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## G = C15×C3⋊C8order 360 = 23·32·5

### Direct product of C15 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C15×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C60 — C3×C60 — C15×C3⋊C8
 Lower central C3 — C15×C3⋊C8
 Upper central C1 — C60

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

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 3A 3B 3C 3D 3E 4A 4B 5A 5B 5C 5D 6A 6B 6C 6D 6E 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 12C 12D 12E ··· 12J 15A ··· 15H 15I ··· 15T 20A ··· 20H 24A ··· 24H 30A ··· 30H 30I ··· 30T 40A ··· 40P 60A ··· 60P 60Q ··· 60AN 120A ··· 120AF order 1 2 3 3 3 3 3 4 4 5 5 5 5 6 6 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 12 ··· 12 15 ··· 15 15 ··· 15 20 ··· 20 24 ··· 24 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C5 C6 C8 C10 C12 C15 C20 C24 C30 C40 C60 C120 S3 Dic3 C3×S3 C3⋊C8 C5×S3 C3×Dic3 C5×Dic3 C3×C3⋊C8 S3×C15 C5×C3⋊C8 Dic3×C15 C15×C3⋊C8 kernel C15×C3⋊C8 C3×C60 C5×C3⋊C8 C3×C30 C3×C3⋊C8 C60 C3×C15 C3×C12 C30 C3⋊C8 C3×C6 C15 C12 C32 C6 C3 C60 C30 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 4 8 8 8 8 16 16 32 1 1 2 2 4 2 4 4 8 8 8 16

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

 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
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

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