direct product, metacyclic, supersoluble, monomial
Aliases: C3×D60, C60⋊1C6, C60⋊3S3, D30⋊1C6, C15⋊5D12, C12⋊3D15, C32⋊4D20, C6.21D30, C30.53D6, C4⋊(C3×D15), C5⋊1(C3×D12), C20⋊1(C3×S3), (C3×C60)⋊2C2, C15⋊4(C3×D4), C12⋊1(C3×D5), C3⋊1(C3×D20), (C3×C12)⋊2D5, (C3×C15)⋊18D4, (C6×D15)⋊1C2, C2.4(C6×D15), C6.10(C6×D5), C10.10(S3×C6), C30.10(C2×C6), (C3×C6).29D10, (C3×C30).39C22, SmallGroup(360,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D60
G = < a,b,c | a3=b60=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 372 in 70 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22 [×2], C5, S3 [×2], C6 [×2], C6 [×3], D4, C32, D5 [×2], C10, C12 [×2], C12, D6 [×2], C2×C6 [×2], C15 [×2], C15, C3×S3 [×2], C3×C6, C20, D10 [×2], D12, C3×D4, C3×D5 [×2], D15 [×2], C30 [×2], C30, C3×C12, S3×C6 [×2], D20, C3×C15, C60 [×2], C60, C6×D5 [×2], D30 [×2], C3×D12, C3×D15 [×2], C3×C30, C3×D20, D60, C3×C60, C6×D15 [×2], C3×D60
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D5, D6, C2×C6, C3×S3, D10, D12, C3×D4, C3×D5, D15, S3×C6, D20, C6×D5, D30, C3×D12, C3×D15, C3×D20, D60, C6×D15, C3×D60
►On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 81 101)(62 82 102)(63 83 103)(64 84 104)(65 85 105)(66 86 106)(67 87 107)(68 88 108)(69 89 109)(70 90 110)(71 91 111)(72 92 112)(73 93 113)(74 94 114)(75 95 115)(76 96 116)(77 97 117)(78 98 118)(79 99 119)(80 100 120)
(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 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)
G:=sub<Sym(120)| (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,81,101)(62,82,102)(63,83,103)(64,84,104)(65,85,105)(66,86,106)(67,87,107)(68,88,108)(69,89,109)(70,90,110)(71,91,111)(72,92,112)(73,93,113)(74,94,114)(75,95,115)(76,96,116)(77,97,117)(78,98,118)(79,99,119)(80,100,120), (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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)>;
G:=Group( (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,81,101)(62,82,102)(63,83,103)(64,84,104)(65,85,105)(66,86,106)(67,87,107)(68,88,108)(69,89,109)(70,90,110)(71,91,111)(72,92,112)(73,93,113)(74,94,114)(75,95,115)(76,96,116)(77,97,117)(78,98,118)(79,99,119)(80,100,120), (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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85) );
G=PermutationGroup([(1,41,21),(2,42,22),(3,43,23),(4,44,24),(5,45,25),(6,46,26),(7,47,27),(8,48,28),(9,49,29),(10,50,30),(11,51,31),(12,52,32),(13,53,33),(14,54,34),(15,55,35),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(61,81,101),(62,82,102),(63,83,103),(64,84,104),(65,85,105),(66,86,106),(67,87,107),(68,88,108),(69,89,109),(70,90,110),(71,91,111),(72,92,112),(73,93,113),(74,94,114),(75,95,115),(76,96,116),(77,97,117),(78,98,118),(79,99,119),(80,100,120)], [(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)], [(1,84),(2,83),(3,82),(4,81),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,120),(26,119),(27,118),(28,117),(29,116),(30,115),(31,114),(32,113),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85)])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 10A | 10B | 12A | ··· | 12H | 15A | ··· | 15P | 20A | 20B | 20C | 20D | 30A | ··· | 30P | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 30 | 30 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D5 | D6 | C3×S3 | D10 | D12 | C3×D4 | C3×D5 | D15 | S3×C6 | D20 | C6×D5 | D30 | C3×D12 | C3×D15 | C3×D20 | D60 | C6×D15 | C3×D60 |
| kernel | C3×D60 | C3×C60 | C6×D15 | D60 | C60 | D30 | C60 | C3×C15 | C3×C12 | C30 | C20 | C3×C6 | C15 | C15 | C12 | C12 | C10 | C32 | C6 | C6 | C5 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of C3×D60 ►in GL2(𝔽61) generated by
| 47 | 0 |
| 0 | 47 |
| 44 | 0 |
| 0 | 43 |
| 0 | 43 |
| 44 | 0 |
G:=sub<GL(2,GF(61))| [47,0,0,47],[44,0,0,43],[0,44,43,0] >;
C3×D60 in GAP, Magma, Sage, TeX
C_3\times D_{60} % in TeX
G:=Group("C3xD60"); // GroupNames label
G:=SmallGroup(360,102);
// by ID
G=gap.SmallGroup(360,102);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,79,1444,10373]);
// Polycyclic
G:=Group<a,b,c|a^3=b^60=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations