direct product, metacyclic, supersoluble, monomial
Aliases: C15×D12, C60⋊4C6, C60⋊7S3, C12⋊1C30, D6⋊1C30, C30.69D6, C4⋊(S3×C15), C20⋊3(C3×S3), C12⋊3(C5×S3), (C3×C60)⋊7C2, C3⋊1(D4×C15), C15⋊6(C3×D4), (C3×C15)⋊20D4, (S3×C30)⋊9C2, (S3×C6)⋊3C10, (S3×C10)⋊4C6, C32⋊4(C5×D4), (C3×C12)⋊2C10, C2.4(S3×C30), C6.3(C2×C30), C6.19(S3×C10), C10.15(S3×C6), C30.26(C2×C6), (C3×C30).49C22, (C3×C6).8(C2×C10), SmallGroup(360,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×D12
G = < a,b,c | a15=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 148 in 70 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22 [×2], C5, S3 [×2], C6 [×2], C6 [×3], D4, C32, C10, C10 [×2], C12 [×2], C12, D6 [×2], C2×C6 [×2], C15 [×2], C15, C3×S3 [×2], C3×C6, C20, C2×C10 [×2], D12, C3×D4, C5×S3 [×2], C30 [×2], C30 [×3], C3×C12, S3×C6 [×2], C5×D4, C3×C15, C60 [×2], C60, S3×C10 [×2], C2×C30 [×2], C3×D12, S3×C15 [×2], C3×C30, C5×D12, D4×C15, C3×C60, S3×C30 [×2], C15×D12
Quotients: C1, C2 [×3], C3, C22, C5, S3, C6 [×3], D4, C10 [×3], D6, C2×C6, C15, C3×S3, C2×C10, D12, C3×D4, C5×S3, C30 [×3], S3×C6, C5×D4, S3×C10, C2×C30, C3×D12, S3×C15, C5×D12, D4×C15, S3×C30, C15×D12
►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)
(1 103 118 36 11 98 113 31 6 93 108 41)(2 104 119 37 12 99 114 32 7 94 109 42)(3 105 120 38 13 100 115 33 8 95 110 43)(4 91 106 39 14 101 116 34 9 96 111 44)(5 92 107 40 15 102 117 35 10 97 112 45)(16 67 80 48 21 72 85 53 26 62 90 58)(17 68 81 49 22 73 86 54 27 63 76 59)(18 69 82 50 23 74 87 55 28 64 77 60)(19 70 83 51 24 75 88 56 29 65 78 46)(20 71 84 52 25 61 89 57 30 66 79 47)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 101)(17 102)(18 103)(19 104)(20 105)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 87)(32 88)(33 89)(34 90)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 119)(47 120)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)
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), (1,103,118,36,11,98,113,31,6,93,108,41)(2,104,119,37,12,99,114,32,7,94,109,42)(3,105,120,38,13,100,115,33,8,95,110,43)(4,91,106,39,14,101,116,34,9,96,111,44)(5,92,107,40,15,102,117,35,10,97,112,45)(16,67,80,48,21,72,85,53,26,62,90,58)(17,68,81,49,22,73,86,54,27,63,76,59)(18,69,82,50,23,74,87,55,28,64,77,60)(19,70,83,51,24,75,88,56,29,65,78,46)(20,71,84,52,25,61,89,57,30,66,79,47), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,101)(17,102)(18,103)(19,104)(20,105)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,87)(32,88)(33,89)(34,90)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,119)(47,120)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)>;
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), (1,103,118,36,11,98,113,31,6,93,108,41)(2,104,119,37,12,99,114,32,7,94,109,42)(3,105,120,38,13,100,115,33,8,95,110,43)(4,91,106,39,14,101,116,34,9,96,111,44)(5,92,107,40,15,102,117,35,10,97,112,45)(16,67,80,48,21,72,85,53,26,62,90,58)(17,68,81,49,22,73,86,54,27,63,76,59)(18,69,82,50,23,74,87,55,28,64,77,60)(19,70,83,51,24,75,88,56,29,65,78,46)(20,71,84,52,25,61,89,57,30,66,79,47), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,101)(17,102)(18,103)(19,104)(20,105)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,87)(32,88)(33,89)(34,90)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,119)(47,120)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118) );
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)], [(1,103,118,36,11,98,113,31,6,93,108,41),(2,104,119,37,12,99,114,32,7,94,109,42),(3,105,120,38,13,100,115,33,8,95,110,43),(4,91,106,39,14,101,116,34,9,96,111,44),(5,92,107,40,15,102,117,35,10,97,112,45),(16,67,80,48,21,72,85,53,26,62,90,58),(17,68,81,49,22,73,86,54,27,63,76,59),(18,69,82,50,23,74,87,55,28,64,77,60),(19,70,83,51,24,75,88,56,29,65,78,46),(20,71,84,52,25,61,89,57,30,66,79,47)], [(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,101),(17,102),(18,103),(19,104),(20,105),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,99),(30,100),(31,87),(32,88),(33,89),(34,90),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,119),(47,120),(48,106),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,113),(56,114),(57,115),(58,116),(59,117),(60,118)])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | ··· | 12H | 15A | ··· | 15H | 15I | ··· | 15T | 20A | 20B | 20C | 20D | 30A | ··· | 30H | 30I | ··· | 30T | 30U | ··· | 30AJ | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
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 | D4 | D6 | C3×S3 | D12 | C3×D4 | C5×S3 | S3×C6 | C5×D4 | S3×C10 | C3×D12 | S3×C15 | C5×D12 | D4×C15 | S3×C30 | C15×D12 |
| kernel | C15×D12 | C3×C60 | S3×C30 | C5×D12 | C3×D12 | C60 | S3×C10 | C3×C12 | S3×C6 | D12 | C12 | D6 | C60 | C3×C15 | C30 | C20 | C15 | C15 | C12 | C10 | C32 | C6 | C5 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of C15×D12 ►in GL2(𝔽61) generated by
| 16 | 0 |
| 0 | 16 |
| 21 | 0 |
| 0 | 32 |
| 0 | 32 |
| 21 | 0 |
G:=sub<GL(2,GF(61))| [16,0,0,16],[21,0,0,32],[0,21,32,0] >;
C15×D12 in GAP, Magma, Sage, TeX
C_{15}\times D_{12} % in TeX
G:=Group("C15xD12"); // GroupNames label
G:=SmallGroup(360,97);
// by ID
G=gap.SmallGroup(360,97);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-3,745,367,8645]);
// Polycyclic
G:=Group<a,b,c|a^15=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations