metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊3D4, D6⋊4D20, (C6×D5)⋊3D4, (C2×C20)⋊1D6, D6⋊C4⋊19D5, C15⋊1C22≀C2, (S3×C10)⋊3D4, (C2×D20)⋊6S3, (C6×D20)⋊16C2, (C2×C12)⋊18D10, C5⋊1(C23⋊2D6), C10.26(S3×D4), C2.28(S3×D20), C6.27(C2×D20), C6.137(D4×D5), (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, (C2×Dic15)⋊7C22, (C10×Dic3)⋊3C22, (C22×S3).50D10, (C22×D15).56C22, (C2×C4)⋊1(S3×D5), (C22×S3×D5)⋊1C2, (D5×C2×C6)⋊1C22, (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 D30⋊3D4
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 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×23], C5, S3 [×4], C6 [×3], C6 [×3], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], D5 [×5], C10 [×3], C10 [×2], Dic3 [×2], C12, D6 [×2], D6 [×14], C2×C6, C2×C6 [×7], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×2], D10 [×17], C2×C10, C2×C10 [×4], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×S3 [×7], C22×C6 [×2], C5×S3 [×2], C3×D5 [×3], D15 [×2], C30 [×3], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5 [×2], C22×D5 [×7], C22×C10, D6⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, C5×Dic3, Dic15, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×5], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, D10⋊C4 [×2], C5×C22⋊C4, C2×D20, C2×D20, C2×C5⋊D4, C23×D5, C23⋊2D6, C15⋊D4 [×2], C3⋊D20 [×2], C3×D20 [×2], C10×Dic3, C2×Dic15, C2×C60, C2×S3×D5 [×6], D5×C2×C6 [×2], 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, D30⋊3D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, D20 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, S3×D5, C2×D20, D4×D5 [×2], C23⋊2D6, C2×S3×D5, C22⋊D20, S3×D20, C20⋊D6, D5×C3⋊D4, D30⋊3D4
►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 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 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(53 60)(54 59)(55 58)(56 57)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(116 120)(117 119)
(1 86 57 111)(2 87 58 112)(3 88 59 113)(4 89 60 114)(5 90 31 115)(6 61 32 116)(7 62 33 117)(8 63 34 118)(9 64 35 119)(10 65 36 120)(11 66 37 91)(12 67 38 92)(13 68 39 93)(14 69 40 94)(15 70 41 95)(16 71 42 96)(17 72 43 97)(18 73 44 98)(19 74 45 99)(20 75 46 100)(21 76 47 101)(22 77 48 102)(23 78 49 103)(24 79 50 104)(25 80 51 105)(26 81 52 106)(27 82 53 107)(28 83 54 108)(29 84 55 109)(30 85 56 110)
(1 111)(2 100)(3 119)(4 108)(5 97)(6 116)(7 105)(8 94)(9 113)(10 102)(11 91)(12 110)(13 99)(14 118)(15 107)(16 96)(17 115)(18 104)(19 93)(20 112)(21 101)(22 120)(23 109)(24 98)(25 117)(26 106)(27 95)(28 114)(29 103)(30 92)(31 72)(32 61)(33 80)(34 69)(35 88)(36 77)(37 66)(38 85)(39 74)(40 63)(41 82)(42 71)(43 90)(44 79)(45 68)(46 87)(47 76)(48 65)(49 84)(50 73)(51 62)(52 81)(53 70)(54 89)(55 78)(56 67)(57 86)(58 75)(59 64)(60 83)
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)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(116,120)(117,119), (1,86,57,111)(2,87,58,112)(3,88,59,113)(4,89,60,114)(5,90,31,115)(6,61,32,116)(7,62,33,117)(8,63,34,118)(9,64,35,119)(10,65,36,120)(11,66,37,91)(12,67,38,92)(13,68,39,93)(14,69,40,94)(15,70,41,95)(16,71,42,96)(17,72,43,97)(18,73,44,98)(19,74,45,99)(20,75,46,100)(21,76,47,101)(22,77,48,102)(23,78,49,103)(24,79,50,104)(25,80,51,105)(26,81,52,106)(27,82,53,107)(28,83,54,108)(29,84,55,109)(30,85,56,110), (1,111)(2,100)(3,119)(4,108)(5,97)(6,116)(7,105)(8,94)(9,113)(10,102)(11,91)(12,110)(13,99)(14,118)(15,107)(16,96)(17,115)(18,104)(19,93)(20,112)(21,101)(22,120)(23,109)(24,98)(25,117)(26,106)(27,95)(28,114)(29,103)(30,92)(31,72)(32,61)(33,80)(34,69)(35,88)(36,77)(37,66)(38,85)(39,74)(40,63)(41,82)(42,71)(43,90)(44,79)(45,68)(46,87)(47,76)(48,65)(49,84)(50,73)(51,62)(52,81)(53,70)(54,89)(55,78)(56,67)(57,86)(58,75)(59,64)(60,83)>;
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)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(116,120)(117,119), (1,86,57,111)(2,87,58,112)(3,88,59,113)(4,89,60,114)(5,90,31,115)(6,61,32,116)(7,62,33,117)(8,63,34,118)(9,64,35,119)(10,65,36,120)(11,66,37,91)(12,67,38,92)(13,68,39,93)(14,69,40,94)(15,70,41,95)(16,71,42,96)(17,72,43,97)(18,73,44,98)(19,74,45,99)(20,75,46,100)(21,76,47,101)(22,77,48,102)(23,78,49,103)(24,79,50,104)(25,80,51,105)(26,81,52,106)(27,82,53,107)(28,83,54,108)(29,84,55,109)(30,85,56,110), (1,111)(2,100)(3,119)(4,108)(5,97)(6,116)(7,105)(8,94)(9,113)(10,102)(11,91)(12,110)(13,99)(14,118)(15,107)(16,96)(17,115)(18,104)(19,93)(20,112)(21,101)(22,120)(23,109)(24,98)(25,117)(26,106)(27,95)(28,114)(29,103)(30,92)(31,72)(32,61)(33,80)(34,69)(35,88)(36,77)(37,66)(38,85)(39,74)(40,63)(41,82)(42,71)(43,90)(44,79)(45,68)(46,87)(47,76)(48,65)(49,84)(50,73)(51,62)(52,81)(53,70)(54,89)(55,78)(56,67)(57,86)(58,75)(59,64)(60,83) );
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),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(53,60),(54,59),(55,58),(56,57),(61,65),(62,64),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(91,115),(92,114),(93,113),(94,112),(95,111),(96,110),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(116,120),(117,119)], [(1,86,57,111),(2,87,58,112),(3,88,59,113),(4,89,60,114),(5,90,31,115),(6,61,32,116),(7,62,33,117),(8,63,34,118),(9,64,35,119),(10,65,36,120),(11,66,37,91),(12,67,38,92),(13,68,39,93),(14,69,40,94),(15,70,41,95),(16,71,42,96),(17,72,43,97),(18,73,44,98),(19,74,45,99),(20,75,46,100),(21,76,47,101),(22,77,48,102),(23,78,49,103),(24,79,50,104),(25,80,51,105),(26,81,52,106),(27,82,53,107),(28,83,54,108),(29,84,55,109),(30,85,56,110)], [(1,111),(2,100),(3,119),(4,108),(5,97),(6,116),(7,105),(8,94),(9,113),(10,102),(11,91),(12,110),(13,99),(14,118),(15,107),(16,96),(17,115),(18,104),(19,93),(20,112),(21,101),(22,120),(23,109),(24,98),(25,117),(26,106),(27,95),(28,114),(29,103),(30,92),(31,72),(32,61),(33,80),(34,69),(35,88),(36,77),(37,66),(38,85),(39,74),(40,63),(41,82),(42,71),(43,90),(44,79),(45,68),(46,87),(47,76),(48,65),(49,84),(50,73),(51,62),(52,81),(53,70),(54,89),(55,78),(56,67),(57,86),(58,75),(59,64),(60,83)])
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 | D30⋊3D4 | 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 D30⋊3D4 ►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] >;
D30⋊3D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_3D_4 % in TeX
G:=Group("D30:3D4"); // 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