direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C60, C60⋊6C6, D6.C30, C12⋊2C30, C30.68D6, Dic3⋊2C30, C3⋊1(C2×C60), (C3×C60)⋊9C2, C15⋊9(C2×C12), (C3×C12)⋊3C10, C2.1(S3×C30), C6.2(C2×C30), C32⋊4(C2×C20), (S3×C10).2C6, (S3×C6).2C10, (S3×C30).4C2, C6.18(S3×C10), C10.14(S3×C6), C30.25(C2×C6), (C5×Dic3)⋊5C6, (C3×Dic3)⋊5C10, (Dic3×C15)⋊11C2, (C3×C30).48C22, (C3×C15)⋊29(C2×C4), (C3×C6).7(C2×C10), SmallGroup(360,96)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C60 |
Generators and relations for S3×C60
G = < a,b,c | a60=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 116 in 70 conjugacy classes, 44 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, C20, C2×C10, C4×S3, C2×C12, C5×S3, C30, C30, C3×Dic3, C3×C12, S3×C6, C2×C20, C3×C15, C5×Dic3, C60, C60, S3×C10, C2×C30, S3×C12, S3×C15, C3×C30, S3×C20, C2×C60, Dic3×C15, C3×C60, S3×C30, S3×C60
Quotients: C1, C2, C3, C4, C22, C5, S3, C6, C2×C4, C10, C12, D6, C2×C6, C15, C3×S3, C20, C2×C10, C4×S3, C2×C12, C5×S3, C30, S3×C6, C2×C20, C60, S3×C10, C2×C30, S3×C12, S3×C15, S3×C20, C2×C60, S3×C30, S3×C60
►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 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 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)
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), (1,41,21)(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), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)>;
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), (1,41,21)(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), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75) );
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)], [(1,41,21),(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)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 15A | ··· | 15H | 15I | ··· | 15T | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30H | 30I | ··· | 30T | 30U | ··· | 30AJ | 60A | ··· | 60P | 60Q | ··· | 60AN | 60AO | ··· | 60BD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C30 | C60 | S3 | D6 | C3×S3 | C4×S3 | C5×S3 | S3×C6 | S3×C10 | S3×C12 | S3×C15 | S3×C20 | S3×C30 | S3×C60 |
| kernel | S3×C60 | Dic3×C15 | C3×C60 | S3×C30 | S3×C20 | S3×C15 | S3×C12 | C5×Dic3 | C60 | S3×C10 | C3×Dic3 | C3×C12 | S3×C6 | C5×S3 | C4×S3 | C3×S3 | Dic3 | C12 | D6 | S3 | C60 | C30 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 8 | 8 | 8 | 32 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of S3×C60 ►in GL3(𝔽61) generated by
| 21 | 0 | 0 |
| 0 | 39 | 0 |
| 0 | 0 | 39 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 47 |
| 60 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(61))| [21,0,0,0,39,0,0,0,39],[1,0,0,0,13,0,0,0,47],[60,0,0,0,0,1,0,1,0] >;
S3×C60 in GAP, Magma, Sage, TeX
S_3\times C_{60} % in TeX
G:=Group("S3xC60"); // GroupNames label
G:=SmallGroup(360,96);
// by ID
G=gap.SmallGroup(360,96);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-3,367,8645]);
// Polycyclic
G:=Group<a,b,c|a^60=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations