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G = D30⋊18D4order 480 = 25·3·5

3rd semidirect product of D30 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D30⋊18D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C3⋊D20 — D30⋊18D4
 Lower central C15 — C2×C30 — D30⋊18D4
 Upper central C1 — C22 — C23

Generators and relations for D3018D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a19, cbc-1=dbd=a3b, dcd=c-1 >

Subgroups: 1884 in 260 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3 [×4], C6, C6 [×2], C6 [×3], C2×C4 [×3], D4 [×6], C23, C23 [×9], D5 [×5], C10, C10 [×2], C10 [×2], Dic3 [×2], C12, D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×19], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×8], C22×C6, C22×C6, C3×D5, D15 [×4], C30, C30 [×2], C30 [×2], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×D5 [×8], C22×C10, D6⋊C4 [×2], C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, C5×Dic3 [×2], C3×Dic5, C6×D5 [×3], D30 [×4], D30 [×12], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, C232D6, C3⋊D20 [×4], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3 [×2], D5×C2×C6, C22×D15 [×2], C22×D15 [×6], C22×C30, C22⋊D20, D304C4 [×2], C5×C6.D4, C2×C3⋊D20 [×2], C6×C5⋊D4, C23×D15, D3018D4
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], C232D6, C3⋊D20 [×2], C2×S3×D5, C22⋊D20, C2×C3⋊D20, D10⋊D6 [×2], D3018D4

Smallest permutation representation of D3018D4
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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 100)(13 99)(14 98)(15 97)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)
(1 54 112 89)(2 43 113 78)(3 32 114 67)(4 51 115 86)(5 40 116 75)(6 59 117 64)(7 48 118 83)(8 37 119 72)(9 56 120 61)(10 45 91 80)(11 34 92 69)(12 53 93 88)(13 42 94 77)(14 31 95 66)(15 50 96 85)(16 39 97 74)(17 58 98 63)(18 47 99 82)(19 36 100 71)(20 55 101 90)(21 44 102 79)(22 33 103 68)(23 52 104 87)(24 41 105 76)(25 60 106 65)(26 49 107 84)(27 38 108 73)(28 57 109 62)(29 46 110 81)(30 35 111 70)
(1 89)(2 78)(3 67)(4 86)(5 75)(6 64)(7 83)(8 72)(9 61)(10 80)(11 69)(12 88)(13 77)(14 66)(15 85)(16 74)(17 63)(18 82)(19 71)(20 90)(21 79)(22 68)(23 87)(24 76)(25 65)(26 84)(27 73)(28 62)(29 81)(30 70)(31 95)(32 114)(33 103)(34 92)(35 111)(36 100)(37 119)(38 108)(39 97)(40 116)(41 105)(42 94)(43 113)(44 102)(45 91)(46 110)(47 99)(48 118)(49 107)(50 96)(51 115)(52 104)(53 93)(54 112)(55 101)(56 120)(57 109)(58 98)(59 117)(60 106)```

`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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,99)(14,98)(15,97)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67), (1,54,112,89)(2,43,113,78)(3,32,114,67)(4,51,115,86)(5,40,116,75)(6,59,117,64)(7,48,118,83)(8,37,119,72)(9,56,120,61)(10,45,91,80)(11,34,92,69)(12,53,93,88)(13,42,94,77)(14,31,95,66)(15,50,96,85)(16,39,97,74)(17,58,98,63)(18,47,99,82)(19,36,100,71)(20,55,101,90)(21,44,102,79)(22,33,103,68)(23,52,104,87)(24,41,105,76)(25,60,106,65)(26,49,107,84)(27,38,108,73)(28,57,109,62)(29,46,110,81)(30,35,111,70), (1,89)(2,78)(3,67)(4,86)(5,75)(6,64)(7,83)(8,72)(9,61)(10,80)(11,69)(12,88)(13,77)(14,66)(15,85)(16,74)(17,63)(18,82)(19,71)(20,90)(21,79)(22,68)(23,87)(24,76)(25,65)(26,84)(27,73)(28,62)(29,81)(30,70)(31,95)(32,114)(33,103)(34,92)(35,111)(36,100)(37,119)(38,108)(39,97)(40,116)(41,105)(42,94)(43,113)(44,102)(45,91)(46,110)(47,99)(48,118)(49,107)(50,96)(51,115)(52,104)(53,93)(54,112)(55,101)(56,120)(57,109)(58,98)(59,117)(60,106)>;`

`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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,99)(14,98)(15,97)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67), (1,54,112,89)(2,43,113,78)(3,32,114,67)(4,51,115,86)(5,40,116,75)(6,59,117,64)(7,48,118,83)(8,37,119,72)(9,56,120,61)(10,45,91,80)(11,34,92,69)(12,53,93,88)(13,42,94,77)(14,31,95,66)(15,50,96,85)(16,39,97,74)(17,58,98,63)(18,47,99,82)(19,36,100,71)(20,55,101,90)(21,44,102,79)(22,33,103,68)(23,52,104,87)(24,41,105,76)(25,60,106,65)(26,49,107,84)(27,38,108,73)(28,57,109,62)(29,46,110,81)(30,35,111,70), (1,89)(2,78)(3,67)(4,86)(5,75)(6,64)(7,83)(8,72)(9,61)(10,80)(11,69)(12,88)(13,77)(14,66)(15,85)(16,74)(17,63)(18,82)(19,71)(20,90)(21,79)(22,68)(23,87)(24,76)(25,65)(26,84)(27,73)(28,62)(29,81)(30,70)(31,95)(32,114)(33,103)(34,92)(35,111)(36,100)(37,119)(38,108)(39,97)(40,116)(41,105)(42,94)(43,113)(44,102)(45,91)(46,110)(47,99)(48,118)(49,107)(50,96)(51,115)(52,104)(53,93)(54,112)(55,101)(56,120)(57,109)(58,98)(59,117)(60,106) );`

`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)], [(1,111),(2,110),(3,109),(4,108),(5,107),(6,106),(7,105),(8,104),(9,103),(10,102),(11,101),(12,100),(13,99),(14,98),(15,97),(16,96),(17,95),(18,94),(19,93),(20,92),(21,91),(22,120),(23,119),(24,118),(25,117),(26,116),(27,115),(28,114),(29,113),(30,112),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,84),(44,83),(45,82),(46,81),(47,80),(48,79),(49,78),(50,77),(51,76),(52,75),(53,74),(54,73),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67)], [(1,54,112,89),(2,43,113,78),(3,32,114,67),(4,51,115,86),(5,40,116,75),(6,59,117,64),(7,48,118,83),(8,37,119,72),(9,56,120,61),(10,45,91,80),(11,34,92,69),(12,53,93,88),(13,42,94,77),(14,31,95,66),(15,50,96,85),(16,39,97,74),(17,58,98,63),(18,47,99,82),(19,36,100,71),(20,55,101,90),(21,44,102,79),(22,33,103,68),(23,52,104,87),(24,41,105,76),(25,60,106,65),(26,49,107,84),(27,38,108,73),(28,57,109,62),(29,46,110,81),(30,35,111,70)], [(1,89),(2,78),(3,67),(4,86),(5,75),(6,64),(7,83),(8,72),(9,61),(10,80),(11,69),(12,88),(13,77),(14,66),(15,85),(16,74),(17,63),(18,82),(19,71),(20,90),(21,79),(22,68),(23,87),(24,76),(25,65),(26,84),(27,73),(28,62),(29,81),(30,70),(31,95),(32,114),(33,103),(34,92),(35,111),(36,100),(37,119),(38,108),(39,97),(40,116),(41,105),(42,94),(43,113),(44,102),(45,91),(46,110),(47,99),(48,118),(49,107),(50,96),(51,115),(52,104),(53,93),(54,112),(55,101),(56,120),(57,109),(58,98),(59,117),(60,106)])`

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 ··· 20H 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 6 6 10 ··· 10 10 10 10 10 12 12 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 2 2 20 30 30 30 30 2 12 12 20 2 2 2 2 2 4 4 20 20 2 ··· 2 4 4 4 4 20 20 4 4 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 D6 D10 D10 C3⋊D4 D20 S3×D4 S3×D5 D4×D5 C3⋊D20 C2×S3×D5 D10⋊D6 kernel D30⋊18D4 D30⋊4C4 C5×C6.D4 C2×C3⋊D20 C6×C5⋊D4 C23×D15 C2×C5⋊D4 D30 C2×C30 C6.D4 C2×Dic5 C22×D5 C22×C10 C2×Dic3 C22×C6 C2×C10 C2×C6 C10 C23 C6 C22 C22 C2 # reps 1 2 1 2 1 1 1 4 2 2 1 1 1 4 2 4 8 2 2 4 4 2 8

Matrix representation of D3018D4 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 1 0 0 0 0 60 0 0 0 0 0 0 0 44 1 0 0 0 0 60 0
,
 1 0 0 0 0 0 7 60 0 0 0 0 0 0 60 0 0 0 0 0 60 1 0 0 0 0 0 0 44 1 0 0 0 0 17 17
,
 39 15 0 0 0 0 49 22 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 29 54 0 0 0 0 59 32
,
 39 15 0 0 0 0 41 22 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 29 54 0 0 0 0 59 32

`G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,44,60,0,0,0,0,1,0],[1,7,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,0,44,17,0,0,0,0,1,17],[39,49,0,0,0,0,15,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,59,0,0,0,0,54,32],[39,41,0,0,0,0,15,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,59,0,0,0,0,54,32] >;`

D3018D4 in GAP, Magma, Sage, TeX

`D_{30}\rtimes_{18}D_4`
`% in TeX`

`G:=Group("D30:18D4");`
`// GroupNames label`

`G:=SmallGroup(480,648);`
`// by ID`

`G=gap.SmallGroup(480,648);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,422,219,1356,18822]);`
`// 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^19,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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