direct product, metacyclic, supersoluble, monomial
Aliases: C15×Dic6, C60.6C6, C12.1C30, C60.15S3, C30.67D6, Dic3.C30, C3⋊(Q8×C15), C4.(S3×C15), C15⋊4(C3×Q8), (C3×C15)⋊11Q8, C32⋊3(C5×Q8), C20.3(C3×S3), C2.3(S3×C30), C12.7(C5×S3), (C3×C60).7C2, C6.1(C2×C30), C10.13(S3×C6), C6.17(S3×C10), (C3×C12).2C10, C30.24(C2×C6), (C5×Dic3).2C6, (C3×C30).47C22, (Dic3×C15).4C2, (C3×Dic3).2C10, (C3×C6).6(C2×C10), SmallGroup(360,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×Dic6
G = < a,b,c | a15=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 84 in 54 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C3, C3, C4, C4, C5, C6, C6, Q8, C32, C10, Dic3, C12, C12, C15, C15, C3×C6, C20, C20, Dic6, C3×Q8, C30, C30, C3×Dic3, C3×C12, C5×Q8, C3×C15, C5×Dic3, C60, C60, C3×Dic6, C3×C30, C5×Dic6, Q8×C15, Dic3×C15, C3×C60, C15×Dic6
Quotients: C1, C2, C3, C22, C5, S3, C6, Q8, C10, D6, C2×C6, C15, C3×S3, C2×C10, Dic6, C3×Q8, C5×S3, C30, S3×C6, C5×Q8, S3×C10, C2×C30, C3×Dic6, S3×C15, C5×Dic6, Q8×C15, S3×C30, C15×Dic6
►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 18 75 99 6 23 65 104 11 28 70 94)(2 19 61 100 7 24 66 105 12 29 71 95)(3 20 62 101 8 25 67 91 13 30 72 96)(4 21 63 102 9 26 68 92 14 16 73 97)(5 22 64 103 10 27 69 93 15 17 74 98)(31 56 77 110 41 51 87 120 36 46 82 115)(32 57 78 111 42 52 88 106 37 47 83 116)(33 58 79 112 43 53 89 107 38 48 84 117)(34 59 80 113 44 54 90 108 39 49 85 118)(35 60 81 114 45 55 76 109 40 50 86 119)
(1 40 65 81)(2 41 66 82)(3 42 67 83)(4 43 68 84)(5 44 69 85)(6 45 70 86)(7 31 71 87)(8 32 72 88)(9 33 73 89)(10 34 74 90)(11 35 75 76)(12 36 61 77)(13 37 62 78)(14 38 63 79)(15 39 64 80)(16 107 102 58)(17 108 103 59)(18 109 104 60)(19 110 105 46)(20 111 91 47)(21 112 92 48)(22 113 93 49)(23 114 94 50)(24 115 95 51)(25 116 96 52)(26 117 97 53)(27 118 98 54)(28 119 99 55)(29 120 100 56)(30 106 101 57)
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), (1,18,75,99,6,23,65,104,11,28,70,94)(2,19,61,100,7,24,66,105,12,29,71,95)(3,20,62,101,8,25,67,91,13,30,72,96)(4,21,63,102,9,26,68,92,14,16,73,97)(5,22,64,103,10,27,69,93,15,17,74,98)(31,56,77,110,41,51,87,120,36,46,82,115)(32,57,78,111,42,52,88,106,37,47,83,116)(33,58,79,112,43,53,89,107,38,48,84,117)(34,59,80,113,44,54,90,108,39,49,85,118)(35,60,81,114,45,55,76,109,40,50,86,119), (1,40,65,81)(2,41,66,82)(3,42,67,83)(4,43,68,84)(5,44,69,85)(6,45,70,86)(7,31,71,87)(8,32,72,88)(9,33,73,89)(10,34,74,90)(11,35,75,76)(12,36,61,77)(13,37,62,78)(14,38,63,79)(15,39,64,80)(16,107,102,58)(17,108,103,59)(18,109,104,60)(19,110,105,46)(20,111,91,47)(21,112,92,48)(22,113,93,49)(23,114,94,50)(24,115,95,51)(25,116,96,52)(26,117,97,53)(27,118,98,54)(28,119,99,55)(29,120,100,56)(30,106,101,57)>;
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), (1,18,75,99,6,23,65,104,11,28,70,94)(2,19,61,100,7,24,66,105,12,29,71,95)(3,20,62,101,8,25,67,91,13,30,72,96)(4,21,63,102,9,26,68,92,14,16,73,97)(5,22,64,103,10,27,69,93,15,17,74,98)(31,56,77,110,41,51,87,120,36,46,82,115)(32,57,78,111,42,52,88,106,37,47,83,116)(33,58,79,112,43,53,89,107,38,48,84,117)(34,59,80,113,44,54,90,108,39,49,85,118)(35,60,81,114,45,55,76,109,40,50,86,119), (1,40,65,81)(2,41,66,82)(3,42,67,83)(4,43,68,84)(5,44,69,85)(6,45,70,86)(7,31,71,87)(8,32,72,88)(9,33,73,89)(10,34,74,90)(11,35,75,76)(12,36,61,77)(13,37,62,78)(14,38,63,79)(15,39,64,80)(16,107,102,58)(17,108,103,59)(18,109,104,60)(19,110,105,46)(20,111,91,47)(21,112,92,48)(22,113,93,49)(23,114,94,50)(24,115,95,51)(25,116,96,52)(26,117,97,53)(27,118,98,54)(28,119,99,55)(29,120,100,56)(30,106,101,57) );
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)], [(1,18,75,99,6,23,65,104,11,28,70,94),(2,19,61,100,7,24,66,105,12,29,71,95),(3,20,62,101,8,25,67,91,13,30,72,96),(4,21,63,102,9,26,68,92,14,16,73,97),(5,22,64,103,10,27,69,93,15,17,74,98),(31,56,77,110,41,51,87,120,36,46,82,115),(32,57,78,111,42,52,88,106,37,47,83,116),(33,58,79,112,43,53,89,107,38,48,84,117),(34,59,80,113,44,54,90,108,39,49,85,118),(35,60,81,114,45,55,76,109,40,50,86,119)], [(1,40,65,81),(2,41,66,82),(3,42,67,83),(4,43,68,84),(5,44,69,85),(6,45,70,86),(7,31,71,87),(8,32,72,88),(9,33,73,89),(10,34,74,90),(11,35,75,76),(12,36,61,77),(13,37,62,78),(14,38,63,79),(15,39,64,80),(16,107,102,58),(17,108,103,59),(18,109,104,60),(19,110,105,46),(20,111,91,47),(21,112,92,48),(22,113,93,49),(23,114,94,50),(24,115,95,51),(25,116,96,52),(26,117,97,53),(27,118,98,54),(28,119,99,55),(29,120,100,56),(30,106,101,57)]])
135 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 10C | 10D | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 15A | ··· | 15H | 15I | ··· | 15T | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30H | 30I | ··· | 30T | 60A | ··· | 60AF | 60AG | ··· | 60AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | S3 | Q8 | D6 | C3×S3 | Dic6 | C3×Q8 | C5×S3 | S3×C6 | C5×Q8 | S3×C10 | C3×Dic6 | S3×C15 | C5×Dic6 | Q8×C15 | S3×C30 | C15×Dic6 |
| kernel | C15×Dic6 | Dic3×C15 | C3×C60 | C5×Dic6 | C3×Dic6 | C5×Dic3 | C60 | C3×Dic3 | C3×C12 | Dic6 | Dic3 | C12 | C60 | C3×C15 | C30 | C20 | C15 | C15 | C12 | C10 | C32 | C6 | C5 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 8 | 16 | 8 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of C15×Dic6 ►in GL2(𝔽61) generated by
| 57 | 0 |
| 0 | 57 |
| 40 | 0 |
| 0 | 29 |
| 0 | 1 |
| 60 | 0 |
G:=sub<GL(2,GF(61))| [57,0,0,57],[40,0,0,29],[0,60,1,0] >;
C15×Dic6 in GAP, Magma, Sage, TeX
C_{15}\times {\rm Dic}_6 % in TeX
G:=Group("C15xDic6"); // GroupNames label
G:=SmallGroup(360,95);
// by ID
G=gap.SmallGroup(360,95);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-3,360,745,367,8645]);
// Polycyclic
G:=Group<a,b,c|a^15=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations