direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C30, C6⋊C60, C30⋊5C12, C30.70D6, C62.1C10, C3⋊2(C2×C60), (C3×C6)⋊2C20, (C3×C30)⋊10C4, C2.2(S3×C30), (C2×C30).9C6, C6.4(C2×C30), (C6×C30).4C2, (C2×C6).3C30, C32⋊6(C2×C20), C22.(S3×C15), C15⋊12(C2×C12), C10.16(S3×C6), C6.20(S3×C10), C30.27(C2×C6), (C2×C30).12S3, (C3×C30).50C22, (C3×C15)⋊34(C2×C4), (C2×C6).6(C5×S3), (C3×C6).9(C2×C10), (C2×C10).2(C3×S3), SmallGroup(360,98)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C30 |
Generators and relations for Dic3×C30
G = < a,b,c | a30=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 108 in 74 conjugacy classes, 52 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, C6, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C15, C3×C6, C3×C6, C20, C2×C10, C2×Dic3, C2×C12, C30, C30, C30, C3×Dic3, C62, C2×C20, C3×C15, C5×Dic3, C60, C2×C30, C2×C30, C6×Dic3, C3×C30, C3×C30, C10×Dic3, C2×C60, Dic3×C15, C6×C30, Dic3×C30
Quotients: C1, C2, C3, C4, C22, C5, S3, C6, C2×C4, C10, Dic3, C12, D6, C2×C6, C15, C3×S3, C20, C2×C10, C2×Dic3, C2×C12, C5×S3, C30, C3×Dic3, S3×C6, C2×C20, C5×Dic3, C60, S3×C10, C2×C30, C6×Dic3, S3×C15, C10×Dic3, C2×C60, Dic3×C15, S3×C30, Dic3×C30
►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 78 21 68 11 88)(2 79 22 69 12 89)(3 80 23 70 13 90)(4 81 24 71 14 61)(5 82 25 72 15 62)(6 83 26 73 16 63)(7 84 27 74 17 64)(8 85 28 75 18 65)(9 86 29 76 19 66)(10 87 30 77 20 67)(31 114 41 94 51 104)(32 115 42 95 52 105)(33 116 43 96 53 106)(34 117 44 97 54 107)(35 118 45 98 55 108)(36 119 46 99 56 109)(37 120 47 100 57 110)(38 91 48 101 58 111)(39 92 49 102 59 112)(40 93 50 103 60 113)
(1 112 68 49)(2 113 69 50)(3 114 70 51)(4 115 71 52)(5 116 72 53)(6 117 73 54)(7 118 74 55)(8 119 75 56)(9 120 76 57)(10 91 77 58)(11 92 78 59)(12 93 79 60)(13 94 80 31)(14 95 81 32)(15 96 82 33)(16 97 83 34)(17 98 84 35)(18 99 85 36)(19 100 86 37)(20 101 87 38)(21 102 88 39)(22 103 89 40)(23 104 90 41)(24 105 61 42)(25 106 62 43)(26 107 63 44)(27 108 64 45)(28 109 65 46)(29 110 66 47)(30 111 67 48)
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), (1,78,21,68,11,88)(2,79,22,69,12,89)(3,80,23,70,13,90)(4,81,24,71,14,61)(5,82,25,72,15,62)(6,83,26,73,16,63)(7,84,27,74,17,64)(8,85,28,75,18,65)(9,86,29,76,19,66)(10,87,30,77,20,67)(31,114,41,94,51,104)(32,115,42,95,52,105)(33,116,43,96,53,106)(34,117,44,97,54,107)(35,118,45,98,55,108)(36,119,46,99,56,109)(37,120,47,100,57,110)(38,91,48,101,58,111)(39,92,49,102,59,112)(40,93,50,103,60,113), (1,112,68,49)(2,113,69,50)(3,114,70,51)(4,115,71,52)(5,116,72,53)(6,117,73,54)(7,118,74,55)(8,119,75,56)(9,120,76,57)(10,91,77,58)(11,92,78,59)(12,93,79,60)(13,94,80,31)(14,95,81,32)(15,96,82,33)(16,97,83,34)(17,98,84,35)(18,99,85,36)(19,100,86,37)(20,101,87,38)(21,102,88,39)(22,103,89,40)(23,104,90,41)(24,105,61,42)(25,106,62,43)(26,107,63,44)(27,108,64,45)(28,109,65,46)(29,110,66,47)(30,111,67,48)>;
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), (1,78,21,68,11,88)(2,79,22,69,12,89)(3,80,23,70,13,90)(4,81,24,71,14,61)(5,82,25,72,15,62)(6,83,26,73,16,63)(7,84,27,74,17,64)(8,85,28,75,18,65)(9,86,29,76,19,66)(10,87,30,77,20,67)(31,114,41,94,51,104)(32,115,42,95,52,105)(33,116,43,96,53,106)(34,117,44,97,54,107)(35,118,45,98,55,108)(36,119,46,99,56,109)(37,120,47,100,57,110)(38,91,48,101,58,111)(39,92,49,102,59,112)(40,93,50,103,60,113), (1,112,68,49)(2,113,69,50)(3,114,70,51)(4,115,71,52)(5,116,72,53)(6,117,73,54)(7,118,74,55)(8,119,75,56)(9,120,76,57)(10,91,77,58)(11,92,78,59)(12,93,79,60)(13,94,80,31)(14,95,81,32)(15,96,82,33)(16,97,83,34)(17,98,84,35)(18,99,85,36)(19,100,86,37)(20,101,87,38)(21,102,88,39)(22,103,89,40)(23,104,90,41)(24,105,61,42)(25,106,62,43)(26,107,63,44)(27,108,64,45)(28,109,65,46)(29,110,66,47)(30,111,67,48) );
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)], [(1,78,21,68,11,88),(2,79,22,69,12,89),(3,80,23,70,13,90),(4,81,24,71,14,61),(5,82,25,72,15,62),(6,83,26,73,16,63),(7,84,27,74,17,64),(8,85,28,75,18,65),(9,86,29,76,19,66),(10,87,30,77,20,67),(31,114,41,94,51,104),(32,115,42,95,52,105),(33,116,43,96,53,106),(34,117,44,97,54,107),(35,118,45,98,55,108),(36,119,46,99,56,109),(37,120,47,100,57,110),(38,91,48,101,58,111),(39,92,49,102,59,112),(40,93,50,103,60,113)], [(1,112,68,49),(2,113,69,50),(3,114,70,51),(4,115,71,52),(5,116,72,53),(6,117,73,54),(7,118,74,55),(8,119,75,56),(9,120,76,57),(10,91,77,58),(11,92,78,59),(12,93,79,60),(13,94,80,31),(14,95,81,32),(15,96,82,33),(16,97,83,34),(17,98,84,35),(18,99,85,36),(19,100,86,37),(20,101,87,38),(21,102,88,39),(22,103,89,40),(23,104,90,41),(24,105,61,42),(25,106,62,43),(26,107,63,44),(27,108,64,45),(28,109,65,46),(29,110,66,47),(30,111,67,48)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | ··· | 6O | 10A | ··· | 10L | 12A | ··· | 12H | 15A | ··· | 15H | 15I | ··· | 15T | 20A | ··· | 20P | 30A | ··· | 30X | 30Y | ··· | 30BH | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | S3 | Dic3 | D6 | C3×S3 | C5×S3 | C3×Dic3 | S3×C6 | C5×Dic3 | S3×C10 | S3×C15 | Dic3×C15 | S3×C30 |
| kernel | Dic3×C30 | Dic3×C15 | C6×C30 | C10×Dic3 | C3×C30 | C6×Dic3 | C5×Dic3 | C2×C30 | C3×Dic3 | C62 | C30 | C2×Dic3 | C3×C6 | Dic3 | C2×C6 | C6 | C2×C30 | C30 | C30 | C2×C10 | C2×C6 | C10 | C10 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 8 | 16 | 16 | 8 | 32 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 8 | 16 | 8 |
Matrix representation of Dic3×C30 ►in GL3(𝔽61) generated by
| 48 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 48 |
| 60 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(61))| [48,0,0,0,36,0,0,0,36],[1,0,0,0,14,0,0,0,48],[60,0,0,0,0,1,0,60,0] >;
Dic3×C30 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{30} % in TeX
G:=Group("Dic3xC30"); // GroupNames label
G:=SmallGroup(360,98);
// by ID
G=gap.SmallGroup(360,98);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-3,360,8645]);
// Polycyclic
G:=Group<a,b,c|a^30=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations