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## G = D30.27D4order 480 = 25·3·5

### 11st non-split extension by D30 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D30.27D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C22×S3×D5 — D30.27D4
 Lower central C15 — C30 — D30.27D4
 Upper central C1 — C22 — C2×C4

Generators and relations for D30.27D4
G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a15c-1 >

Subgroups: 1676 in 264 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×4], C22, C22 [×22], C5, S3 [×6], C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], C23 [×11], D5 [×6], C10 [×3], C10 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×16], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×2], D10 [×16], C2×C10, C2×C10 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3 [×9], C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×4], C30 [×3], C2×C22⋊C4, C4×D5 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5 [×9], C22×C10, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4 [×2], S3×C23, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×6], C2×C30, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5 [×2], C23×D5, S3×C22⋊C4, D30.C2 [×2], C6×Dic5, C10×Dic3, C4×D15 [×2], C2×Dic15, C2×C60, C2×S3×D5 [×4], C2×S3×D5 [×4], D5×C2×C6, S3×C2×C10, C22×D15, D5×C22⋊C4, D10⋊Dic3, D6⋊Dic5, C3×D10⋊C4, C5×D6⋊C4, C2×D30.C2, C2×C4×D15, C22×S3×D5, D30.27D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C2×C22⋊C4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4 [×2], S3×D5, C2×C4×D5, D4×D5 [×2], S3×C22⋊C4, C2×S3×D5, D5×C22⋊C4, C4×S3×D5, C20⋊D6, D10⋊D6, D30.27D4

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 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 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 10 10 10 10 12 12 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 15 15 15 15 2 2 2 6 6 10 10 30 30 2 2 2 2 2 20 20 2 ··· 2 12 12 12 12 4 4 20 20 4 4 4 4 4 4 12 12 12 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 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 C4 S3 D4 D5 D6 D6 D6 D10 D10 D10 C4×S3 C4×D5 S3×D4 S3×D5 D4×D5 C2×S3×D5 C4×S3×D5 C20⋊D6 D10⋊D6 kernel D30.27D4 D10⋊Dic3 D6⋊Dic5 C3×D10⋊C4 C5×D6⋊C4 C2×D30.C2 C2×C4×D15 C22×S3×D5 C2×S3×D5 D10⋊C4 D30 D6⋊C4 C2×Dic5 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 8 1 4 2 1 1 1 2 2 2 4 8 2 2 4 2 4 4 4

Matrix representation of D30.27D4 in GL6(𝔽61)

 60 15 0 0 0 0 12 2 0 0 0 0 0 0 60 60 0 0 0 0 19 18 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 15 0 0 0 0 0 1 0 0 0 0 0 0 0 17 0 0 0 0 18 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 20 0 0 0 0 0 60
,
 50 0 0 0 0 0 10 11 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 20 0 0 0 0 6 60

`G:=sub<GL(6,GF(61))| [60,12,0,0,0,0,15,2,0,0,0,0,0,0,60,19,0,0,0,0,60,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,15,1,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,20,60],[50,10,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,6,0,0,0,0,20,60] >;`

D30.27D4 in GAP, Magma, Sage, TeX

`D_{30}._{27}D_4`
`% in TeX`

`G:=Group("D30.27D4");`
`// GroupNames label`

`G:=SmallGroup(480,549);`
`// by ID`

`G=gap.SmallGroup(480,549);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,219,58,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^15*c^-1>;`
`// generators/relations`

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