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## G = D30.S3order 360 = 23·32·5

### The non-split extension by D30 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — D30.S3
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C6×D15 — D30.S3
 Lower central C3×C15 — D30.S3
 Upper central C1 — C2

Generators and relations for D30.S3
G = < a,b,c,d | a30=b2=c3=1, d2=a15, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 356 in 70 conjugacy classes, 27 normal (23 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, C20, D10, C4×S3, C2×Dic3, C3×D5, D15, C30, C30, C3×Dic3, C3⋊Dic3, S3×C6, C4×D5, C3×C15, C5×Dic3, C3×Dic5, Dic15, C6×D5, D30, S3×Dic3, C3×D15, C3×C30, D5×Dic3, D30.C2, C3×Dic15, C5×C3⋊Dic3, C6×D15, D30.S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, D10, C4×S3, C2×Dic3, S32, C4×D5, S3×D5, S3×Dic3, D5×Dic3, D30.C2, D15⋊S3, D30.S3

Smallest permutation representation of D30.S3
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 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 90)(16 89)(17 88)(18 87)(19 86)(20 85)(21 84)(22 83)(23 82)(24 81)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 120)(39 119)(40 118)(41 117)(42 116)(43 115)(44 114)(45 113)(46 112)(47 111)(48 110)(49 109)(50 108)(51 107)(52 106)(53 105)(54 104)(55 103)(56 102)(57 101)(58 100)(59 99)(60 98)
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 28)(9 19 29)(10 20 30)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(1 46 16 31)(2 57 17 42)(3 38 18 53)(4 49 19 34)(5 60 20 45)(6 41 21 56)(7 52 22 37)(8 33 23 48)(9 44 24 59)(10 55 25 40)(11 36 26 51)(12 47 27 32)(13 58 28 43)(14 39 29 54)(15 50 30 35)(61 109 76 94)(62 120 77 105)(63 101 78 116)(64 112 79 97)(65 93 80 108)(66 104 81 119)(67 115 82 100)(68 96 83 111)(69 107 84 92)(70 118 85 103)(71 99 86 114)(72 110 87 95)(73 91 88 106)(74 102 89 117)(75 113 90 98)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 10A 10B 12A 12B 15A ··· 15H 20A 20B 20C 20D 30A ··· 30H order 1 2 2 2 3 3 3 4 4 4 4 5 5 6 6 6 6 6 10 10 12 12 15 ··· 15 20 20 20 20 30 ··· 30 size 1 1 15 15 2 2 4 9 9 15 15 2 2 2 2 4 30 30 2 2 30 30 4 ··· 4 18 18 18 18 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + - + + + + - - + image C1 C2 C2 C2 C4 S3 S3 D5 Dic3 D6 D10 C4×S3 C4×D5 S32 S3×D5 S3×Dic3 D5×Dic3 D30.C2 D15⋊S3 D30.S3 kernel D30.S3 C3×Dic15 C5×C3⋊Dic3 C6×D15 C3×D15 Dic15 D30 C3⋊Dic3 D15 C30 C3×C6 C15 C32 C10 C6 C5 C3 C3 C2 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 2 4 1 4 1 2 2 4 4

Matrix representation of D30.S3 in GL6(𝔽61)

 44 60 0 0 0 0 45 60 0 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 47 0 0 0 0 32 49 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 11 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(61))| [44,45,0,0,0,0,60,60,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,32,0,0,0,0,47,49,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D30.S3 in GAP, Magma, Sage, TeX

`D_{30}.S_3`
`% in TeX`

`G:=Group("D30.S3");`
`// GroupNames label`

`G:=SmallGroup(360,84);`
`// by ID`

`G=gap.SmallGroup(360,84);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,31,387,201,730,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=c^3=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=c^-1>;`
`// generators/relations`

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