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

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

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

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

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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