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## G = C20.D6order 240 = 24·3·5

### 4th non-split extension by C20 of D6 acting via D6/C3=C22

Aliases: C20.4D6, C30.5D4, C152SD16, D12.2D5, C12.4D10, Dic102S3, C60.24C22, C153C88C2, C32(D4.D5), C4.17(S3×D5), (C5×D12).2C2, C52(Q82S3), C6.9(C5⋊D4), (C3×Dic10)⋊4C2, C2.6(C15⋊D4), C10.9(C3⋊D4), SmallGroup(240,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C20.D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — C20.D6
 Lower central C15 — C30 — C60 — C20.D6
 Upper central C1 — C2 — C4

Generators and relations for C20.D6
G = < a,b,c | a30=1, b4=c2=a15, bab-1=a-1, cac-1=a19, cbc-1=b3 >

Character table of C20.D6

 class 1 2A 2B 3 4A 4B 5A 5B 6 8A 8B 10A 10B 10C 10D 10E 10F 12A 12B 12C 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D size 1 1 12 2 2 20 2 2 2 30 30 2 2 12 12 12 12 4 20 20 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 -1 2 2 2 2 -1 0 0 2 2 0 0 0 0 -1 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 2 -2 0 2 2 2 0 0 2 2 0 0 0 0 -2 0 0 2 2 -2 -2 2 2 -2 -2 -2 -2 orthogonal lifted from D4 ρ7 2 2 0 -1 2 -2 2 2 -1 0 0 2 2 0 0 0 0 -1 1 1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ8 2 2 -2 2 2 0 -1-√5/2 -1+√5/2 2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ9 2 2 2 2 2 0 -1-√5/2 -1+√5/2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ10 2 2 -2 2 2 0 -1+√5/2 -1-√5/2 2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ11 2 2 2 2 2 0 -1+√5/2 -1-√5/2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ12 2 2 0 2 -2 0 -1-√5/2 -1+√5/2 2 0 0 -1-√5/2 -1+√5/2 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 -2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ13 2 2 0 -1 -2 0 2 2 -1 0 0 2 2 0 0 0 0 1 √-3 -√-3 -1 -1 -2 -2 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ14 2 2 0 -1 -2 0 2 2 -1 0 0 2 2 0 0 0 0 1 -√-3 √-3 -1 -1 -2 -2 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 2 -2 0 -1+√5/2 -1-√5/2 2 0 0 -1+√5/2 -1-√5/2 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 -2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ16 2 2 0 2 -2 0 -1-√5/2 -1+√5/2 2 0 0 -1-√5/2 -1+√5/2 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 -2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ17 2 2 0 2 -2 0 -1+√5/2 -1-√5/2 2 0 0 -1+√5/2 -1-√5/2 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 -2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ18 2 -2 0 2 0 0 2 2 -2 √-2 -√-2 -2 -2 0 0 0 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 complex lifted from SD16 ρ19 2 -2 0 2 0 0 2 2 -2 -√-2 √-2 -2 -2 0 0 0 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 complex lifted from SD16 ρ20 4 -4 0 -2 0 0 4 4 2 0 0 -4 -4 0 0 0 0 0 0 0 -2 -2 0 0 2 2 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ21 4 4 0 -2 4 0 -1-√5 -1+√5 -2 0 0 -1-√5 -1+√5 0 0 0 0 -2 0 0 1-√5/2 1+√5/2 -1+√5 -1-√5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ22 4 4 0 -2 4 0 -1+√5 -1-√5 -2 0 0 -1+√5 -1-√5 0 0 0 0 -2 0 0 1+√5/2 1-√5/2 -1-√5 -1+√5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ23 4 -4 0 4 0 0 -1+√5 -1-√5 -4 0 0 1-√5 1+√5 0 0 0 0 0 0 0 -1-√5 -1+√5 0 0 1-√5 1+√5 0 0 0 0 symplectic lifted from D4.D5, Schur index 2 ρ24 4 4 0 -2 -4 0 -1+√5 -1-√5 -2 0 0 -1+√5 -1-√5 0 0 0 0 2 0 0 1+√5/2 1-√5/2 1+√5 1-√5 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ25 4 4 0 -2 -4 0 -1-√5 -1+√5 -2 0 0 -1-√5 -1+√5 0 0 0 0 2 0 0 1-√5/2 1+√5/2 1-√5 1+√5 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ26 4 -4 0 4 0 0 -1-√5 -1+√5 -4 0 0 1+√5 1-√5 0 0 0 0 0 0 0 -1+√5 -1-√5 0 0 1+√5 1-√5 0 0 0 0 symplectic lifted from D4.D5, Schur index 2 ρ27 4 -4 0 -2 0 0 -1+√5 -1-√5 2 0 0 1-√5 1+√5 0 0 0 0 0 0 0 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 complex faithful ρ28 4 -4 0 -2 0 0 -1-√5 -1+√5 2 0 0 1+√5 1-√5 0 0 0 0 0 0 0 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 complex faithful ρ29 4 -4 0 -2 0 0 -1+√5 -1-√5 2 0 0 1-√5 1+√5 0 0 0 0 0 0 0 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 complex faithful ρ30 4 -4 0 -2 0 0 -1-√5 -1+√5 2 0 0 1+√5 1-√5 0 0 0 0 0 0 0 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 complex faithful

Smallest permutation representation of C20.D6
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)
(1 116 36 86 16 101 51 71)(2 115 37 85 17 100 52 70)(3 114 38 84 18 99 53 69)(4 113 39 83 19 98 54 68)(5 112 40 82 20 97 55 67)(6 111 41 81 21 96 56 66)(7 110 42 80 22 95 57 65)(8 109 43 79 23 94 58 64)(9 108 44 78 24 93 59 63)(10 107 45 77 25 92 60 62)(11 106 46 76 26 91 31 61)(12 105 47 75 27 120 32 90)(13 104 48 74 28 119 33 89)(14 103 49 73 29 118 34 88)(15 102 50 72 30 117 35 87)
(1 86 16 71)(2 75 17 90)(3 64 18 79)(4 83 19 68)(5 72 20 87)(6 61 21 76)(7 80 22 65)(8 69 23 84)(9 88 24 73)(10 77 25 62)(11 66 26 81)(12 85 27 70)(13 74 28 89)(14 63 29 78)(15 82 30 67)(31 111 46 96)(32 100 47 115)(33 119 48 104)(34 108 49 93)(35 97 50 112)(36 116 51 101)(37 105 52 120)(38 94 53 109)(39 113 54 98)(40 102 55 117)(41 91 56 106)(42 110 57 95)(43 99 58 114)(44 118 59 103)(45 107 60 92)```

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

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

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

C20.D6 is a maximal subgroup of
C4014D6  D246D5  D245D5  D30.4D4  D20.34D6  D2021D6  D12.37D10  Dic103D6  S3×D4.D5  D12.24D10  D12.9D10  D5×Q82S3  Dic10.26D6  D12.27D10  Dic10.27D6
C20.D6 is a maximal quotient of
D12⋊Dic5  C30.Q16  C30.SD16

Matrix representation of C20.D6 in GL6(𝔽241)

 240 240 0 0 0 0 1 0 0 0 0 0 0 0 87 0 0 0 0 0 71 205 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 240 240 0 0 0 0 0 0 136 2 0 0 0 0 30 105 0 0 0 0 0 0 0 62 0 0 0 0 35 203
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 136 2 0 0 0 0 31 105 0 0 0 0 0 0 0 62 0 0 0 0 206 0

`G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,87,71,0,0,0,0,0,205,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,240,0,0,0,0,0,240,0,0,0,0,0,0,136,30,0,0,0,0,2,105,0,0,0,0,0,0,0,35,0,0,0,0,62,203],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,136,31,0,0,0,0,2,105,0,0,0,0,0,0,0,206,0,0,0,0,62,0] >;`

C20.D6 in GAP, Magma, Sage, TeX

`C_{20}.D_6`
`% in TeX`

`G:=Group("C20.D6");`
`// GroupNames label`

`G:=SmallGroup(240,17);`
`// by ID`

`G=gap.SmallGroup(240,17);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,218,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=1,b^4=c^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^19,c*b*c^-1=b^3>;`
`// generators/relations`

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