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## G = D4⋊D30order 480 = 25·3·5

### 2nd semidirect product of D4 and D30 acting via D30/C30=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D4⋊D30
 Chief series C1 — C5 — C15 — C30 — C60 — D60 — C2×D60 — D4⋊D30
 Lower central C15 — C30 — C60 — D4⋊D30
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4⋊D30
G = < a,b,c,d | a4=c30=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab, dcd=c-1 >

Subgroups: 932 in 136 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], C12 [×2], C12, D6 [×4], C2×C6, C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, D15 [×2], C30, C30 [×2], C8⋊C22, C52C8 [×2], D20 [×3], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, C60 [×2], C60, D30 [×4], C2×C30, C2×C30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, D4⋊D6, C153C8 [×2], D60 [×2], D60, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C22×D15, D4⋊D10, C60.7C4, D4⋊D15 [×2], Q82D15 [×2], C2×D60, C15×C4○D4, D4⋊D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, D30 [×3], C2×C5⋊D4, D4⋊D6, C157D4 [×2], C22×D15, D4⋊D10, C2×C157D4, D4⋊D30

Smallest permutation representation of D4⋊D30
On 120 points
Generators in S120
```(1 51 21 31)(2 52 22 32)(3 53 23 33)(4 54 24 34)(5 55 25 35)(6 56 26 36)(7 57 27 37)(8 58 28 38)(9 59 29 39)(10 60 30 40)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(61 116 76 101)(62 117 77 102)(63 118 78 103)(64 119 79 104)(65 120 80 105)(66 91 81 106)(67 92 82 107)(68 93 83 108)(69 94 84 109)(70 95 85 110)(71 96 86 111)(72 97 87 112)(73 98 88 113)(74 99 89 114)(75 100 90 115)
(1 86 21 71)(2 72 22 87)(3 88 23 73)(4 74 24 89)(5 90 25 75)(6 76 26 61)(7 62 27 77)(8 78 28 63)(9 64 29 79)(10 80 30 65)(11 66 16 81)(12 82 17 67)(13 68 18 83)(14 84 19 69)(15 70 20 85)(31 111 51 96)(32 97 52 112)(33 113 53 98)(34 99 54 114)(35 115 55 100)(36 101 56 116)(37 117 57 102)(38 103 58 118)(39 119 59 104)(40 105 60 120)(41 91 46 106)(42 107 47 92)(43 93 48 108)(44 109 49 94)(45 95 50 110)
(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 5)(2 4)(6 15)(7 14)(8 13)(9 12)(10 11)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 60)(42 59)(43 58)(44 57)(45 56)(61 95)(62 94)(63 93)(64 92)(65 91)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)```

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 10C ··· 10H 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20J 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 6 6 8 8 10 10 10 ··· 10 12 12 12 12 12 15 15 15 15 20 20 20 20 20 ··· 20 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 4 60 60 2 2 2 4 2 2 2 4 4 4 60 60 2 2 4 ··· 4 2 2 4 4 4 2 2 2 2 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 D15 C5⋊D4 C5⋊D4 D30 D30 D30 C15⋊7D4 C15⋊7D4 C8⋊C22 D4⋊D6 D4⋊D10 D4⋊D30 kernel D4⋊D30 C60.7C4 D4⋊D15 Q8⋊2D15 C2×D60 C15×C4○D4 C5×C4○D4 C60 C2×C30 C3×C4○D4 C2×C20 C5×D4 C5×Q8 C2×C12 C3×D4 C3×Q8 C20 C2×C10 C4○D4 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 1 1 1 1 2 1 1 1 2 2 2 2 2 4 4 4 4 4 4 8 8 1 2 4 8

Matrix representation of D4⋊D30 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 200 156 0 0 0 0 85 41 0 0 0 0 0 0 41 85 0 0 0 0 156 200
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 240 0 0 0 0 1 0 0 0 0 0 0 0 189 240 0 0 0 0 1 0 0 0 0 0 0 0 52 1 0 0 0 0 240 0
,
 1 0 0 0 0 0 1 240 0 0 0 0 0 0 189 240 0 0 0 0 52 52 0 0 0 0 0 0 122 200 0 0 0 0 122 119

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,200,85,0,0,0,0,156,41,0,0,0,0,0,0,41,156,0,0,0,0,85,200],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,240,0,0],[1,1,0,0,0,0,240,0,0,0,0,0,0,0,189,1,0,0,0,0,240,0,0,0,0,0,0,0,52,240,0,0,0,0,1,0],[1,1,0,0,0,0,0,240,0,0,0,0,0,0,189,52,0,0,0,0,240,52,0,0,0,0,0,0,122,122,0,0,0,0,200,119] >;`

D4⋊D30 in GAP, Magma, Sage, TeX

`D_4\rtimes D_{30}`
`% in TeX`

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

`G:=SmallGroup(480,914);`
`// by ID`

`G=gap.SmallGroup(480,914);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,675,185,80,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^30=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a*b,d*c*d=c^-1>;`
`// generators/relations`

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