metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊4D20, D30⋊3D4, (C6×D5)⋊3D4, (C2×C20)⋊1D6, D6⋊C4⋊19D5, C15⋊1C22≀C2, (C2×D20)⋊6S3, (S3×C10)⋊3D4, (C6×D20)⋊16C2, (C2×C12)⋊18D10, C5⋊1(C23⋊2D6), C2.28(S3×D20), C6.27(C2×D20), C6.137(D4×D5), C10.26(S3×D4), (C22×D5)⋊2D6, D10⋊5(C3⋊D4), C3⋊3(C22⋊D20), (C2×C60)⋊19C22, (C2×Dic3)⋊1D10, C30.162(C2×D4), D30⋊3C4⋊17C2, D10⋊Dic3⋊24C2, C2.28(C20⋊D6), (C2×C30).164C23, (C10×Dic3)⋊3C22, (C2×Dic15)⋊7C22, (C22×S3).50D10, (C22×D15).56C22, (C2×C4)⋊1(S3×D5), (D5×C2×C6)⋊1C22, (C22×S3×D5)⋊1C2, (C5×D6⋊C4)⋊19C2, (C2×C15⋊D4)⋊9C2, (C2×C3⋊D20)⋊9C2, C2.19(D5×C3⋊D4), C10.40(C2×C3⋊D4), C22.212(C2×S3×D5), (S3×C2×C10).42C22, (C2×C6).176(C22×D5), (C2×C10).176(C22×S3), SmallGroup(480,550)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊4D20
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, ac=ca, dad=a19, cbc-1=a15b, dbd=a3b, dcd=c-1 >
Subgroups: 1820 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, D6⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, Dic15, C60, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, D10⋊C4, C5×C22⋊C4, C2×D20, C2×D20, C2×C5⋊D4, C23×D5, C23⋊2D6, C15⋊D4, C3⋊D20, C3×D20, C10×Dic3, C2×Dic15, C2×C60, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C22⋊D20, D10⋊Dic3, C5×D6⋊C4, D30⋊3C4, C2×C15⋊D4, C2×C3⋊D20, C6×D20, C22×S3×D5, D6⋊4D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22≀C2, D20, C22×D5, S3×D4, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, C23⋊2D6, C2×S3×D5, C22⋊D20, S3×D20, C20⋊D6, D5×C3⋊D4, D6⋊4D20
►On 120 points
Generators in S120
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 35)(32 34)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(91 101)(92 100)(93 99)(94 98)(95 97)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)
(1 56 76 104)(2 57 77 105)(3 58 78 106)(4 59 79 107)(5 60 80 108)(6 31 81 109)(7 32 82 110)(8 33 83 111)(9 34 84 112)(10 35 85 113)(11 36 86 114)(12 37 87 115)(13 38 88 116)(14 39 89 117)(15 40 90 118)(16 41 61 119)(17 42 62 120)(18 43 63 91)(19 44 64 92)(20 45 65 93)(21 46 66 94)(22 47 67 95)(23 48 68 96)(24 49 69 97)(25 50 70 98)(26 51 71 99)(27 52 72 100)(28 53 73 101)(29 54 74 102)(30 55 75 103)
(1 104)(2 93)(3 112)(4 101)(5 120)(6 109)(7 98)(8 117)(9 106)(10 95)(11 114)(12 103)(13 92)(14 111)(15 100)(16 119)(17 108)(18 97)(19 116)(20 105)(21 94)(22 113)(23 102)(24 91)(25 110)(26 99)(27 118)(28 107)(29 96)(30 115)(31 81)(32 70)(33 89)(34 78)(35 67)(36 86)(37 75)(38 64)(39 83)(40 72)(41 61)(42 80)(43 69)(44 88)(45 77)(46 66)(47 85)(48 74)(49 63)(50 82)(51 71)(52 90)(53 79)(54 68)(55 87)(56 76)(57 65)(58 84)(59 73)(60 62)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,35)(32,34)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,101)(92,100)(93,99)(94,98)(95,97)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112), (1,56,76,104)(2,57,77,105)(3,58,78,106)(4,59,79,107)(5,60,80,108)(6,31,81,109)(7,32,82,110)(8,33,83,111)(9,34,84,112)(10,35,85,113)(11,36,86,114)(12,37,87,115)(13,38,88,116)(14,39,89,117)(15,40,90,118)(16,41,61,119)(17,42,62,120)(18,43,63,91)(19,44,64,92)(20,45,65,93)(21,46,66,94)(22,47,67,95)(23,48,68,96)(24,49,69,97)(25,50,70,98)(26,51,71,99)(27,52,72,100)(28,53,73,101)(29,54,74,102)(30,55,75,103), (1,104)(2,93)(3,112)(4,101)(5,120)(6,109)(7,98)(8,117)(9,106)(10,95)(11,114)(12,103)(13,92)(14,111)(15,100)(16,119)(17,108)(18,97)(19,116)(20,105)(21,94)(22,113)(23,102)(24,91)(25,110)(26,99)(27,118)(28,107)(29,96)(30,115)(31,81)(32,70)(33,89)(34,78)(35,67)(36,86)(37,75)(38,64)(39,83)(40,72)(41,61)(42,80)(43,69)(44,88)(45,77)(46,66)(47,85)(48,74)(49,63)(50,82)(51,71)(52,90)(53,79)(54,68)(55,87)(56,76)(57,65)(58,84)(59,73)(60,62)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,35)(32,34)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,101)(92,100)(93,99)(94,98)(95,97)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112), (1,56,76,104)(2,57,77,105)(3,58,78,106)(4,59,79,107)(5,60,80,108)(6,31,81,109)(7,32,82,110)(8,33,83,111)(9,34,84,112)(10,35,85,113)(11,36,86,114)(12,37,87,115)(13,38,88,116)(14,39,89,117)(15,40,90,118)(16,41,61,119)(17,42,62,120)(18,43,63,91)(19,44,64,92)(20,45,65,93)(21,46,66,94)(22,47,67,95)(23,48,68,96)(24,49,69,97)(25,50,70,98)(26,51,71,99)(27,52,72,100)(28,53,73,101)(29,54,74,102)(30,55,75,103), (1,104)(2,93)(3,112)(4,101)(5,120)(6,109)(7,98)(8,117)(9,106)(10,95)(11,114)(12,103)(13,92)(14,111)(15,100)(16,119)(17,108)(18,97)(19,116)(20,105)(21,94)(22,113)(23,102)(24,91)(25,110)(26,99)(27,118)(28,107)(29,96)(30,115)(31,81)(32,70)(33,89)(34,78)(35,67)(36,86)(37,75)(38,64)(39,83)(40,72)(41,61)(42,80)(43,69)(44,88)(45,77)(46,66)(47,85)(48,74)(49,63)(50,82)(51,71)(52,90)(53,79)(54,68)(55,87)(56,76)(57,65)(58,84)(59,73)(60,62) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,35),(32,34),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(91,101),(92,100),(93,99),(94,98),(95,97),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112)], [(1,56,76,104),(2,57,77,105),(3,58,78,106),(4,59,79,107),(5,60,80,108),(6,31,81,109),(7,32,82,110),(8,33,83,111),(9,34,84,112),(10,35,85,113),(11,36,86,114),(12,37,87,115),(13,38,88,116),(14,39,89,117),(15,40,90,118),(16,41,61,119),(17,42,62,120),(18,43,63,91),(19,44,64,92),(20,45,65,93),(21,46,66,94),(22,47,67,95),(23,48,68,96),(24,49,69,97),(25,50,70,98),(26,51,71,99),(27,52,72,100),(28,53,73,101),(29,54,74,102),(30,55,75,103)], [(1,104),(2,93),(3,112),(4,101),(5,120),(6,109),(7,98),(8,117),(9,106),(10,95),(11,114),(12,103),(13,92),(14,111),(15,100),(16,119),(17,108),(18,97),(19,116),(20,105),(21,94),(22,113),(23,102),(24,91),(25,110),(26,99),(27,118),(28,107),(29,96),(30,115),(31,81),(32,70),(33,89),(34,78),(35,67),(36,86),(37,75),(38,64),(39,83),(40,72),(41,61),(42,80),(43,69),(44,88),(45,77),(46,66),(47,85),(48,74),(49,63),(50,82),(51,71),(52,90),(53,79),(54,68),(55,87),(56,76),(57,65),(58,84),(59,73),(60,62)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 20 | 30 | 30 | 2 | 4 | 12 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 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 | S3 | D4 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | D20 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | S3×D20 | C20⋊D6 | D5×C3⋊D4 |
| kernel | D6⋊4D20 | D10⋊Dic3 | C5×D6⋊C4 | D30⋊3C4 | C2×C15⋊D4 | C2×C3⋊D20 | C6×D20 | C22×S3×D5 | C2×D20 | C6×D5 | S3×C10 | D30 | D6⋊C4 | C2×C20 | C22×D5 | C2×Dic3 | C2×C12 | C22×S3 | D10 | D6 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊4D20 ►in GL8(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 46 | 34 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 46 | 34 |
G:=sub<GL(8,GF(61))| [0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,44,45,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,53,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,44,0,0,0,0,0,0,43,0,0,0,0,0,0,0,0,0,60,11,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,53,0,0,0,0,0,0,46,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,16,34],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,46,1,0,0,0,0,0,0,0,0,17,17,0,0,0,0,0,0,1,44,0,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,16,34] >;
D6⋊4D20 in GAP, Magma, Sage, TeX
D_6\rtimes_4D_{20} % in TeX
G:=Group("D6:4D20"); // GroupNames label
G:=SmallGroup(480,550);
// by ID
G=gap.SmallGroup(480,550);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^19,c*b*c^-1=a^15*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations