metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊19D4, C15⋊9C22≀C2, (C2×C10)⋊5D12, (C2×C30)⋊12D4, C6.93(D4×D5), C5⋊4(D6⋊D4), (C2×Dic5)⋊6D6, C10.95(S3×D4), C3⋊2(C23⋊D10), (C2×Dic3)⋊6D10, C10.71(C2×D12), C30.253(C2×D4), (C23×D15)⋊7C2, (C22×S3)⋊3D10, C23.D5⋊14S3, D30⋊4C4⋊37C2, C23.56(S3×D5), C22⋊4(C5⋊D12), (C22×C10).64D6, (C22×C6).49D10, (C2×C30).215C23, (C6×Dic5)⋊10C22, C2.45(D10⋊D6), (C10×Dic3)⋊10C22, (C22×C30).77C22, (C22×D15).115C22, (C2×C3⋊D4)⋊9D5, (C10×C3⋊D4)⋊9C2, (C2×C6)⋊4(C5⋊D4), (S3×C2×C10)⋊3C22, C6.25(C2×C5⋊D4), (C2×C5⋊D12)⋊16C2, C2.26(C2×C5⋊D12), C22.244(C2×S3×D5), (C3×C23.D5)⋊16C2, (C2×C6).227(C22×D5), (C2×C10).227(C22×S3), SmallGroup(480,649)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊19D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a25b, dcd=c-1 >
Subgroups: 1836 in 260 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3 [×5], C6, C6 [×2], C6 [×2], C2×C4 [×3], D4 [×6], C23, C23 [×9], D5 [×4], C10, C10 [×2], C10 [×3], Dic3, C12 [×2], D6 [×19], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×2], C20, D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×5], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×S3 [×8], C22×C6, C5×S3, D15 [×4], C30, C30 [×2], C30 [×2], C22≀C2, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5 [×8], C22×C10, C22×C10, D6⋊C4 [×2], C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, C5×Dic3, C3×Dic5 [×2], S3×C10 [×3], D30 [×4], D30 [×12], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, D6⋊D4, C5⋊D12 [×4], C6×Dic5 [×2], C10×Dic3, C5×C3⋊D4 [×2], S3×C2×C10, C22×D15 [×2], C22×D15 [×6], C22×C30, C23⋊D10, D30⋊4C4 [×2], C3×C23.D5, C2×C5⋊D12 [×2], C10×C3⋊D4, C23×D15, D30⋊19D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×2], C22×S3, C22≀C2, C5⋊D4 [×2], C22×D5, C2×D12, S3×D4 [×2], S3×D5, D4×D5 [×2], C2×C5⋊D4, D6⋊D4, C5⋊D12 [×2], C2×S3×D5, C23⋊D10, C2×C5⋊D12, D10⋊D6 [×2], D30⋊19D4
►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 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 90)(29 89)(30 88)(31 118)(32 117)(33 116)(34 115)(35 114)(36 113)(37 112)(38 111)(39 110)(40 109)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 100)(50 99)(51 98)(52 97)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 120)(60 119)
(1 53 88 112)(2 34 89 93)(3 45 90 104)(4 56 61 115)(5 37 62 96)(6 48 63 107)(7 59 64 118)(8 40 65 99)(9 51 66 110)(10 32 67 91)(11 43 68 102)(12 54 69 113)(13 35 70 94)(14 46 71 105)(15 57 72 116)(16 38 73 97)(17 49 74 108)(18 60 75 119)(19 41 76 100)(20 52 77 111)(21 33 78 92)(22 44 79 103)(23 55 80 114)(24 36 81 95)(25 47 82 106)(26 58 83 117)(27 39 84 98)(28 50 85 109)(29 31 86 120)(30 42 87 101)
(1 112)(2 93)(3 104)(4 115)(5 96)(6 107)(7 118)(8 99)(9 110)(10 91)(11 102)(12 113)(13 94)(14 105)(15 116)(16 97)(17 108)(18 119)(19 100)(20 111)(21 92)(22 103)(23 114)(24 95)(25 106)(26 117)(27 98)(28 109)(29 120)(30 101)(31 86)(32 67)(33 78)(34 89)(35 70)(36 81)(37 62)(38 73)(39 84)(40 65)(41 76)(42 87)(43 68)(44 79)(45 90)(46 71)(47 82)(48 63)(49 74)(50 85)(51 66)(52 77)(53 88)(54 69)(55 80)(56 61)(57 72)(58 83)(59 64)(60 75)
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), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,90)(29,89)(30,88)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,120)(60,119), (1,53,88,112)(2,34,89,93)(3,45,90,104)(4,56,61,115)(5,37,62,96)(6,48,63,107)(7,59,64,118)(8,40,65,99)(9,51,66,110)(10,32,67,91)(11,43,68,102)(12,54,69,113)(13,35,70,94)(14,46,71,105)(15,57,72,116)(16,38,73,97)(17,49,74,108)(18,60,75,119)(19,41,76,100)(20,52,77,111)(21,33,78,92)(22,44,79,103)(23,55,80,114)(24,36,81,95)(25,47,82,106)(26,58,83,117)(27,39,84,98)(28,50,85,109)(29,31,86,120)(30,42,87,101), (1,112)(2,93)(3,104)(4,115)(5,96)(6,107)(7,118)(8,99)(9,110)(10,91)(11,102)(12,113)(13,94)(14,105)(15,116)(16,97)(17,108)(18,119)(19,100)(20,111)(21,92)(22,103)(23,114)(24,95)(25,106)(26,117)(27,98)(28,109)(29,120)(30,101)(31,86)(32,67)(33,78)(34,89)(35,70)(36,81)(37,62)(38,73)(39,84)(40,65)(41,76)(42,87)(43,68)(44,79)(45,90)(46,71)(47,82)(48,63)(49,74)(50,85)(51,66)(52,77)(53,88)(54,69)(55,80)(56,61)(57,72)(58,83)(59,64)(60,75)>;
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), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,90)(29,89)(30,88)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,120)(60,119), (1,53,88,112)(2,34,89,93)(3,45,90,104)(4,56,61,115)(5,37,62,96)(6,48,63,107)(7,59,64,118)(8,40,65,99)(9,51,66,110)(10,32,67,91)(11,43,68,102)(12,54,69,113)(13,35,70,94)(14,46,71,105)(15,57,72,116)(16,38,73,97)(17,49,74,108)(18,60,75,119)(19,41,76,100)(20,52,77,111)(21,33,78,92)(22,44,79,103)(23,55,80,114)(24,36,81,95)(25,47,82,106)(26,58,83,117)(27,39,84,98)(28,50,85,109)(29,31,86,120)(30,42,87,101), (1,112)(2,93)(3,104)(4,115)(5,96)(6,107)(7,118)(8,99)(9,110)(10,91)(11,102)(12,113)(13,94)(14,105)(15,116)(16,97)(17,108)(18,119)(19,100)(20,111)(21,92)(22,103)(23,114)(24,95)(25,106)(26,117)(27,98)(28,109)(29,120)(30,101)(31,86)(32,67)(33,78)(34,89)(35,70)(36,81)(37,62)(38,73)(39,84)(40,65)(41,76)(42,87)(43,68)(44,79)(45,90)(46,71)(47,82)(48,63)(49,74)(50,85)(51,66)(52,77)(53,88)(54,69)(55,80)(56,61)(57,72)(58,83)(59,64)(60,75) );
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)], [(1,87),(2,86),(3,85),(4,84),(5,83),(6,82),(7,81),(8,80),(9,79),(10,78),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,70),(19,69),(20,68),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,90),(29,89),(30,88),(31,118),(32,117),(33,116),(34,115),(35,114),(36,113),(37,112),(38,111),(39,110),(40,109),(41,108),(42,107),(43,106),(44,105),(45,104),(46,103),(47,102),(48,101),(49,100),(50,99),(51,98),(52,97),(53,96),(54,95),(55,94),(56,93),(57,92),(58,91),(59,120),(60,119)], [(1,53,88,112),(2,34,89,93),(3,45,90,104),(4,56,61,115),(5,37,62,96),(6,48,63,107),(7,59,64,118),(8,40,65,99),(9,51,66,110),(10,32,67,91),(11,43,68,102),(12,54,69,113),(13,35,70,94),(14,46,71,105),(15,57,72,116),(16,38,73,97),(17,49,74,108),(18,60,75,119),(19,41,76,100),(20,52,77,111),(21,33,78,92),(22,44,79,103),(23,55,80,114),(24,36,81,95),(25,47,82,106),(26,58,83,117),(27,39,84,98),(28,50,85,109),(29,31,86,120),(30,42,87,101)], [(1,112),(2,93),(3,104),(4,115),(5,96),(6,107),(7,118),(8,99),(9,110),(10,91),(11,102),(12,113),(13,94),(14,105),(15,116),(16,97),(17,108),(18,119),(19,100),(20,111),(21,92),(22,103),(23,114),(24,95),(25,106),(26,117),(27,98),(28,109),(29,120),(30,101),(31,86),(32,67),(33,78),(34,89),(35,70),(36,81),(37,62),(38,73),(39,84),(40,65),(41,76),(42,87),(43,68),(44,79),(45,90),(46,71),(47,82),(48,63),(49,74),(50,85),(51,66),(52,77),(53,88),(54,69),(55,80),(56,61),(57,72),(58,83),(59,64),(60,75)])
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 | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 30 | 30 | 30 | 30 | 2 | 12 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | D12 | C5⋊D4 | S3×D4 | S3×D5 | D4×D5 | C5⋊D12 | C2×S3×D5 | D10⋊D6 |
| kernel | D30⋊19D4 | D30⋊4C4 | C3×C23.D5 | C2×C5⋊D12 | C10×C3⋊D4 | C23×D15 | C23.D5 | D30 | C2×C30 | C2×C3⋊D4 | C2×Dic5 | C22×C10 | C2×Dic3 | C22×S3 | C22×C6 | C2×C10 | C2×C6 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of D30⋊19D4 ►in GL8(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 17 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 0 | 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 | 14 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 49 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 52 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 60 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 0 | 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 | 47 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 52 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(8,GF(61))| [0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,44,44,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,60,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,14,49,0,0,0,0,0,0,1,47,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,52,60],[0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,47,12,0,0,0,0,0,0,60,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,52,60] >;
D30⋊19D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_{19}D_4 % in TeX
G:=Group("D30:19D4"); // GroupNames label
G:=SmallGroup(480,649);
// by ID
G=gap.SmallGroup(480,649);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^25*b,d*c*d=c^-1>;
// generators/relations