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G = S3×C60order 360 = 23·32·5

Direct product of C60 and S3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: S3×C60, C606C6, D6.C30, C122C30, C30.68D6, Dic32C30, C31(C2×C60), (C3×C60)⋊9C2, C159(C2×C12), (C3×C12)⋊3C10, C2.1(S3×C30), C6.2(C2×C30), C324(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

C1C3 — S3×C60
C1C3C6C30C3×C30S3×C30 — S3×C60
C3 — S3×C60
C1C60

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 [×2], C3 [×2], C3, C4, C4, C22, C5, S3 [×2], C6 [×2], C6 [×3], C2×C4, C32, C10, C10 [×2], Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15 [×2], C15, C3×S3 [×2], C3×C6, C20, C20, C2×C10, C4×S3, C2×C12, C5×S3 [×2], C30 [×2], C30 [×3], C3×Dic3, C3×C12, S3×C6, C2×C20, C3×C15, C5×Dic3, C60 [×2], C60 [×2], S3×C10, C2×C30, S3×C12, S3×C15 [×2], C3×C30, S3×C20, C2×C60, Dic3×C15, C3×C60, S3×C30, S3×C60
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, S3, C6 [×3], C2×C4, C10 [×3], C12 [×2], D6, C2×C6, C15, C3×S3, C20 [×2], C2×C10, C4×S3, C2×C12, C5×S3, C30 [×3], S3×C6, C2×C20, C60 [×2], 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 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)

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

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

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

180 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D5A5B5C5D6A6B6C6D6E6F6G6H6I10A10B10C10D10E···10L12A12B12C12D12E···12J12K12L12M12N15A···15H15I···15T20A···20H20I···20P30A···30H30I···30T30U···30AJ60A···60P60Q···60AN60AO···60BD
order122233333444455556666666661010101010···101212121212···121212121215···1515···1520···2020···2030···3030···3030···3060···6060···6060···60
size1133112221133111111222333311113···311112···233331···12···21···13···31···12···23···31···12···23···3

180 irreducible representations

dim11111111111111111111222222222222
type++++++
imageC1C2C2C2C3C4C5C6C6C6C10C10C10C12C15C20C30C30C30C60S3D6C3×S3C4×S3C5×S3S3×C6S3×C10S3×C12S3×C15S3×C20S3×C30S3×C60
kernelS3×C60Dic3×C15C3×C60S3×C30S3×C20S3×C15S3×C12C5×Dic3C60S3×C10C3×Dic3C3×C12S3×C6C5×S3C4×S3C3×S3Dic3C12D6S3C60C30C20C15C12C10C6C5C4C3C2C1
# reps11112442224448816888321122424488816

Matrix representation of S3×C60 in GL3(𝔽61) generated by

2100
0390
0039
,
100
0130
0047
,
6000
001
010
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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