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## G = S3×C60order 360 = 23·32·5

### Direct product of C60 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C60
 Chief series C1 — C3 — C6 — C30 — C3×C30 — S3×C30 — S3×C60
 Lower central C3 — S3×C60
 Upper central C1 — 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

Smallest permutation representation of 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

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