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