metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.3D6, Dic5.5D6, C30.16C23, D30.3C22, Dic3.12D10, Dic15.5C22, C15⋊Q8⋊4C2, C5⋊D4⋊3S3, C3⋊4(C4○D20), C15⋊8(C4○D4), C3⋊D20⋊4C2, C15⋊7D4⋊3C2, (C2×C6).1D10, C5⋊3(D4⋊2S3), D30.C2⋊3C2, (C2×Dic3)⋊3D5, (D5×Dic3)⋊3C2, (C2×C10).13D6, C22.1(S3×D5), (C10×Dic3)⋊4C2, (C6×D5).3C22, C6.16(C22×D5), C10.16(C22×S3), (C2×C30).10C22, (C3×Dic5).5C22, (C5×Dic3).11C22, C2.18(C2×S3×D5), (C3×C5⋊D4)⋊1C2, SmallGroup(240,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.D6
G = < a,b,c,d | a10=c6=1, b2=d2=a5, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >
Subgroups: 344 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, C2×C6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C3×D4, C3×D5, D15, C30, C30, Dic10, C4×D5, D20, C5⋊D4, C5⋊D4, C2×C20, D4⋊2S3, C5×Dic3, C3×Dic5, Dic15, C6×D5, D30, C2×C30, C4○D20, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, C3×C5⋊D4, C10×Dic3, C15⋊7D4, Dic5.D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, D4⋊2S3, S3×D5, C4○D20, C2×S3×D5, Dic5.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 89 6 84)(2 88 7 83)(3 87 8 82)(4 86 9 81)(5 85 10 90)(11 46 16 41)(12 45 17 50)(13 44 18 49)(14 43 19 48)(15 42 20 47)(21 97 26 92)(22 96 27 91)(23 95 28 100)(24 94 29 99)(25 93 30 98)(31 80 36 75)(32 79 37 74)(33 78 38 73)(34 77 39 72)(35 76 40 71)(51 112 56 117)(52 111 57 116)(53 120 58 115)(54 119 59 114)(55 118 60 113)(61 105 66 110)(62 104 67 109)(63 103 68 108)(64 102 69 107)(65 101 70 106)
(1 19 33 119 25 108)(2 18 34 118 26 107)(3 17 35 117 27 106)(4 16 36 116 28 105)(5 15 37 115 29 104)(6 14 38 114 30 103)(7 13 39 113 21 102)(8 12 40 112 22 101)(9 11 31 111 23 110)(10 20 32 120 24 109)(41 80 52 95 66 81)(42 79 53 94 67 90)(43 78 54 93 68 89)(44 77 55 92 69 88)(45 76 56 91 70 87)(46 75 57 100 61 86)(47 74 58 99 62 85)(48 73 59 98 63 84)(49 72 60 97 64 83)(50 71 51 96 65 82)
(1 63 6 68)(2 62 7 67)(3 61 8 66)(4 70 9 65)(5 69 10 64)(11 96 16 91)(12 95 17 100)(13 94 18 99)(14 93 19 98)(15 92 20 97)(21 42 26 47)(22 41 27 46)(23 50 28 45)(24 49 29 44)(25 48 30 43)(31 51 36 56)(32 60 37 55)(33 59 38 54)(34 58 39 53)(35 57 40 52)(71 116 76 111)(72 115 77 120)(73 114 78 119)(74 113 79 118)(75 112 80 117)(81 106 86 101)(82 105 87 110)(83 104 88 109)(84 103 89 108)(85 102 90 107)
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), (1,89,6,84)(2,88,7,83)(3,87,8,82)(4,86,9,81)(5,85,10,90)(11,46,16,41)(12,45,17,50)(13,44,18,49)(14,43,19,48)(15,42,20,47)(21,97,26,92)(22,96,27,91)(23,95,28,100)(24,94,29,99)(25,93,30,98)(31,80,36,75)(32,79,37,74)(33,78,38,73)(34,77,39,72)(35,76,40,71)(51,112,56,117)(52,111,57,116)(53,120,58,115)(54,119,59,114)(55,118,60,113)(61,105,66,110)(62,104,67,109)(63,103,68,108)(64,102,69,107)(65,101,70,106), (1,19,33,119,25,108)(2,18,34,118,26,107)(3,17,35,117,27,106)(4,16,36,116,28,105)(5,15,37,115,29,104)(6,14,38,114,30,103)(7,13,39,113,21,102)(8,12,40,112,22,101)(9,11,31,111,23,110)(10,20,32,120,24,109)(41,80,52,95,66,81)(42,79,53,94,67,90)(43,78,54,93,68,89)(44,77,55,92,69,88)(45,76,56,91,70,87)(46,75,57,100,61,86)(47,74,58,99,62,85)(48,73,59,98,63,84)(49,72,60,97,64,83)(50,71,51,96,65,82), (1,63,6,68)(2,62,7,67)(3,61,8,66)(4,70,9,65)(5,69,10,64)(11,96,16,91)(12,95,17,100)(13,94,18,99)(14,93,19,98)(15,92,20,97)(21,42,26,47)(22,41,27,46)(23,50,28,45)(24,49,29,44)(25,48,30,43)(31,51,36,56)(32,60,37,55)(33,59,38,54)(34,58,39,53)(35,57,40,52)(71,116,76,111)(72,115,77,120)(73,114,78,119)(74,113,79,118)(75,112,80,117)(81,106,86,101)(82,105,87,110)(83,104,88,109)(84,103,89,108)(85,102,90,107)>;
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), (1,89,6,84)(2,88,7,83)(3,87,8,82)(4,86,9,81)(5,85,10,90)(11,46,16,41)(12,45,17,50)(13,44,18,49)(14,43,19,48)(15,42,20,47)(21,97,26,92)(22,96,27,91)(23,95,28,100)(24,94,29,99)(25,93,30,98)(31,80,36,75)(32,79,37,74)(33,78,38,73)(34,77,39,72)(35,76,40,71)(51,112,56,117)(52,111,57,116)(53,120,58,115)(54,119,59,114)(55,118,60,113)(61,105,66,110)(62,104,67,109)(63,103,68,108)(64,102,69,107)(65,101,70,106), (1,19,33,119,25,108)(2,18,34,118,26,107)(3,17,35,117,27,106)(4,16,36,116,28,105)(5,15,37,115,29,104)(6,14,38,114,30,103)(7,13,39,113,21,102)(8,12,40,112,22,101)(9,11,31,111,23,110)(10,20,32,120,24,109)(41,80,52,95,66,81)(42,79,53,94,67,90)(43,78,54,93,68,89)(44,77,55,92,69,88)(45,76,56,91,70,87)(46,75,57,100,61,86)(47,74,58,99,62,85)(48,73,59,98,63,84)(49,72,60,97,64,83)(50,71,51,96,65,82), (1,63,6,68)(2,62,7,67)(3,61,8,66)(4,70,9,65)(5,69,10,64)(11,96,16,91)(12,95,17,100)(13,94,18,99)(14,93,19,98)(15,92,20,97)(21,42,26,47)(22,41,27,46)(23,50,28,45)(24,49,29,44)(25,48,30,43)(31,51,36,56)(32,60,37,55)(33,59,38,54)(34,58,39,53)(35,57,40,52)(71,116,76,111)(72,115,77,120)(73,114,78,119)(74,113,79,118)(75,112,80,117)(81,106,86,101)(82,105,87,110)(83,104,88,109)(84,103,89,108)(85,102,90,107) );
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)], [(1,89,6,84),(2,88,7,83),(3,87,8,82),(4,86,9,81),(5,85,10,90),(11,46,16,41),(12,45,17,50),(13,44,18,49),(14,43,19,48),(15,42,20,47),(21,97,26,92),(22,96,27,91),(23,95,28,100),(24,94,29,99),(25,93,30,98),(31,80,36,75),(32,79,37,74),(33,78,38,73),(34,77,39,72),(35,76,40,71),(51,112,56,117),(52,111,57,116),(53,120,58,115),(54,119,59,114),(55,118,60,113),(61,105,66,110),(62,104,67,109),(63,103,68,108),(64,102,69,107),(65,101,70,106)], [(1,19,33,119,25,108),(2,18,34,118,26,107),(3,17,35,117,27,106),(4,16,36,116,28,105),(5,15,37,115,29,104),(6,14,38,114,30,103),(7,13,39,113,21,102),(8,12,40,112,22,101),(9,11,31,111,23,110),(10,20,32,120,24,109),(41,80,52,95,66,81),(42,79,53,94,67,90),(43,78,54,93,68,89),(44,77,55,92,69,88),(45,76,56,91,70,87),(46,75,57,100,61,86),(47,74,58,99,62,85),(48,73,59,98,63,84),(49,72,60,97,64,83),(50,71,51,96,65,82)], [(1,63,6,68),(2,62,7,67),(3,61,8,66),(4,70,9,65),(5,69,10,64),(11,96,16,91),(12,95,17,100),(13,94,18,99),(14,93,19,98),(15,92,20,97),(21,42,26,47),(22,41,27,46),(23,50,28,45),(24,49,29,44),(25,48,30,43),(31,51,36,56),(32,60,37,55),(33,59,38,54),(34,58,39,53),(35,57,40,52),(71,116,76,111),(72,115,77,120),(73,114,78,119),(74,113,79,118),(75,112,80,117),(81,106,86,101),(82,105,87,110),(83,104,88,109),(84,103,89,108),(85,102,90,107)]])
Dic5.D6 is a maximal subgroup of
D20.38D6 S3×C4○D20 C15⋊2- 1+4 D5×D4⋊2S3 D30.C23 D20⋊14D6 C15⋊2+ 1+4
Dic5.D6 is a maximal quotient of
Dic3⋊5Dic10 Dic5.2Dic6 Dic15.Q8 C4⋊Dic3⋊D5 Dic3.D20 D30.D4 (C2×C12).D10 (C4×Dic3)⋊D5 Dic3.3Dic10 C10.D4⋊S3 (D5×Dic3)⋊C4 Dic3⋊4D20 D30.23(C2×C4) D10.16D12 D30⋊2Q8 Dic15.D4 Dic15.19D4 C23.26(S3×D5) C23.13(S3×D5) C23.14(S3×D5) D30⋊6D4 C6.(D4×D5) (C2×C30).D4 C10.(C2×D12) C23.17(S3×D5) Dic15⋊3D4 Dic3×C5⋊D4 C15⋊28(C4×D4) D30.16D4 (C2×C6)⋊D20 (C2×C10)⋊8Dic6
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12 | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 10 | 30 | 2 | 3 | 3 | 6 | 10 | 30 | 2 | 2 | 2 | 4 | 20 | 2 | ··· | 2 | 20 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
39 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 | D6 | C4○D4 | D10 | D10 | C4○D20 | D4⋊2S3 | S3×D5 | C2×S3×D5 | Dic5.D6 |
| kernel | Dic5.D6 | D5×Dic3 | D30.C2 | C3⋊D20 | C15⋊Q8 | C3×C5⋊D4 | C10×Dic3 | C15⋊7D4 | C5⋊D4 | C2×Dic3 | Dic5 | D10 | C2×C10 | C15 | Dic3 | C2×C6 | C3 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of Dic5.D6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 1 | 0 | 0 |
| 0 | 0 | 16 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 27 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 43 | 0 | 0 |
| 0 | 0 | 16 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 11 | 18 | 0 | 0 | 0 | 0 |
| 34 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 18 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 60 | 15 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 43 | 0 | 0 |
| 0 | 0 | 16 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,16,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,27,0,0,0,0,0,11,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[11,34,0,0,0,0,18,50,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[60,8,0,0,0,0,15,1,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,1,0,0,0,0,0,1,60] >;
Dic5.D6 in GAP, Magma, Sage, TeX
{\rm Dic}_5.D_6 % in TeX
G:=Group("Dic5.D6"); // GroupNames label
G:=SmallGroup(240,140);
// by ID
G=gap.SmallGroup(240,140);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^6=1,b^2=d^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations