direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C12×D15, C60⋊2C6, C60⋊5S3, D30.2C6, C30.52D6, C6.20D30, Dic15⋊5C6, C5⋊3(S3×C12), C20⋊2(C3×S3), (C3×C60)⋊4C2, C3⋊2(D5×C12), (C3×C12)⋊3D5, C12⋊2(C3×D5), C6.9(C6×D5), C15⋊14(C4×S3), C32⋊7(C4×D5), C15⋊7(C2×C12), C10.9(S3×C6), C30.9(C2×C6), C2.1(C6×D15), (C6×D15).4C2, (C3×C6).28D10, (C3×Dic15)⋊11C2, (C3×C30).38C22, (C3×C15)⋊27(C2×C4), SmallGroup(360,101)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C12×D15 |
Generators and relations for C12×D15
G = < a,b,c | a12=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 276 in 70 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, C20, D10, C4×S3, C2×C12, C3×D5, D15, C30, C30, C3×Dic3, C3×C12, S3×C6, C4×D5, C3×C15, C3×Dic5, Dic15, C60, C60, C6×D5, D30, S3×C12, C3×D15, C3×C30, D5×C12, C4×D15, C3×Dic15, C3×C60, C6×D15, C12×D15
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D5, C12, D6, C2×C6, C3×S3, D10, C4×S3, C2×C12, C3×D5, D15, S3×C6, C4×D5, C6×D5, D30, S3×C12, C3×D15, D5×C12, C4×D15, C6×D15, C12×D15
►On 120 points
Generators in S120
(1 59 26 32 6 49 16 37 11 54 21 42)(2 60 27 33 7 50 17 38 12 55 22 43)(3 46 28 34 8 51 18 39 13 56 23 44)(4 47 29 35 9 52 19 40 14 57 24 45)(5 48 30 36 10 53 20 41 15 58 25 31)(61 114 88 92 71 109 83 102 66 119 78 97)(62 115 89 93 72 110 84 103 67 120 79 98)(63 116 90 94 73 111 85 104 68 106 80 99)(64 117 76 95 74 112 86 105 69 107 81 100)(65 118 77 96 75 113 87 91 70 108 82 101)
(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 81)(2 80)(3 79)(4 78)(5 77)(6 76)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 84)(14 83)(15 82)(16 74)(17 73)(18 72)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 75)(31 118)(32 117)(33 116)(34 115)(35 114)(36 113)(37 112)(38 111)(39 110)(40 109)(41 108)(42 107)(43 106)(44 120)(45 119)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 105)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)
G:=sub<Sym(120)| (1,59,26,32,6,49,16,37,11,54,21,42)(2,60,27,33,7,50,17,38,12,55,22,43)(3,46,28,34,8,51,18,39,13,56,23,44)(4,47,29,35,9,52,19,40,14,57,24,45)(5,48,30,36,10,53,20,41,15,58,25,31)(61,114,88,92,71,109,83,102,66,119,78,97)(62,115,89,93,72,110,84,103,67,120,79,98)(63,116,90,94,73,111,85,104,68,106,80,99)(64,117,76,95,74,112,86,105,69,107,81,100)(65,118,77,96,75,113,87,91,70,108,82,101), (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), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,75)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,120)(45,119)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)>;
G:=Group( (1,59,26,32,6,49,16,37,11,54,21,42)(2,60,27,33,7,50,17,38,12,55,22,43)(3,46,28,34,8,51,18,39,13,56,23,44)(4,47,29,35,9,52,19,40,14,57,24,45)(5,48,30,36,10,53,20,41,15,58,25,31)(61,114,88,92,71,109,83,102,66,119,78,97)(62,115,89,93,72,110,84,103,67,120,79,98)(63,116,90,94,73,111,85,104,68,106,80,99)(64,117,76,95,74,112,86,105,69,107,81,100)(65,118,77,96,75,113,87,91,70,108,82,101), (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), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,75)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,120)(45,119)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99) );
G=PermutationGroup([[(1,59,26,32,6,49,16,37,11,54,21,42),(2,60,27,33,7,50,17,38,12,55,22,43),(3,46,28,34,8,51,18,39,13,56,23,44),(4,47,29,35,9,52,19,40,14,57,24,45),(5,48,30,36,10,53,20,41,15,58,25,31),(61,114,88,92,71,109,83,102,66,119,78,97),(62,115,89,93,72,110,84,103,67,120,79,98),(63,116,90,94,73,111,85,104,68,106,80,99),(64,117,76,95,74,112,86,105,69,107,81,100),(65,118,77,96,75,113,87,91,70,108,82,101)], [(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)], [(1,81),(2,80),(3,79),(4,78),(5,77),(6,76),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,84),(14,83),(15,82),(16,74),(17,73),(18,72),(19,71),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,75),(31,118),(32,117),(33,116),(34,115),(35,114),(36,113),(37,112),(38,111),(39,110),(40,109),(41,108),(42,107),(43,106),(44,120),(45,119),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,105),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 10A | 10B | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 15A | ··· | 15P | 20A | 20B | 20C | 20D | 30A | ··· | 30P | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 15 | 15 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 15 | 15 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 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 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D5 | D6 | C3×S3 | D10 | C4×S3 | C3×D5 | D15 | S3×C6 | C4×D5 | C6×D5 | D30 | S3×C12 | C3×D15 | D5×C12 | C4×D15 | C6×D15 | C12×D15 |
| kernel | C12×D15 | C3×Dic15 | C3×C60 | C6×D15 | C4×D15 | C3×D15 | Dic15 | C60 | D30 | D15 | C60 | C3×C12 | C30 | C20 | C3×C6 | C15 | C12 | C12 | C10 | C32 | C6 | C6 | C5 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of C12×D15 ►in GL4(𝔽61) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 47 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 6 | 34 |
| 0 | 48 | 0 | 0 |
| 14 | 0 | 0 | 0 |
| 0 | 0 | 5 | 31 |
| 0 | 0 | 13 | 56 |
G:=sub<GL(4,GF(61))| [40,0,0,0,0,40,0,0,0,0,11,0,0,0,0,11],[47,0,0,0,0,13,0,0,0,0,9,6,0,0,0,34],[0,14,0,0,48,0,0,0,0,0,5,13,0,0,31,56] >;
C12×D15 in GAP, Magma, Sage, TeX
C_{12}\times D_{15} % in TeX
G:=Group("C12xD15"); // GroupNames label
G:=SmallGroup(360,101);
// by ID
G=gap.SmallGroup(360,101);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,79,1444,10373]);
// Polycyclic
G:=Group<a,b,c|a^12=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations