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G = C20.D6order 240 = 24·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C60 — C20.D6
C1C5C15C30C60C3×Dic10 — C20.D6
C15C30C60 — C20.D6
C1C2C4

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 >

12C2
6C22
10C4
4S3
12C10
3D4
5Q8
15C8
2D6
10C12
2Dic5
6C2×C10
4C5×S3
15SD16
5C3×Q8
5C3⋊C8
3C52C8
3C5×D4
2S3×C10
2C3×Dic5
5Q82S3
3D4.D5

Character table of C20.D6

 class 12A2B34A4B5A5B68A8B10A10B10C10D10E10F12A12B12C15A15B20A20B30A30B60A60B60C60D
 size 1112222022230302212121212420204444444444
ρ1111111111111111111111111111111    trivial
ρ211-1111111-1-111-1-1-1-11111111111111    linear of order 2
ρ311-111-11111111-1-1-1-11-1-11111111111    linear of order 2
ρ411111-1111-1-11111111-1-11111111111    linear of order 2
ρ5220-12222-100220000-1-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ62202-2022200220000-20022-2-222-2-2-2-2    orthogonal lifted from D4
ρ7220-12-222-100220000-111-1-122-1-1-1-1-1-1    orthogonal lifted from D6
ρ822-2220-1-5/2-1+5/2200-1-5/2-1+5/21-5/21+5/21+5/21-5/2200-1+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
ρ9222220-1-5/2-1+5/2200-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2200-1+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
ρ1022-2220-1+5/2-1-5/2200-1+5/2-1-5/21+5/21-5/21-5/21+5/2200-1-5/2-1+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
ρ11222220-1+5/2-1-5/2200-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2200-1-5/2-1+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
ρ122202-20-1-5/2-1+5/2200-1-5/2-1+5/2ζ5455352ζ5352545-200-1+5/2-1-5/21-5/21+5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ13220-1-2022-1002200001-3--3-1-1-2-2-1-11111    complex lifted from C3⋊D4
ρ14220-1-2022-1002200001--3-3-1-1-2-2-1-11111    complex lifted from C3⋊D4
ρ152202-20-1+5/2-1-5/2200-1+5/2-1-5/25352545ζ545ζ5352-200-1-5/2-1+5/21+5/21-5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ162202-20-1-5/2-1+5/2200-1-5/2-1+5/2545ζ53525352ζ545-200-1+5/2-1-5/21-5/21+5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ172202-20-1+5/2-1-5/2200-1+5/2-1-5/2ζ5352ζ5455455352-200-1-5/2-1+5/21+5/21-5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ182-2020022-2-2--2-2-200000002200-2-20000    complex lifted from SD16
ρ192-2020022-2--2-2-2-200000002200-2-20000    complex lifted from SD16
ρ204-40-20044200-4-40000000-2-200220000    orthogonal lifted from Q82S3
ρ21440-240-1-5-1+5-200-1-5-1+50000-2001-5/21+5/2-1+5-1-51+5/21-5/21+5/21-5/21-5/21+5/2    orthogonal lifted from S3×D5
ρ22440-240-1+5-1-5-200-1+5-1-50000-2001+5/21-5/2-1-5-1+51-5/21+5/21-5/21+5/21+5/21-5/2    orthogonal lifted from S3×D5
ρ234-40400-1+5-1-5-4001-51+50000000-1-5-1+5001-51+50000    symplectic lifted from D4.D5, Schur index 2
ρ24440-2-40-1+5-1-5-200-1+5-1-500002001+5/21-5/21+51-51-5/21+5/2-1+5/2-1-5/2-1-5/2-1+5/2    symplectic lifted from C15⋊D4, Schur index 2
ρ25440-2-40-1-5-1+5-200-1-5-1+500002001-5/21+5/21-51+51+5/21-5/2-1-5/2-1+5/2-1+5/2-1-5/2    symplectic lifted from C15⋊D4, Schur index 2
ρ264-40400-1-5-1+5-4001+51-50000000-1+5-1-5001+51-50000    symplectic lifted from D4.D5, Schur index 2
ρ274-40-200-1+5-1-52001-51+500000001+5/21-5/200-1+5/2-1-5/243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ52-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ524ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5    complex faithful
ρ284-40-200-1-5-1+52001+51-500000001-5/21+5/200-1-5/2-1+5/2-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ5243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ54ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ52    complex faithful
ρ294-40-200-1+5-1-52001-51+500000001+5/21-5/200-1+5/2-1-5/24ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ52-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ5243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5    complex faithful
ρ304-40-200-1-5-1+52001+51-500000001-5/21+5/200-1-5/2-1+5/2-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ524ζ3ζ54-2ζ4ζ3ζ54ζ544ζ543ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ52    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)
(1 101 46 81 16 116 31 66)(2 100 47 80 17 115 32 65)(3 99 48 79 18 114 33 64)(4 98 49 78 19 113 34 63)(5 97 50 77 20 112 35 62)(6 96 51 76 21 111 36 61)(7 95 52 75 22 110 37 90)(8 94 53 74 23 109 38 89)(9 93 54 73 24 108 39 88)(10 92 55 72 25 107 40 87)(11 91 56 71 26 106 41 86)(12 120 57 70 27 105 42 85)(13 119 58 69 28 104 43 84)(14 118 59 68 29 103 44 83)(15 117 60 67 30 102 45 82)
(1 81 16 66)(2 70 17 85)(3 89 18 74)(4 78 19 63)(5 67 20 82)(6 86 21 71)(7 75 22 90)(8 64 23 79)(9 83 24 68)(10 72 25 87)(11 61 26 76)(12 80 27 65)(13 69 28 84)(14 88 29 73)(15 77 30 62)(31 116 46 101)(32 105 47 120)(33 94 48 109)(34 113 49 98)(35 102 50 117)(36 91 51 106)(37 110 52 95)(38 99 53 114)(39 118 54 103)(40 107 55 92)(41 96 56 111)(42 115 57 100)(43 104 58 119)(44 93 59 108)(45 112 60 97)

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

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

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

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)

2402400000
100000
0087000
007120500
00002400
00000240
,
100000
2402400000
00136200
003010500
0000062
000035203
,
100000
010000
00136200
003110500
0000062
00002060

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

Export

Subgroup lattice of C20.D6 in TeX
Character table of C20.D6 in TeX

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