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G = C15×C3⋊C8order 360 = 23·32·5

Direct product of C15 and C3⋊C8

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

Aliases: C15×C3⋊C8, C3⋊C120, C6.C60, C155C24, C323C40, C60.10C6, C12.2C30, C30.9C12, C60.16S3, C30.12Dic3, (C3×C15)⋊13C8, C20.4(C3×S3), C12.8(C5×S3), C4.2(S3×C15), (C3×C60).9C2, (C3×C6).2C20, C2.(Dic3×C15), (C3×C12).3C10, (C3×C30).10C4, C6.4(C5×Dic3), C10.3(C3×Dic3), SmallGroup(360,34)

Series: Derived Chief Lower central Upper central

C1C3 — C15×C3⋊C8
C1C3C6C12C60C3×C60 — C15×C3⋊C8
C3 — C15×C3⋊C8
C1C60

Generators and relations for C15×C3⋊C8
 G = < a,b,c | a15=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
2C15
3C8
2C12
2C30
3C24
3C40
2C60
3C120

Smallest permutation representation of C15×C3⋊C8
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 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
(1 67 101 59 83 16 119 37)(2 68 102 60 84 17 120 38)(3 69 103 46 85 18 106 39)(4 70 104 47 86 19 107 40)(5 71 105 48 87 20 108 41)(6 72 91 49 88 21 109 42)(7 73 92 50 89 22 110 43)(8 74 93 51 90 23 111 44)(9 75 94 52 76 24 112 45)(10 61 95 53 77 25 113 31)(11 62 96 54 78 26 114 32)(12 63 97 55 79 27 115 33)(13 64 98 56 80 28 116 34)(14 65 99 57 81 29 117 35)(15 66 100 58 82 30 118 36)

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

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

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

180 conjugacy classes

class 1  2 3A3B3C3D3E4A4B5A5B5C5D6A6B6C6D6E8A8B8C8D10A10B10C10D12A12B12C12D12E···12J15A···15H15I···15T20A···20H24A···24H30A···30H30I···30T40A···40P60A···60P60Q···60AN120A···120AF
order1233333445555666668888101010101212121212···1215···1515···1520···2024···2430···3030···3040···4060···6060···60120···120
size1111222111111112223333111111112···21···12···21···13···31···12···23···31···12···23···3

180 irreducible representations

dim1111111111111111222222222222
type+++-
imageC1C2C3C4C5C6C8C10C12C15C20C24C30C40C60C120S3Dic3C3×S3C3⋊C8C5×S3C3×Dic3C5×Dic3C3×C3⋊C8S3×C15C5×C3⋊C8Dic3×C15C15×C3⋊C8
kernelC15×C3⋊C8C3×C60C5×C3⋊C8C3×C30C3×C3⋊C8C60C3×C15C3×C12C30C3⋊C8C3×C6C15C12C32C6C3C60C30C20C15C12C10C6C5C4C3C2C1
# reps11224244488881616321122424488816

Matrix representation of C15×C3⋊C8 in GL4(𝔽241) generated by

225000
018300
002250
000225
,
1000
0100
002250
00015
,
8000
06400
0001
0010
G:=sub<GL(4,GF(241))| [225,0,0,0,0,183,0,0,0,0,225,0,0,0,0,225],[1,0,0,0,0,1,0,0,0,0,225,0,0,0,0,15],[8,0,0,0,0,64,0,0,0,0,0,1,0,0,1,0] >;

C15×C3⋊C8 in GAP, Magma, Sage, TeX

C_{15}\times C_3\rtimes C_8
% in TeX

G:=Group("C15xC3:C8");
// GroupNames label

G:=SmallGroup(360,34);
// by ID

G=gap.SmallGroup(360,34);
# by ID

G:=PCGroup([6,-2,-3,-5,-2,-2,-3,180,69,8645]);
// Polycyclic

G:=Group<a,b,c|a^15=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C15×C3⋊C8 in TeX

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