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## G = C3×C15⋊3C8order 360 = 23·32·5

### Direct product of C3 and C15⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C3×C15⋊3C8
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C60 — C3×C15⋊3C8
 Lower central C15 — C3×C15⋊3C8
 Upper central C1 — C12

Generators and relations for C3×C153C8
G = < a,b,c | a3=b15=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×C153C8
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)
(1 110 47 84 19 92 45 73)(2 109 48 83 20 91 31 72)(3 108 49 82 21 105 32 71)(4 107 50 81 22 104 33 70)(5 106 51 80 23 103 34 69)(6 120 52 79 24 102 35 68)(7 119 53 78 25 101 36 67)(8 118 54 77 26 100 37 66)(9 117 55 76 27 99 38 65)(10 116 56 90 28 98 39 64)(11 115 57 89 29 97 40 63)(12 114 58 88 30 96 41 62)(13 113 59 87 16 95 42 61)(14 112 60 86 17 94 43 75)(15 111 46 85 18 93 44 74)

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

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

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

108 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 5A 5B 6A 6B 6C 6D 6E 8A 8B 8C 8D 10A 10B 12A 12B 12C 12D 12E ··· 12J 15A ··· 15P 20A 20B 20C 20D 24A ··· 24H 30A ··· 30P 60A ··· 60AF order 1 2 3 3 3 3 3 4 4 5 5 6 6 6 6 6 8 8 8 8 10 10 12 12 12 12 12 ··· 12 15 ··· 15 20 20 20 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 1 1 2 2 1 1 2 2 2 15 15 15 15 2 2 1 1 1 1 2 ··· 2 2 ··· 2 2 2 2 2 15 ··· 15 2 ··· 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - - + - image C1 C2 C3 C4 C6 C8 C12 C24 S3 D5 Dic3 C3×S3 Dic5 C3⋊C8 C3×D5 D15 C3×Dic3 C5⋊2C8 C3×Dic5 Dic15 C3×C3⋊C8 C3×D15 C3×C5⋊2C8 C15⋊3C8 C3×Dic15 C3×C15⋊3C8 kernel C3×C15⋊3C8 C3×C60 C15⋊3C8 C3×C30 C60 C3×C15 C30 C15 C60 C3×C12 C30 C20 C3×C6 C15 C12 C12 C10 C32 C6 C6 C5 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 1 2 1 2 2 2 4 4 2 4 4 4 4 8 8 8 8 16

Matrix representation of C3×C153C8 in GL5(𝔽241)

 15 0 0 0 0 0 15 0 0 0 0 0 15 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 15 0 0 0 0 0 225 0 0 0 0 0 178 190 0 0 0 179 11
,
 233 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 224 91 0 0 0 18 17

G:=sub<GL(5,GF(241))| [15,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,15,0,0,0,0,0,225,0,0,0,0,0,178,179,0,0,0,190,11],[233,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,224,18,0,0,0,91,17] >;

C3×C153C8 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes_3C_8
% in TeX

G:=Group("C3xC15:3C8");
// GroupNames label

G:=SmallGroup(360,35);
// by ID

G=gap.SmallGroup(360,35);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-5,36,50,1444,10373]);
// 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^-1>;
// generators/relations

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