metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊Dic5, C30.14D4, C10.10D12, C5⋊4(D6⋊C4), (S3×C10)⋊3C4, (C2×C6).7D10, (C2×C10).7D6, (C22×S3).D5, C10.20(C4×S3), C15⋊5(C22⋊C4), C30.30(C2×C4), (C2×Dic5)⋊1S3, (C6×Dic5)⋊1C2, C2.4(S3×Dic5), C6.4(C2×Dic5), C3⋊1(C23.D5), C22.6(S3×D5), (C2×Dic15)⋊6C2, C6.12(C5⋊D4), C2.2(C15⋊D4), C2.1(C5⋊D12), (C2×C30).4C22, C10.12(C3⋊D4), (S3×C2×C10).1C2, SmallGroup(240,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊Dic5
G = < a,b,c,d | a6=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 240 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, Dic5, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C2×Dic5, C2×Dic5, C22×C10, D6⋊C4, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C23.D5, C6×Dic5, C2×Dic15, S3×C2×C10, D6⋊Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, Dic5, D10, C4×S3, D12, C3⋊D4, C2×Dic5, C5⋊D4, D6⋊C4, S3×D5, C23.D5, S3×Dic5, C15⋊D4, C5⋊D12, D6⋊Dic5
►On 120 points
(1 14 45 59 28 65)(2 15 46 60 29 66)(3 16 47 51 30 67)(4 17 48 52 21 68)(5 18 49 53 22 69)(6 19 50 54 23 70)(7 20 41 55 24 61)(8 11 42 56 25 62)(9 12 43 57 26 63)(10 13 44 58 27 64)(31 76 95 110 116 82)(32 77 96 101 117 83)(33 78 97 102 118 84)(34 79 98 103 119 85)(35 80 99 104 120 86)(36 71 100 105 111 87)(37 72 91 106 112 88)(38 73 92 107 113 89)(39 74 93 108 114 90)(40 75 94 109 115 81)
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 30)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 60)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)(49 58)(50 59)(71 110)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)
(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 76 6 71)(2 75 7 80)(3 74 8 79)(4 73 9 78)(5 72 10 77)(11 98 16 93)(12 97 17 92)(13 96 18 91)(14 95 19 100)(15 94 20 99)(21 89 26 84)(22 88 27 83)(23 87 28 82)(24 86 29 81)(25 85 30 90)(31 70 36 65)(32 69 37 64)(33 68 38 63)(34 67 39 62)(35 66 40 61)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)
G:=sub<Sym(120)| (1,14,45,59,28,65)(2,15,46,60,29,66)(3,16,47,51,30,67)(4,17,48,52,21,68)(5,18,49,53,22,69)(6,19,50,54,23,70)(7,20,41,55,24,61)(8,11,42,56,25,62)(9,12,43,57,26,63)(10,13,44,58,27,64)(31,76,95,110,116,82)(32,77,96,101,117,83)(33,78,97,102,118,84)(34,79,98,103,119,85)(35,80,99,104,120,86)(36,71,100,105,111,87)(37,72,91,106,112,88)(38,73,92,107,113,89)(39,74,93,108,114,90)(40,75,94,109,115,81), (1,70)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,30)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,60)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(50,59)(71,110)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100), (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), (1,76,6,71)(2,75,7,80)(3,74,8,79)(4,73,9,78)(5,72,10,77)(11,98,16,93)(12,97,17,92)(13,96,18,91)(14,95,19,100)(15,94,20,99)(21,89,26,84)(22,88,27,83)(23,87,28,82)(24,86,29,81)(25,85,30,90)(31,70,36,65)(32,69,37,64)(33,68,38,63)(34,67,39,62)(35,66,40,61)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,114,56,119)(52,113,57,118)(53,112,58,117)(54,111,59,116)(55,120,60,115)>;
G:=Group( (1,14,45,59,28,65)(2,15,46,60,29,66)(3,16,47,51,30,67)(4,17,48,52,21,68)(5,18,49,53,22,69)(6,19,50,54,23,70)(7,20,41,55,24,61)(8,11,42,56,25,62)(9,12,43,57,26,63)(10,13,44,58,27,64)(31,76,95,110,116,82)(32,77,96,101,117,83)(33,78,97,102,118,84)(34,79,98,103,119,85)(35,80,99,104,120,86)(36,71,100,105,111,87)(37,72,91,106,112,88)(38,73,92,107,113,89)(39,74,93,108,114,90)(40,75,94,109,115,81), (1,70)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,30)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,60)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(50,59)(71,110)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100), (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), (1,76,6,71)(2,75,7,80)(3,74,8,79)(4,73,9,78)(5,72,10,77)(11,98,16,93)(12,97,17,92)(13,96,18,91)(14,95,19,100)(15,94,20,99)(21,89,26,84)(22,88,27,83)(23,87,28,82)(24,86,29,81)(25,85,30,90)(31,70,36,65)(32,69,37,64)(33,68,38,63)(34,67,39,62)(35,66,40,61)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,114,56,119)(52,113,57,118)(53,112,58,117)(54,111,59,116)(55,120,60,115) );
G=PermutationGroup([[(1,14,45,59,28,65),(2,15,46,60,29,66),(3,16,47,51,30,67),(4,17,48,52,21,68),(5,18,49,53,22,69),(6,19,50,54,23,70),(7,20,41,55,24,61),(8,11,42,56,25,62),(9,12,43,57,26,63),(10,13,44,58,27,64),(31,76,95,110,116,82),(32,77,96,101,117,83),(33,78,97,102,118,84),(34,79,98,103,119,85),(35,80,99,104,120,86),(36,71,100,105,111,87),(37,72,91,106,112,88),(38,73,92,107,113,89),(39,74,93,108,114,90),(40,75,94,109,115,81)], [(1,70),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,30),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,60),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,57),(49,58),(50,59),(71,110),(72,101),(73,102),(74,103),(75,104),(76,105),(77,106),(78,107),(79,108),(80,109),(81,86),(82,87),(83,88),(84,89),(85,90),(91,96),(92,97),(93,98),(94,99),(95,100)], [(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)], [(1,76,6,71),(2,75,7,80),(3,74,8,79),(4,73,9,78),(5,72,10,77),(11,98,16,93),(12,97,17,92),(13,96,18,91),(14,95,19,100),(15,94,20,99),(21,89,26,84),(22,88,27,83),(23,87,28,82),(24,86,29,81),(25,85,30,90),(31,70,36,65),(32,69,37,64),(33,68,38,63),(34,67,39,62),(35,66,40,61),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(51,114,56,119),(52,113,57,118),(53,112,58,117),(54,111,59,116),(55,120,60,115)]])
D6⋊Dic5 is a maximal subgroup of
(C2×C20).D6 (S3×C20)⋊5C4 D6⋊C4.D5 C60⋊5C4⋊C2 Dic5.8D12 D6⋊Dic5⋊C2 D6⋊Dic10 D30.35D4 C60.45D4 D6⋊Dic5.C2 C60.46D4 C60.89D4 (S3×C20)⋊7C4 C5⋊(C42⋊3S3) C60.69D4 D6.(C4×D5) (S3×Dic5)⋊C4 D6⋊1Dic10 D10.17D12 Dic5×D12 Dic5⋊D12 D6⋊2Dic10 (C2×D12).D5 D6⋊3Dic10 Dic15⋊8D4 D6⋊4Dic10 D30.7D4 C4×C15⋊D4 C15⋊17(C4×D4) C4×C5⋊D12 C15⋊22(C4×D4) D6⋊C4⋊D5 C60⋊D4 D10⋊C4⋊S3 Dic15⋊2D4 Dic15.10D4 Dic15.31D4 C20⋊2D12 D5×D6⋊C4 D30.27D4 D30⋊4D4 C23.D5⋊S3 C30.(C2×D4) (C2×C10).D12 (C6×D5)⋊D4 Dic5×C3⋊D4 S3×C23.D5 (S3×C10).D4 D30⋊7D4 Dic15⋊17D4 (C2×C10)⋊4D12 C15⋊C22≀C2 (C2×C10)⋊11D12 Dic15⋊18D4 D30⋊8D4
D6⋊Dic5 is a maximal quotient of
C60.94D4 C20.5D12 C60.54D4 D12⋊Dic5 C10.D24 Dic6⋊Dic5 C10.Dic12 C60.98D4 C60.99D4 C30.24C42 C15⋊8(C23⋊C4)
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | Dic5 | D10 | C4×S3 | D12 | C3⋊D4 | C5⋊D4 | S3×D5 | S3×Dic5 | C15⋊D4 | C5⋊D12 |
| kernel | D6⋊Dic5 | C6×Dic5 | C2×Dic15 | S3×C2×C10 | S3×C10 | C2×Dic5 | C30 | C22×S3 | C2×C10 | D6 | C2×C6 | C10 | C10 | C10 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 2 |
Matrix representation of D6⋊Dic5 ►in GL4(𝔽61) generated by
| 60 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 1 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 14 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 41 | 0 |
| 0 | 0 | 22 | 3 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 50 | 19 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(61))| [60,1,0,0,60,0,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,1,60,0,0,0,0,1,14,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,41,22,0,0,0,3],[50,0,0,0,0,50,0,0,0,0,50,0,0,0,19,11] >;
D6⋊Dic5 in GAP, Magma, Sage, TeX
D_6\rtimes {\rm Dic}_5 % in TeX
G:=Group("D6:Dic5"); // GroupNames label
G:=SmallGroup(240,27);
// by ID
G=gap.SmallGroup(240,27);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^10=1,d^2=c^5,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^-1>;
// generators/relations