metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic30, C4.D15, C15⋊2Q8, C5⋊2Dic6, C60.1C2, C20.1S3, C10.8D6, C12.1D5, C2.3D30, C6.8D10, C3⋊2Dic10, C30.8C22, Dic15.1C2, SmallGroup(120,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic30
G = < a,b | a60=1, b2=a30, bab-1=a-1 >
Smallest permutation representation of Dic30
►Regular action 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 83 31 113)(2 82 32 112)(3 81 33 111)(4 80 34 110)(5 79 35 109)(6 78 36 108)(7 77 37 107)(8 76 38 106)(9 75 39 105)(10 74 40 104)(11 73 41 103)(12 72 42 102)(13 71 43 101)(14 70 44 100)(15 69 45 99)(16 68 46 98)(17 67 47 97)(18 66 48 96)(19 65 49 95)(20 64 50 94)(21 63 51 93)(22 62 52 92)(23 61 53 91)(24 120 54 90)(25 119 55 89)(26 118 56 88)(27 117 57 87)(28 116 58 86)(29 115 59 85)(30 114 60 84)
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), (1,83,31,113)(2,82,32,112)(3,81,33,111)(4,80,34,110)(5,79,35,109)(6,78,36,108)(7,77,37,107)(8,76,38,106)(9,75,39,105)(10,74,40,104)(11,73,41,103)(12,72,42,102)(13,71,43,101)(14,70,44,100)(15,69,45,99)(16,68,46,98)(17,67,47,97)(18,66,48,96)(19,65,49,95)(20,64,50,94)(21,63,51,93)(22,62,52,92)(23,61,53,91)(24,120,54,90)(25,119,55,89)(26,118,56,88)(27,117,57,87)(28,116,58,86)(29,115,59,85)(30,114,60,84)>;
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), (1,83,31,113)(2,82,32,112)(3,81,33,111)(4,80,34,110)(5,79,35,109)(6,78,36,108)(7,77,37,107)(8,76,38,106)(9,75,39,105)(10,74,40,104)(11,73,41,103)(12,72,42,102)(13,71,43,101)(14,70,44,100)(15,69,45,99)(16,68,46,98)(17,67,47,97)(18,66,48,96)(19,65,49,95)(20,64,50,94)(21,63,51,93)(22,62,52,92)(23,61,53,91)(24,120,54,90)(25,119,55,89)(26,118,56,88)(27,117,57,87)(28,116,58,86)(29,115,59,85)(30,114,60,84) );
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)], [(1,83,31,113),(2,82,32,112),(3,81,33,111),(4,80,34,110),(5,79,35,109),(6,78,36,108),(7,77,37,107),(8,76,38,106),(9,75,39,105),(10,74,40,104),(11,73,41,103),(12,72,42,102),(13,71,43,101),(14,70,44,100),(15,69,45,99),(16,68,46,98),(17,67,47,97),(18,66,48,96),(19,65,49,95),(20,64,50,94),(21,63,51,93),(22,62,52,92),(23,61,53,91),(24,120,54,90),(25,119,55,89),(26,118,56,88),(27,117,57,87),(28,116,58,86),(29,115,59,85),(30,114,60,84)]])
Dic30 is a maximal subgroup of
C6.D20 D12.D5 C3⋊Dic20 C5⋊Dic12 C24⋊D5 Dic60 D4.D15 C15⋊7Q16 D5×Dic6 D20⋊5S3 S3×Dic10 D12⋊5D5 D60⋊11C2 D4⋊2D15 Q8×D15 Dic90 C3⋊Dic30 C12.D15 C20.1S4 C20.2S4
Dic30 is a maximal quotient of
C30.4Q8 C60⋊5C4 Dic90 C3⋊Dic30 C12.D15 C20.1S4
33 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
33 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D5 | D6 | D10 | Dic6 | D15 | Dic10 | D30 | Dic30 |
| kernel | Dic30 | Dic15 | C60 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of Dic30 ►in GL4(𝔽61) generated by
| 27 | 25 | 0 | 0 |
| 36 | 4 | 0 | 0 |
| 0 | 0 | 23 | 53 |
| 0 | 0 | 8 | 45 |
| 55 | 52 | 0 | 0 |
| 38 | 6 | 0 | 0 |
| 0 | 0 | 8 | 45 |
| 0 | 0 | 23 | 53 |
G:=sub<GL(4,GF(61))| [27,36,0,0,25,4,0,0,0,0,23,8,0,0,53,45],[55,38,0,0,52,6,0,0,0,0,8,23,0,0,45,53] >;
Dic30 in GAP, Magma, Sage, TeX
{\rm Dic}_{30} % in TeX
G:=Group("Dic30"); // GroupNames label
G:=SmallGroup(120,26);
// by ID
G=gap.SmallGroup(120,26);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,20,61,26,323,2404]);
// Polycyclic
G:=Group<a,b|a^60=1,b^2=a^30,b*a*b^-1=a^-1>;
// generators/relations
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