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## G = D6.D10order 240 = 24·3·5

### 3rd non-split extension by D6 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D6.D10
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — D6.D10
 Lower central C15 — C30 — D6.D10
 Upper central C1 — C4

Generators and relations for D6.D10
G = < a,b,c,d | a6=b2=1, c10=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c9 >

Subgroups: 352 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, D15, C30, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C2×C20, C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C4○D20, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D6.D10
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, C4○D12, S3×D5, C4○D20, C2×S3×D5, D6.D10

Smallest permutation representation of D6.D10
On 120 points
Generators in S120
```(1 40 100 11 30 90)(2 21 81 12 31 91)(3 22 82 13 32 92)(4 23 83 14 33 93)(5 24 84 15 34 94)(6 25 85 16 35 95)(7 26 86 17 36 96)(8 27 87 18 37 97)(9 28 88 19 38 98)(10 29 89 20 39 99)(41 104 70 51 114 80)(42 105 71 52 115 61)(43 106 72 53 116 62)(44 107 73 54 117 63)(45 108 74 55 118 64)(46 109 75 56 119 65)(47 110 76 57 120 66)(48 111 77 58 101 67)(49 112 78 59 102 68)(50 113 79 60 103 69)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(99 113)(100 114)
(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)
(1 108 11 118)(2 117 12 107)(3 106 13 116)(4 115 14 105)(5 104 15 114)(6 113 16 103)(7 102 17 112)(8 111 18 101)(9 120 19 110)(10 109 20 119)(21 63 31 73)(22 72 32 62)(23 61 33 71)(24 70 34 80)(25 79 35 69)(26 68 36 78)(27 77 37 67)(28 66 38 76)(29 75 39 65)(30 64 40 74)(41 84 51 94)(42 93 52 83)(43 82 53 92)(44 91 54 81)(45 100 55 90)(46 89 56 99)(47 98 57 88)(48 87 58 97)(49 96 59 86)(50 85 60 95)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 60A 60B 60C 60D order 1 2 2 2 2 3 4 4 4 4 4 5 5 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 60 60 size 1 1 6 10 30 2 1 1 6 10 30 2 2 2 10 10 2 2 6 6 6 6 2 2 10 10 4 4 2 2 2 2 6 6 6 6 4 4 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 C4○D12 C4○D20 S3×D5 C2×S3×D5 D6.D10 kernel D6.D10 C15⋊D4 C3⋊D20 C5⋊D12 C15⋊Q8 D5×C12 S3×C20 C4×D15 C4×D5 C4×S3 Dic5 C20 D10 C15 Dic3 C12 D6 C5 C3 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 4 8 2 2 4

Matrix representation of D6.D10 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 2 46 0 0 49 60
,
 60 0 0 0 0 60 0 0 0 0 0 21 0 0 32 0
,
 43 44 0 0 18 0 0 0 0 0 11 0 0 0 0 11
,
 43 60 0 0 18 18 0 0 0 0 53 19 0 0 3 8
`G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,2,49,0,0,46,60],[60,0,0,0,0,60,0,0,0,0,0,32,0,0,21,0],[43,18,0,0,44,0,0,0,0,0,11,0,0,0,0,11],[43,18,0,0,60,18,0,0,0,0,53,3,0,0,19,8] >;`

D6.D10 in GAP, Magma, Sage, TeX

`D_6.D_{10}`
`% in TeX`

`G:=Group("D6.D10");`
`// GroupNames label`

`G:=SmallGroup(240,132);`
`// by ID`

`G=gap.SmallGroup(240,132);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=1,c^10=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^9>;`
`// generators/relations`

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