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G = C15×D12order 360 = 23·32·5

Direct product of C15 and D12

direct product, metacyclic, supersoluble, monomial

Aliases: C15×D12, C604C6, C607S3, C121C30, D61C30, C30.69D6, C4⋊(S3×C15), C203(C3×S3), C123(C5×S3), (C3×C60)⋊7C2, C31(D4×C15), C156(C3×D4), (C3×C15)⋊20D4, (S3×C30)⋊9C2, (S3×C6)⋊3C10, (S3×C10)⋊4C6, C324(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

C1C6 — C15×D12
C1C3C6C30C3×C30S3×C30 — C15×D12
C3C6 — C15×D12
C1C30C60

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 [×2], C3 [×2], C3, C4, C22 [×2], C5, S3 [×2], C6 [×2], C6 [×3], D4, C32, C10, C10 [×2], C12 [×2], C12, D6 [×2], C2×C6 [×2], C15 [×2], C15, C3×S3 [×2], C3×C6, C20, C2×C10 [×2], D12, C3×D4, C5×S3 [×2], C30 [×2], C30 [×3], C3×C12, S3×C6 [×2], C5×D4, C3×C15, C60 [×2], C60, S3×C10 [×2], C2×C30 [×2], C3×D12, S3×C15 [×2], C3×C30, C5×D12, D4×C15, C3×C60, S3×C30 [×2], C15×D12
Quotients: C1, C2 [×3], C3, C22, C5, S3, C6 [×3], D4, C10 [×3], D6, C2×C6, C15, C3×S3, C2×C10, D12, C3×D4, C5×S3, C30 [×3], S3×C6, C5×D4, S3×C10, C2×C30, C3×D12, S3×C15, C5×D12, D4×C15, S3×C30, C15×D12

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

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

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

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

135 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 5A5B5C5D6A6B6C6D6E6F6G6H6I10A10B10C10D10E···10L12A···12H15A···15H15I···15T20A20B20C20D30A···30H30I···30T30U···30AJ60A···60AF
order122233333455556666666661010101010···1012···1215···1515···152020202030···3030···3030···3060···60
size1166112222111111222666611116···62···21···12···222221···12···26···62···2

135 irreducible representations

dim1111111111112222222222222222
type+++++++
imageC1C2C2C3C5C6C6C10C10C15C30C30S3D4D6C3×S3D12C3×D4C5×S3S3×C6C5×D4S3×C10C3×D12S3×C15C5×D12D4×C15S3×C30C15×D12
kernelC15×D12C3×C60S3×C30C5×D12C3×D12C60S3×C10C3×C12S3×C6D12C12D6C60C3×C15C30C20C15C15C12C10C32C6C5C4C3C3C2C1
# reps112242448881611122242444888816

Matrix representation of C15×D12 in GL2(𝔽61) generated by

160
016
,
210
032
,
032
210
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

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