D15 | D4×D5 | S3×D4 | C4×D5 | D20 | C5⋊D4 | C5×D4 | C22×D5 | C4×S3 | D12 | C3⋊D4 | C3×D4 | C22×S3 | C15 | D4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{13}D_6 % in TeX
G:=Group("D20:13D6"); // GroupNames label
G:=SmallGroup(480,1101);
// by ID
G=gap.SmallGroup(480,1101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,185,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations