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, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, C10, C10, C12, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, C2×C10, D12, C3×D4, C5×S3, C30, C30, C3×C12, S3×C6, C5×D4, C3×C15, C60, C60, S3×C10, C2×C30, C3×D12, S3×C15, C3×C30, C5×D12, D4×C15, C3×C60, S3×C30, C15×D12
Quotients: C1, C2, C3, C22, C5, S3, C6, D4, C10, D6, C2×C6, C15, C3×S3, C2×C10, D12, C3×D4, C5×S3, C30, 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 35 61 83 11 45 71 78 6 40 66 88)(2 36 62 84 12 31 72 79 7 41 67 89)(3 37 63 85 13 32 73 80 8 42 68 90)(4 38 64 86 14 33 74 81 9 43 69 76)(5 39 65 87 15 34 75 82 10 44 70 77)(16 93 54 116 21 98 59 106 26 103 49 111)(17 94 55 117 22 99 60 107 27 104 50 112)(18 95 56 118 23 100 46 108 28 105 51 113)(19 96 57 119 24 101 47 109 29 91 52 114)(20 97 58 120 25 102 48 110 30 92 53 115)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 74)(17 75)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
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), (1,35,61,83,11,45,71,78,6,40,66,88)(2,36,62,84,12,31,72,79,7,41,67,89)(3,37,63,85,13,32,73,80,8,42,68,90)(4,38,64,86,14,33,74,81,9,43,69,76)(5,39,65,87,15,34,75,82,10,44,70,77)(16,93,54,116,21,98,59,106,26,103,49,111)(17,94,55,117,22,99,60,107,27,104,50,112)(18,95,56,118,23,100,46,108,28,105,51,113)(19,96,57,119,24,101,47,109,29,91,52,114)(20,97,58,120,25,102,48,110,30,92,53,115), (1,56)(2,57)(3,58)(4,59)(5,60)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,74)(17,75)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)>;
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), (1,35,61,83,11,45,71,78,6,40,66,88)(2,36,62,84,12,31,72,79,7,41,67,89)(3,37,63,85,13,32,73,80,8,42,68,90)(4,38,64,86,14,33,74,81,9,43,69,76)(5,39,65,87,15,34,75,82,10,44,70,77)(16,93,54,116,21,98,59,106,26,103,49,111)(17,94,55,117,22,99,60,107,27,104,50,112)(18,95,56,118,23,100,46,108,28,105,51,113)(19,96,57,119,24,101,47,109,29,91,52,114)(20,97,58,120,25,102,48,110,30,92,53,115), (1,56)(2,57)(3,58)(4,59)(5,60)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,74)(17,75)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120) );
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)], [(1,35,61,83,11,45,71,78,6,40,66,88),(2,36,62,84,12,31,72,79,7,41,67,89),(3,37,63,85,13,32,73,80,8,42,68,90),(4,38,64,86,14,33,74,81,9,43,69,76),(5,39,65,87,15,34,75,82,10,44,70,77),(16,93,54,116,21,98,59,106,26,103,49,111),(17,94,55,117,22,99,60,107,27,104,50,112),(18,95,56,118,23,100,46,108,28,105,51,113),(19,96,57,119,24,101,47,109,29,91,52,114),(20,97,58,120,25,102,48,110,30,92,53,115)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,74),(17,75),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120)]])
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