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, (S3×D4)⋊4D5, (C4×D5)⋊3D6, C5⋊D4⋊5D6, (C5×D4)⋊13D6, D4⋊10(S3×D5), (C4×S3)⋊3D10, C3⋊D4⋊5D10, (C3×D4)⋊13D10, C20⋊D6⋊5C2, C5⋊2(D4⋊6D6), (C22×D5)⋊6D6, C15⋊Q8⋊12C22, D4⋊2D15⋊7C2, D12⋊5D5⋊5C2, D20⋊5S3⋊5C2, C3⋊2(D4⋊6D10), (S3×C20)⋊4C22, (D5×C12)⋊4C22, (C3×D20)⋊9C22, (C22×S3)⋊5D10, (C4×D15)⋊4C22, (C5×D12)⋊9C22, C15⋊7D4⋊5C22, (C2×C30).5C23, C6.29(C23×D5), C30.C23⋊5C2, D6.D10⋊2C2, (D4×C15)⋊11C22, C5⋊D12⋊14C22, C3⋊D20⋊14C22, C15⋊D4⋊14C22, C10.29(S3×C23), C20.53(C22×S3), (D5×Dic3)⋊2C22, (S3×Dic5)⋊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, (D5×C2×C6)⋊8C22, (C2×S3×D5)⋊4C22, C22.5(C2×S3×D5), (S3×C2×C10)⋊8C22, (C2×C15⋊D4)⋊20C2, C2.32(C22×S3×D5), (C5×C3⋊D4)⋊5C22, (C3×C5⋊D4)⋊5C22, (C2×C6).5(C22×D5), (C2×C10).5(C22×S3), SmallGroup(480,1101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊13D6
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 >
Subgroups: 1740 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C15, C2×D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, 2+ 1+4, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C4○D12, S3×D4, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, S3×D5, C6×D5, C6×D5, C6×D5, S3×C10, S3×C10, S3×C10, D30, C2×C30, C4○D20, D4×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, D4×C10, D4⋊6D6, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, C2×S3×D5, D5×C2×C6, S3×C2×C10, D4⋊6D10, 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, D20⋊13D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2+ 1+4, C22×D5, S3×C23, S3×D5, C23×D5, D4⋊6D6, C2×S3×D5, D4⋊6D10, C22×S3×D5, D20⋊13D6
►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 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)
(1 77 119 6 72 104)(2 68 120 17 73 115)(3 79 101 8 74 106)(4 70 102 19 75 117)(5 61 103 10 76 108)(7 63 105 12 78 110)(9 65 107 14 80 112)(11 67 109 16 62 114)(13 69 111 18 64 116)(15 71 113 20 66 118)(21 97 58 36 82 53)(22 88 59 27 83 44)(23 99 60 38 84 55)(24 90 41 29 85 46)(25 81 42 40 86 57)(26 92 43 31 87 48)(28 94 45 33 89 50)(30 96 47 35 91 52)(32 98 49 37 93 54)(34 100 51 39 95 56)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 41)(20 42)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(61 96)(62 97)(63 98)(64 99)(65 100)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)
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)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119), (1,77,119,6,72,104)(2,68,120,17,73,115)(3,79,101,8,74,106)(4,70,102,19,75,117)(5,61,103,10,76,108)(7,63,105,12,78,110)(9,65,107,14,80,112)(11,67,109,16,62,114)(13,69,111,18,64,116)(15,71,113,20,66,118)(21,97,58,36,82,53)(22,88,59,27,83,44)(23,99,60,38,84,55)(24,90,41,29,85,46)(25,81,42,40,86,57)(26,92,43,31,87,48)(28,94,45,33,89,50)(30,96,47,35,91,52)(32,98,49,37,93,54)(34,100,51,39,95,56), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,41)(20,42)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)>;
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)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119), (1,77,119,6,72,104)(2,68,120,17,73,115)(3,79,101,8,74,106)(4,70,102,19,75,117)(5,61,103,10,76,108)(7,63,105,12,78,110)(9,65,107,14,80,112)(11,67,109,16,62,114)(13,69,111,18,64,116)(15,71,113,20,66,118)(21,97,58,36,82,53)(22,88,59,27,83,44)(23,99,60,38,84,55)(24,90,41,29,85,46)(25,81,42,40,86,57)(26,92,43,31,87,48)(28,94,45,33,89,50)(30,96,47,35,91,52)(32,98,49,37,93,54)(34,100,51,39,95,56), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,41)(20,42)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95) );
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),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,82),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109),(117,120),(118,119)], [(1,77,119,6,72,104),(2,68,120,17,73,115),(3,79,101,8,74,106),(4,70,102,19,75,117),(5,61,103,10,76,108),(7,63,105,12,78,110),(9,65,107,14,80,112),(11,67,109,16,62,114),(13,69,111,18,64,116),(15,71,113,20,66,118),(21,97,58,36,82,53),(22,88,59,27,83,44),(23,99,60,38,84,55),(24,90,41,29,85,46),(25,81,42,40,86,57),(26,92,43,31,87,48),(28,94,45,33,89,50),(30,96,47,35,91,52),(32,98,49,37,93,54),(34,100,51,39,95,56)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,41),(20,42),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,101),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,113),(61,96),(62,97),(63,98),(64,99),(65,100),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95)]])
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 |
Matrix representation of D20⋊13D6 ►in 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] >;
D20⋊13D6 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