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## G = Dic3×C30order 360 = 23·32·5

### Direct product of C30 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C30
 Chief series C1 — C3 — C6 — C30 — C3×C30 — Dic3×C15 — Dic3×C30
 Lower central C3 — Dic3×C30
 Upper central C1 — C2×C30

Generators and relations for Dic3×C30
G = < a,b,c | a30=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 108 in 74 conjugacy classes, 52 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, C6, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C15, C3×C6, C3×C6, C20, C2×C10, C2×Dic3, C2×C12, C30, C30, C30, C3×Dic3, C62, C2×C20, C3×C15, C5×Dic3, C60, C2×C30, C2×C30, C6×Dic3, C3×C30, C3×C30, C10×Dic3, C2×C60, Dic3×C15, C6×C30, Dic3×C30
Quotients: C1, C2, C3, C4, C22, C5, S3, C6, C2×C4, C10, Dic3, C12, D6, C2×C6, C15, C3×S3, C20, C2×C10, C2×Dic3, C2×C12, C5×S3, C30, C3×Dic3, S3×C6, C2×C20, C5×Dic3, C60, S3×C10, C2×C30, C6×Dic3, S3×C15, C10×Dic3, C2×C60, Dic3×C15, S3×C30, Dic3×C30

Smallest permutation representation of Dic3×C30
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 78 21 68 11 88)(2 79 22 69 12 89)(3 80 23 70 13 90)(4 81 24 71 14 61)(5 82 25 72 15 62)(6 83 26 73 16 63)(7 84 27 74 17 64)(8 85 28 75 18 65)(9 86 29 76 19 66)(10 87 30 77 20 67)(31 114 41 94 51 104)(32 115 42 95 52 105)(33 116 43 96 53 106)(34 117 44 97 54 107)(35 118 45 98 55 108)(36 119 46 99 56 109)(37 120 47 100 57 110)(38 91 48 101 58 111)(39 92 49 102 59 112)(40 93 50 103 60 113)
(1 112 68 49)(2 113 69 50)(3 114 70 51)(4 115 71 52)(5 116 72 53)(6 117 73 54)(7 118 74 55)(8 119 75 56)(9 120 76 57)(10 91 77 58)(11 92 78 59)(12 93 79 60)(13 94 80 31)(14 95 81 32)(15 96 82 33)(16 97 83 34)(17 98 84 35)(18 99 85 36)(19 100 86 37)(20 101 87 38)(21 102 88 39)(22 103 89 40)(23 104 90 41)(24 105 61 42)(25 106 62 43)(26 107 63 44)(27 108 64 45)(28 109 65 46)(29 110 66 47)(30 111 67 48)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6F 6G ··· 6O 10A ··· 10L 12A ··· 12H 15A ··· 15H 15I ··· 15T 20A ··· 20P 30A ··· 30X 30Y ··· 30BH 60A ··· 60AF order 1 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 5 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 15 ··· 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 2 2 3 3 3 3 1 1 1 1 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 C2 C3 C4 C5 C6 C6 C10 C10 C12 C15 C20 C30 C30 C60 S3 Dic3 D6 C3×S3 C5×S3 C3×Dic3 S3×C6 C5×Dic3 S3×C10 S3×C15 Dic3×C15 S3×C30 kernel Dic3×C30 Dic3×C15 C6×C30 C10×Dic3 C3×C30 C6×Dic3 C5×Dic3 C2×C30 C3×Dic3 C62 C30 C2×Dic3 C3×C6 Dic3 C2×C6 C6 C2×C30 C30 C30 C2×C10 C2×C6 C10 C10 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 4 2 8 4 8 8 16 16 8 32 1 2 1 2 4 4 2 8 4 8 16 8

Matrix representation of Dic3×C30 in GL3(𝔽61) generated by

 48 0 0 0 36 0 0 0 36
,
 1 0 0 0 14 0 0 0 48
,
 60 0 0 0 0 60 0 1 0
G:=sub<GL(3,GF(61))| [48,0,0,0,36,0,0,0,36],[1,0,0,0,14,0,0,0,48],[60,0,0,0,0,1,0,60,0] >;

Dic3×C30 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{30}
% in TeX

G:=Group("Dic3xC30");
// GroupNames label

G:=SmallGroup(360,98);
// by ID

G=gap.SmallGroup(360,98);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-3,360,8645]);
// Polycyclic

G:=Group<a,b,c|a^30=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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