metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.4D10, C30.19C23, Dic5.13D6, Dic3.6D10, D30.4C22, Dic15.6C22, C15⋊Q8⋊6C2, C3⋊D4⋊3D5, C5⋊4(C4○D12), C5⋊D12⋊4C2, C15⋊7D4⋊4C2, (C2×C10).2D6, C15⋊10(C4○D4), C3⋊3(D4⋊2D5), D30.C2⋊4C2, (C6×Dic5)⋊4C2, (S3×Dic5)⋊4C2, (C2×Dic5)⋊3S3, (C2×C6).14D10, C22.2(S3×D5), C6.19(C22×D5), (S3×C10).4C22, C10.19(C22×S3), (C2×C30).13C22, (C5×Dic3).6C22, (C3×Dic5).15C22, C2.20(C2×S3×D5), (C5×C3⋊D4)⋊2C2, SmallGroup(240,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3.D10
G = < a,b,c,d | a6=c10=1, b2=d2=a3, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 336 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C3⋊D4, C2×C12, C5×S3, D15, C30, C30, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C5×D4, C4○D12, C5×Dic3, C3×Dic5, Dic15, S3×C10, D30, C2×C30, D4⋊2D5, S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C6×Dic5, C5×C3⋊D4, C15⋊7D4, Dic3.D10
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, C4○D12, S3×D5, D4⋊2D5, C2×S3×D5, Dic3.D10
►On 120 points
Generators in S120
(1 103 93 50 73 30)(2 21 74 41 94 104)(3 105 95 42 75 22)(4 23 76 43 96 106)(5 107 97 44 77 24)(6 25 78 45 98 108)(7 109 99 46 79 26)(8 27 80 47 100 110)(9 101 91 48 71 28)(10 29 72 49 92 102)(11 35 118 55 61 88)(12 89 62 56 119 36)(13 37 120 57 63 90)(14 81 64 58 111 38)(15 39 112 59 65 82)(16 83 66 60 113 40)(17 31 114 51 67 84)(18 85 68 52 115 32)(19 33 116 53 69 86)(20 87 70 54 117 34)
(1 118 50 88)(2 89 41 119)(3 120 42 90)(4 81 43 111)(5 112 44 82)(6 83 45 113)(7 114 46 84)(8 85 47 115)(9 116 48 86)(10 87 49 117)(11 30 55 93)(12 94 56 21)(13 22 57 95)(14 96 58 23)(15 24 59 97)(16 98 60 25)(17 26 51 99)(18 100 52 27)(19 28 53 91)(20 92 54 29)(31 79 67 109)(32 110 68 80)(33 71 69 101)(34 102 70 72)(35 73 61 103)(36 104 62 74)(37 75 63 105)(38 106 64 76)(39 77 65 107)(40 108 66 78)
(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 87 50 117)(2 86 41 116)(3 85 42 115)(4 84 43 114)(5 83 44 113)(6 82 45 112)(7 81 46 111)(8 90 47 120)(9 89 48 119)(10 88 49 118)(11 72 55 102)(12 71 56 101)(13 80 57 110)(14 79 58 109)(15 78 59 108)(16 77 60 107)(17 76 51 106)(18 75 52 105)(19 74 53 104)(20 73 54 103)(21 69 94 33)(22 68 95 32)(23 67 96 31)(24 66 97 40)(25 65 98 39)(26 64 99 38)(27 63 100 37)(28 62 91 36)(29 61 92 35)(30 70 93 34)
G:=sub<Sym(120)| (1,103,93,50,73,30)(2,21,74,41,94,104)(3,105,95,42,75,22)(4,23,76,43,96,106)(5,107,97,44,77,24)(6,25,78,45,98,108)(7,109,99,46,79,26)(8,27,80,47,100,110)(9,101,91,48,71,28)(10,29,72,49,92,102)(11,35,118,55,61,88)(12,89,62,56,119,36)(13,37,120,57,63,90)(14,81,64,58,111,38)(15,39,112,59,65,82)(16,83,66,60,113,40)(17,31,114,51,67,84)(18,85,68,52,115,32)(19,33,116,53,69,86)(20,87,70,54,117,34), (1,118,50,88)(2,89,41,119)(3,120,42,90)(4,81,43,111)(5,112,44,82)(6,83,45,113)(7,114,46,84)(8,85,47,115)(9,116,48,86)(10,87,49,117)(11,30,55,93)(12,94,56,21)(13,22,57,95)(14,96,58,23)(15,24,59,97)(16,98,60,25)(17,26,51,99)(18,100,52,27)(19,28,53,91)(20,92,54,29)(31,79,67,109)(32,110,68,80)(33,71,69,101)(34,102,70,72)(35,73,61,103)(36,104,62,74)(37,75,63,105)(38,106,64,76)(39,77,65,107)(40,108,66,78), (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), (1,87,50,117)(2,86,41,116)(3,85,42,115)(4,84,43,114)(5,83,44,113)(6,82,45,112)(7,81,46,111)(8,90,47,120)(9,89,48,119)(10,88,49,118)(11,72,55,102)(12,71,56,101)(13,80,57,110)(14,79,58,109)(15,78,59,108)(16,77,60,107)(17,76,51,106)(18,75,52,105)(19,74,53,104)(20,73,54,103)(21,69,94,33)(22,68,95,32)(23,67,96,31)(24,66,97,40)(25,65,98,39)(26,64,99,38)(27,63,100,37)(28,62,91,36)(29,61,92,35)(30,70,93,34)>;
G:=Group( (1,103,93,50,73,30)(2,21,74,41,94,104)(3,105,95,42,75,22)(4,23,76,43,96,106)(5,107,97,44,77,24)(6,25,78,45,98,108)(7,109,99,46,79,26)(8,27,80,47,100,110)(9,101,91,48,71,28)(10,29,72,49,92,102)(11,35,118,55,61,88)(12,89,62,56,119,36)(13,37,120,57,63,90)(14,81,64,58,111,38)(15,39,112,59,65,82)(16,83,66,60,113,40)(17,31,114,51,67,84)(18,85,68,52,115,32)(19,33,116,53,69,86)(20,87,70,54,117,34), (1,118,50,88)(2,89,41,119)(3,120,42,90)(4,81,43,111)(5,112,44,82)(6,83,45,113)(7,114,46,84)(8,85,47,115)(9,116,48,86)(10,87,49,117)(11,30,55,93)(12,94,56,21)(13,22,57,95)(14,96,58,23)(15,24,59,97)(16,98,60,25)(17,26,51,99)(18,100,52,27)(19,28,53,91)(20,92,54,29)(31,79,67,109)(32,110,68,80)(33,71,69,101)(34,102,70,72)(35,73,61,103)(36,104,62,74)(37,75,63,105)(38,106,64,76)(39,77,65,107)(40,108,66,78), (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), (1,87,50,117)(2,86,41,116)(3,85,42,115)(4,84,43,114)(5,83,44,113)(6,82,45,112)(7,81,46,111)(8,90,47,120)(9,89,48,119)(10,88,49,118)(11,72,55,102)(12,71,56,101)(13,80,57,110)(14,79,58,109)(15,78,59,108)(16,77,60,107)(17,76,51,106)(18,75,52,105)(19,74,53,104)(20,73,54,103)(21,69,94,33)(22,68,95,32)(23,67,96,31)(24,66,97,40)(25,65,98,39)(26,64,99,38)(27,63,100,37)(28,62,91,36)(29,61,92,35)(30,70,93,34) );
G=PermutationGroup([[(1,103,93,50,73,30),(2,21,74,41,94,104),(3,105,95,42,75,22),(4,23,76,43,96,106),(5,107,97,44,77,24),(6,25,78,45,98,108),(7,109,99,46,79,26),(8,27,80,47,100,110),(9,101,91,48,71,28),(10,29,72,49,92,102),(11,35,118,55,61,88),(12,89,62,56,119,36),(13,37,120,57,63,90),(14,81,64,58,111,38),(15,39,112,59,65,82),(16,83,66,60,113,40),(17,31,114,51,67,84),(18,85,68,52,115,32),(19,33,116,53,69,86),(20,87,70,54,117,34)], [(1,118,50,88),(2,89,41,119),(3,120,42,90),(4,81,43,111),(5,112,44,82),(6,83,45,113),(7,114,46,84),(8,85,47,115),(9,116,48,86),(10,87,49,117),(11,30,55,93),(12,94,56,21),(13,22,57,95),(14,96,58,23),(15,24,59,97),(16,98,60,25),(17,26,51,99),(18,100,52,27),(19,28,53,91),(20,92,54,29),(31,79,67,109),(32,110,68,80),(33,71,69,101),(34,102,70,72),(35,73,61,103),(36,104,62,74),(37,75,63,105),(38,106,64,76),(39,77,65,107),(40,108,66,78)], [(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)], [(1,87,50,117),(2,86,41,116),(3,85,42,115),(4,84,43,114),(5,83,44,113),(6,82,45,112),(7,81,46,111),(8,90,47,120),(9,89,48,119),(10,88,49,118),(11,72,55,102),(12,71,56,101),(13,80,57,110),(14,79,58,109),(15,78,59,108),(16,77,60,107),(17,76,51,106),(18,75,52,105),(19,74,53,104),(20,73,54,103),(21,69,94,33),(22,68,95,32),(23,67,96,31),(24,66,97,40),(25,65,98,39),(26,64,99,38),(27,63,100,37),(28,62,91,36),(29,61,92,35),(30,70,93,34)]])
Dic3.D10 is a maximal subgroup of
C5⋊C8.D6 D15⋊C8⋊C2 C30.C24 D5×C4○D12 C15⋊2- 1+4 S3×D4⋊2D5 D30.C23 D12⋊14D10 C15⋊2+ 1+4
Dic3.D10 is a maximal quotient of
Dic5⋊5Dic6 Dic15.2Q8 D6⋊C4.D5 C4⋊Dic5⋊S3 Dic3.2Dic10 Dic5.8D12 C5⋊(C42⋊3S3) Dic5.7Dic6 (C4×Dic5)⋊S3 D6.(C4×D5) D30.C2⋊C4 Dic5⋊4D12 D6.D20 D30⋊4Q8 D6⋊4Dic10 D30.7D4 C23.D5⋊S3 Dic15.19D4 (C6×Dic5)⋊7C4 C23.13(S3×D5) C23.14(S3×D5) C30.(C2×D4) C6.D4⋊D5 Dic5×C3⋊D4 C15⋊26(C4×D4) (S3×C10).D4 D30⋊7D4 Dic15⋊4D4 D30.16D4 (C2×C10)⋊4D12 (C2×C30)⋊Q8
36 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 | 30A | ··· | 30F |
| 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 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 6 | 30 | 2 | 5 | 5 | 6 | 10 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 10 | 10 | 10 | 10 | 4 | 4 | 12 | 12 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | S3×D5 | D4⋊2D5 | C2×S3×D5 | Dic3.D10 |
| kernel | Dic3.D10 | S3×Dic5 | D30.C2 | C5⋊D12 | C15⋊Q8 | C6×Dic5 | C5×C3⋊D4 | C15⋊7D4 | C2×Dic5 | C3⋊D4 | Dic5 | C2×C10 | C15 | Dic3 | D6 | C2×C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of Dic3.D10 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 33 | 50 |
| 1 | 43 | 0 | 0 | 0 | 0 |
| 18 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 48 |
| 0 | 0 | 0 | 0 | 28 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 18 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 21 |
| 0 | 0 | 0 | 0 | 58 | 1 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,11,33,0,0,0,0,0,50],[1,18,0,0,0,0,43,43,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,50,28,0,0,0,0,48,11],[1,18,0,0,0,0,0,60,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,60,58,0,0,0,0,21,1] >;
Dic3.D10 in GAP, Magma, Sage, TeX
{\rm Dic}_3.D_{10} % in TeX
G:=Group("Dic3.D10"); // GroupNames label
G:=SmallGroup(240,143);
// by ID
G=gap.SmallGroup(240,143);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,116,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^10=1,b^2=d^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations