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## G = C3×C15⋊C8order 360 = 23·32·5

### Direct product of C3 and C15⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C3×C15⋊C8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C32×Dic5 — C3×C15⋊C8
 Lower central C15 — C3×C15⋊C8
 Upper central C1 — C6

Generators and relations for C3×C15⋊C8
G = < a,b,c | a3=b15=c8=1, ab=ba, ac=ca, cbc-1=b2 >

Smallest permutation representation of C3×C15⋊C8
On 120 points
Generators in S120
(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 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(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)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(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)
(1 120 54 76 21 105 41 71)(2 113 58 78 22 98 45 73)(3 106 47 80 23 91 34 75)(4 114 51 82 24 99 38 62)(5 107 55 84 25 92 42 64)(6 115 59 86 26 100 31 66)(7 108 48 88 27 93 35 68)(8 116 52 90 28 101 39 70)(9 109 56 77 29 94 43 72)(10 117 60 79 30 102 32 74)(11 110 49 81 16 95 36 61)(12 118 53 83 17 103 40 63)(13 111 57 85 18 96 44 65)(14 119 46 87 19 104 33 67)(15 112 50 89 20 97 37 69)

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

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

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

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 5 6A 6B 6C 6D 6E 8A 8B 8C 8D 10 12A 12B 12C 12D 12E ··· 12J 15A ··· 15H 24A ··· 24H 30A ··· 30H order 1 2 3 3 3 3 3 4 4 5 6 6 6 6 6 8 8 8 8 10 12 12 12 12 12 ··· 12 15 ··· 15 24 ··· 24 30 ··· 30 size 1 1 1 1 2 2 2 5 5 4 1 1 2 2 2 15 15 15 15 4 5 5 5 5 10 ··· 10 4 ··· 4 15 ··· 15 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + - + - image C1 C2 C3 C4 C6 C8 C12 C24 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3×C3⋊C8 F5 C5⋊C8 C3×F5 C3⋊F5 C3×C5⋊C8 C15⋊C8 C3×C3⋊F5 C3×C15⋊C8 kernel C3×C15⋊C8 C32×Dic5 C15⋊C8 C3×C30 C3×Dic5 C3×C15 C30 C15 C3×Dic5 C30 Dic5 C15 C10 C5 C3×C6 C32 C6 C6 C3 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 1 1 2 2 2 4 1 1 2 2 2 2 4 4

Matrix representation of C3×C15⋊C8 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 225 0 0 0 0 0 0 15 0 0 0 0 0 0 0 93 0 0 0 0 132 109 0 0 0 0 0 0 57 226 0 0 0 0 72 226
,
 0 30 0 0 0 0 211 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 187 234 0 0 0 0 141 54 0 0

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[225,0,0,0,0,0,0,15,0,0,0,0,0,0,0,132,0,0,0,0,93,109,0,0,0,0,0,0,57,72,0,0,0,0,226,226],[0,211,0,0,0,0,30,0,0,0,0,0,0,0,0,0,187,141,0,0,0,0,234,54,0,0,1,0,0,0,0,0,0,1,0,0] >;

C3×C15⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC15:C8");
// GroupNames label

G:=SmallGroup(360,53);
// by ID

G=gap.SmallGroup(360,53);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-5,36,50,1444,7781,1745]);
// Polycyclic

G:=Group<a,b,c|a^3=b^15=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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