metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.9D6, D6.7D10, C20.35D6, C12.35D10, C30.8C23, C60.35C22, Dic5.12D6, Dic3.8D10, D30.11C22, Dic15.13C22, C15⋊Q8⋊7C2, (C4×S3)⋊4D5, (C4×D5)⋊4S3, (S3×C20)⋊4C2, (D5×C12)⋊4C2, (C4×D15)⋊8C2, C5⋊2(C4○D12), C15⋊5(C4○D4), C3⋊2(C4○D20), C3⋊D20⋊7C2, C5⋊D12⋊7C2, C15⋊D4⋊7C2, C4.28(S3×D5), C6.8(C22×D5), C10.8(C22×S3), (S3×C10).8C22, (C6×D5).10C22, (C5×Dic3).10C22, (C3×Dic5).12C22, C2.12(C2×S3×D5), SmallGroup(240,132)
Series: Derived ►Chief ►Lower central ►Upper central
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
►On 120 points
(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)]])
D6.D10 is a maximal subgroup of
C40.54D6 C40.34D6 C40.55D6 C40.35D6 D5×C4○D12 S3×C4○D20 D20⋊24D6 C15⋊2- 1+4 D20⋊13D6 D20⋊14D6 D12⋊14D10 D20.29D6 C30.33C24 D12.29D10 D20⋊17D6
D6.D10 is a maximal quotient of
Dic3⋊C4⋊D5 D10⋊Dic6 Dic5.8D12 D6⋊Dic5⋊C2 D6⋊Dic10 Dic3.D20 D30.34D4 D30.35D4 (D5×C12)⋊C4 (C4×Dic3)⋊D5 (C4×Dic15)⋊C2 D6⋊Dic5.C2 (S3×C20)⋊7C4 C5⋊(C42⋊3S3) D30⋊8Q8 Dic5.7Dic6 Dic3.3Dic10 C10.D4⋊S3 Dic15.4Q8 (C4×D15)⋊10C4 (C4×Dic5)⋊S3 C4×C15⋊D4 C4×C3⋊D20 C4×C5⋊D12 D6⋊C4⋊D5 D10⋊D12 D10⋊C4⋊S3 D6⋊D20 D30⋊12D4 Dic15.31D4 C4×C15⋊Q8
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