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G = C15×Dic6order 360 = 23·32·5

Direct product of C15 and Dic6

direct product, metacyclic, supersoluble, monomial

Aliases: C15×Dic6, C60.6C6, C12.1C30, C60.15S3, C30.67D6, Dic3.C30, C3⋊(Q8×C15), C4.(S3×C15), C154(C3×Q8), (C3×C15)⋊11Q8, C323(C5×Q8), C20.3(C3×S3), C2.3(S3×C30), C12.7(C5×S3), (C3×C60).7C2, C6.1(C2×C30), C10.13(S3×C6), C6.17(S3×C10), (C3×C12).2C10, C30.24(C2×C6), (C5×Dic3).2C6, (C3×C30).47C22, (Dic3×C15).4C2, (C3×Dic3).2C10, (C3×C6).6(C2×C10), SmallGroup(360,95)

Series: Derived Chief Lower central Upper central

C1C6 — C15×Dic6
C1C3C6C30C3×C30Dic3×C15 — C15×Dic6
C3C6 — C15×Dic6
C1C30C60

Generators and relations for C15×Dic6
 G = < a,b,c | a15=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 84 in 54 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C3 [×2], C3, C4, C4 [×2], C5, C6 [×2], C6, Q8, C32, C10, Dic3 [×2], C12 [×2], C12 [×3], C15 [×2], C15, C3×C6, C20, C20 [×2], Dic6, C3×Q8, C30 [×2], C30, C3×Dic3 [×2], C3×C12, C5×Q8, C3×C15, C5×Dic3 [×2], C60 [×2], C60 [×3], C3×Dic6, C3×C30, C5×Dic6, Q8×C15, Dic3×C15 [×2], C3×C60, C15×Dic6
Quotients: C1, C2 [×3], C3, C22, C5, S3, C6 [×3], Q8, C10 [×3], D6, C2×C6, C15, C3×S3, C2×C10, Dic6, C3×Q8, C5×S3, C30 [×3], S3×C6, C5×Q8, S3×C10, C2×C30, C3×Dic6, S3×C15, C5×Dic6, Q8×C15, S3×C30, C15×Dic6

Smallest permutation representation of C15×Dic6
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 78 109 62 6 83 114 67 11 88 119 72)(2 79 110 63 7 84 115 68 12 89 120 73)(3 80 111 64 8 85 116 69 13 90 106 74)(4 81 112 65 9 86 117 70 14 76 107 75)(5 82 113 66 10 87 118 71 15 77 108 61)(16 103 35 52 26 98 45 47 21 93 40 57)(17 104 36 53 27 99 31 48 22 94 41 58)(18 105 37 54 28 100 32 49 23 95 42 59)(19 91 38 55 29 101 33 50 24 96 43 60)(20 92 39 56 30 102 34 51 25 97 44 46)
(1 21 114 35)(2 22 115 36)(3 23 116 37)(4 24 117 38)(5 25 118 39)(6 26 119 40)(7 27 120 41)(8 28 106 42)(9 29 107 43)(10 30 108 44)(11 16 109 45)(12 17 110 31)(13 18 111 32)(14 19 112 33)(15 20 113 34)(46 66 102 77)(47 67 103 78)(48 68 104 79)(49 69 105 80)(50 70 91 81)(51 71 92 82)(52 72 93 83)(53 73 94 84)(54 74 95 85)(55 75 96 86)(56 61 97 87)(57 62 98 88)(58 63 99 89)(59 64 100 90)(60 65 101 76)

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

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

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

135 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C5A5B5C5D6A6B6C6D6E10A10B10C10D12A···12H12I12J12K12L15A···15H15I···15T20A20B20C20D20E···20L30A···30H30I···30T60A···60AF60AG···60AV
order12333334445555666661010101012···121212121215···1515···152020202020···2030···3030···3060···6060···60
size111122226611111122211112···266661···12···222226···61···12···22···26···6

135 irreducible representations

dim1111111111112222222222222222
type++++-+-
imageC1C2C2C3C5C6C6C10C10C15C30C30S3Q8D6C3×S3Dic6C3×Q8C5×S3S3×C6C5×Q8S3×C10C3×Dic6S3×C15C5×Dic6Q8×C15S3×C30C15×Dic6
kernelC15×Dic6Dic3×C15C3×C60C5×Dic6C3×Dic6C5×Dic3C60C3×Dic3C3×C12Dic6Dic3C12C60C3×C15C30C20C15C15C12C10C32C6C5C4C3C3C2C1
# reps121244284816811122242444888816

Matrix representation of C15×Dic6 in GL2(𝔽61) generated by

570
057
,
400
029
,
01
600
G:=sub<GL(2,GF(61))| [57,0,0,57],[40,0,0,29],[0,60,1,0] >;

C15×Dic6 in GAP, Magma, Sage, TeX

C_{15}\times {\rm Dic}_6
% in TeX

G:=Group("C15xDic6");
// GroupNames label

G:=SmallGroup(360,95);
// by ID

G=gap.SmallGroup(360,95);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-3,360,745,367,8645]);
// Polycyclic

G:=Group<a,b,c|a^15=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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