metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊14D6, D60⋊9C22, Dic6⋊13D10, C60.54C23, C30.30C24, C15⋊52+ 1+4, D30.15C23, Dic15.17C23, (C4×D5)⋊4D6, (D4×D5)⋊5S3, C5⋊D4⋊6D6, (S3×D20)⋊5C2, (D4×D15)⋊6C2, (C4×S3)⋊4D10, (C5×D4)⋊14D6, D4⋊11(S3×D5), C3⋊D4⋊6D10, (C3×D4)⋊14D10, D4⋊2S3⋊5D5, C5⋊3(D4⋊6D6), (C22×D5)⋊7D6, C15⋊Q8⋊13C22, D10⋊D6⋊4C2, C12.28D10⋊5C2, D20⋊S3⋊5C2, C3⋊3(D4⋊8D10), (S3×C20)⋊5C22, (C2×Dic3)⋊8D10, (C4×D15)⋊5C22, (D5×C12)⋊5C22, C15⋊7D4⋊6C22, (C2×C30).6C23, C6.30(C23×D5), Dic5.D6⋊5C2, D6.D10⋊3C2, (C3×D20)⋊10C22, (D4×C15)⋊12C22, C15⋊D4⋊15C22, C3⋊D20⋊21C22, C5⋊D12⋊15C22, C10.30(S3×C23), C20.54(C22×S3), D30.C2⋊2C22, (C5×Dic6)⋊9C22, (D5×Dic3)⋊3C22, D6.14(C22×D5), (C6×D5).46C23, C12.54(C22×D5), (S3×C10).15C23, D10.15(C22×S3), (C10×Dic3)⋊14C22, (C22×D15)⋊12C22, Dic3.27(C22×D5), (C3×Dic5).15C23, Dic5.16(C22×S3), (C5×Dic3).30C23, (C3×D4×D5)⋊7C2, C4.54(C2×S3×D5), (D5×C3⋊D4)⋊5C2, (D5×C2×C6)⋊9C22, (C2×S3×D5)⋊5C22, C22.6(C2×S3×D5), (C5×D4⋊2S3)⋊7C2, (C2×C3⋊D20)⋊20C2, C2.33(C22×S3×D5), (C3×C5⋊D4)⋊6C22, (C5×C3⋊D4)⋊6C22, (C2×C6).6(C22×D5), (C2×C10).6(C22×S3), SmallGroup(480,1102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊14D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=a11, dad=a9, cbc-1=a10b, dbd=a18b, dcd=c-1 >
Subgroups: 1932 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, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C2×D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, 2+ 1+4, Dic10, C4×D5, C4×D5, D20, D20, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C4○D12, S3×D4, D4⋊2S3, D4⋊2S3, C2×C3⋊D4, C6×D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, C6×D5, C6×D5, S3×C10, D30, D30, D30, C2×C30, C2×D20, C4○D20, D4×D5, D4×D5, Q8⋊2D5, C5×C4○D4, D4⋊6D6, D5×Dic3, D30.C2, C15⋊D4, C3⋊D20, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, C4×D15, D60, C15⋊7D4, D4×C15, C2×S3×D5, D5×C2×C6, C22×D15, D4⋊8D10, D20⋊S3, D6.D10, C12.28D10, S3×D20, Dic5.D6, C2×C3⋊D20, D5×C3⋊D4, D10⋊D6, C3×D4×D5, C5×D4⋊2S3, D4×D15, D20⋊14D6
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⋊8D10, C22×S3×D5, D20⋊14D6
►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 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 76 86)(2 67 87 12 77 97)(3 78 88)(4 69 89 14 79 99)(5 80 90)(6 71 91 16 61 81)(7 62 92)(8 73 93 18 63 83)(9 64 94)(10 75 95 20 65 85)(11 66 96)(13 68 98)(15 70 100)(17 72 82)(19 74 84)(21 114 57 31 104 47)(22 105 58)(23 116 59 33 106 49)(24 107 60)(25 118 41 35 108 51)(26 109 42)(27 120 43 37 110 53)(28 111 44)(29 102 45 39 112 55)(30 113 46)(32 115 48)(34 117 50)(36 119 52)(38 101 54)(40 103 56)
(1 60)(2 49)(3 58)(4 47)(5 56)(6 45)(7 54)(8 43)(9 52)(10 41)(11 50)(12 59)(13 48)(14 57)(15 46)(16 55)(17 44)(18 53)(19 42)(20 51)(21 99)(22 88)(23 97)(24 86)(25 95)(26 84)(27 93)(28 82)(29 91)(30 100)(31 89)(32 98)(33 87)(34 96)(35 85)(36 94)(37 83)(38 92)(39 81)(40 90)(61 112)(62 101)(63 110)(64 119)(65 108)(66 117)(67 106)(68 115)(69 104)(70 113)(71 102)(72 111)(73 120)(74 109)(75 118)(76 107)(77 116)(78 105)(79 114)(80 103)
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)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,76,86)(2,67,87,12,77,97)(3,78,88)(4,69,89,14,79,99)(5,80,90)(6,71,91,16,61,81)(7,62,92)(8,73,93,18,63,83)(9,64,94)(10,75,95,20,65,85)(11,66,96)(13,68,98)(15,70,100)(17,72,82)(19,74,84)(21,114,57,31,104,47)(22,105,58)(23,116,59,33,106,49)(24,107,60)(25,118,41,35,108,51)(26,109,42)(27,120,43,37,110,53)(28,111,44)(29,102,45,39,112,55)(30,113,46)(32,115,48)(34,117,50)(36,119,52)(38,101,54)(40,103,56), (1,60)(2,49)(3,58)(4,47)(5,56)(6,45)(7,54)(8,43)(9,52)(10,41)(11,50)(12,59)(13,48)(14,57)(15,46)(16,55)(17,44)(18,53)(19,42)(20,51)(21,99)(22,88)(23,97)(24,86)(25,95)(26,84)(27,93)(28,82)(29,91)(30,100)(31,89)(32,98)(33,87)(34,96)(35,85)(36,94)(37,83)(38,92)(39,81)(40,90)(61,112)(62,101)(63,110)(64,119)(65,108)(66,117)(67,106)(68,115)(69,104)(70,113)(71,102)(72,111)(73,120)(74,109)(75,118)(76,107)(77,116)(78,105)(79,114)(80,103)>;
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)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,76,86)(2,67,87,12,77,97)(3,78,88)(4,69,89,14,79,99)(5,80,90)(6,71,91,16,61,81)(7,62,92)(8,73,93,18,63,83)(9,64,94)(10,75,95,20,65,85)(11,66,96)(13,68,98)(15,70,100)(17,72,82)(19,74,84)(21,114,57,31,104,47)(22,105,58)(23,116,59,33,106,49)(24,107,60)(25,118,41,35,108,51)(26,109,42)(27,120,43,37,110,53)(28,111,44)(29,102,45,39,112,55)(30,113,46)(32,115,48)(34,117,50)(36,119,52)(38,101,54)(40,103,56), (1,60)(2,49)(3,58)(4,47)(5,56)(6,45)(7,54)(8,43)(9,52)(10,41)(11,50)(12,59)(13,48)(14,57)(15,46)(16,55)(17,44)(18,53)(19,42)(20,51)(21,99)(22,88)(23,97)(24,86)(25,95)(26,84)(27,93)(28,82)(29,91)(30,100)(31,89)(32,98)(33,87)(34,96)(35,85)(36,94)(37,83)(38,92)(39,81)(40,90)(61,112)(62,101)(63,110)(64,119)(65,108)(66,117)(67,106)(68,115)(69,104)(70,113)(71,102)(72,111)(73,120)(74,109)(75,118)(76,107)(77,116)(78,105)(79,114)(80,103) );
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),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)], [(1,76,86),(2,67,87,12,77,97),(3,78,88),(4,69,89,14,79,99),(5,80,90),(6,71,91,16,61,81),(7,62,92),(8,73,93,18,63,83),(9,64,94),(10,75,95,20,65,85),(11,66,96),(13,68,98),(15,70,100),(17,72,82),(19,74,84),(21,114,57,31,104,47),(22,105,58),(23,116,59,33,106,49),(24,107,60),(25,118,41,35,108,51),(26,109,42),(27,120,43,37,110,53),(28,111,44),(29,102,45,39,112,55),(30,113,46),(32,115,48),(34,117,50),(36,119,52),(38,101,54),(40,103,56)], [(1,60),(2,49),(3,58),(4,47),(5,56),(6,45),(7,54),(8,43),(9,52),(10,41),(11,50),(12,59),(13,48),(14,57),(15,46),(16,55),(17,44),(18,53),(19,42),(20,51),(21,99),(22,88),(23,97),(24,86),(25,95),(26,84),(27,93),(28,82),(29,91),(30,100),(31,89),(32,98),(33,87),(34,96),(35,85),(36,94),(37,83),(38,92),(39,81),(40,90),(61,112),(62,101),(63,110),(64,119),(65,108),(66,117),(67,106),(68,115),(69,104),(70,113),(71,102),(72,111),(73,120),(74,109),(75,118),(76,107),(77,116),(78,105),(79,114),(80,103)]])
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 | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 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 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 10 | 10 | 10 | 30 | 30 | 30 | 2 | 2 | 6 | 6 | 6 | 10 | 30 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 20 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 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⋊8D10 | D20⋊14D6 |
| kernel | D20⋊14D6 | D20⋊S3 | D6.D10 | C12.28D10 | S3×D20 | Dic5.D6 | C2×C3⋊D20 | D5×C3⋊D4 | D10⋊D6 | C3×D4×D5 | C5×D4⋊2S3 | D4×D15 | D4×D5 | D4⋊2S3 | C4×D5 | D20 | C5⋊D4 | C5×D4 | C22×D5 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | 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 | 4 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20⋊14D6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 17 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 44 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 27 | 24 | 0 | 0 | 0 | 0 |
| 51 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 54 | 0 | 0 |
| 0 | 0 | 59 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 54 |
| 0 | 0 | 0 | 0 | 59 | 32 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,60,44,0,0,0,60,0,0,0,0,1,17,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[0,60,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[27,51,0,0,0,0,24,34,0,0,0,0,0,0,29,59,0,0,0,0,54,32,0,0,0,0,0,0,29,59,0,0,0,0,54,32] >;
D20⋊14D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{14}D_6 % in TeX
G:=Group("D20:14D6"); // GroupNames label
G:=SmallGroup(480,1102);
// by ID
G=gap.SmallGroup(480,1102);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,185,80,1356,18822]);
// 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,d*a*d=a^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations