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## G = C3×Dic30order 360 = 23·32·5

### Direct product of C3 and Dic30

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C3×Dic30
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C3×Dic15 — C3×Dic30
 Lower central C15 — C30 — C3×Dic30
 Upper central C1 — C6 — C12

Generators and relations for C3×Dic30
G = < a,b,c | a3=b60=1, c2=b30, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 180 in 54 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C3, C3, C4, C4, C5, C6, C6, Q8, C32, C10, Dic3, C12, C12, C15, C15, C3×C6, Dic5, C20, Dic6, C3×Q8, C30, C30, C3×Dic3, C3×C12, Dic10, C3×C15, C3×Dic5, Dic15, C60, C60, C3×Dic6, C3×C30, C3×Dic10, Dic30, C3×Dic15, C3×C60, C3×Dic30
Quotients: C1, C2, C3, C22, S3, C6, Q8, D5, D6, C2×C6, C3×S3, D10, Dic6, C3×Q8, C3×D5, D15, S3×C6, Dic10, C6×D5, D30, C3×Dic6, C3×D15, C3×Dic10, Dic30, C6×D15, C3×Dic30

Smallest permutation representation of C3×Dic30
On 120 points
Generators in S120
(1 21 41)(2 22 42)(3 23 43)(4 24 44)(5 25 45)(6 26 46)(7 27 47)(8 28 48)(9 29 49)(10 30 50)(11 31 51)(12 32 52)(13 33 53)(14 34 54)(15 35 55)(16 36 56)(17 37 57)(18 38 58)(19 39 59)(20 40 60)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(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)
(1 101 31 71)(2 100 32 70)(3 99 33 69)(4 98 34 68)(5 97 35 67)(6 96 36 66)(7 95 37 65)(8 94 38 64)(9 93 39 63)(10 92 40 62)(11 91 41 61)(12 90 42 120)(13 89 43 119)(14 88 44 118)(15 87 45 117)(16 86 46 116)(17 85 47 115)(18 84 48 114)(19 83 49 113)(20 82 50 112)(21 81 51 111)(22 80 52 110)(23 79 53 109)(24 78 54 108)(25 77 55 107)(26 76 56 106)(27 75 57 105)(28 74 58 104)(29 73 59 103)(30 72 60 102)

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

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

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

99 conjugacy classes

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

99 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + + - + - + - image C1 C2 C2 C3 C6 C6 S3 Q8 D5 D6 C3×S3 D10 Dic6 C3×Q8 C3×D5 D15 S3×C6 Dic10 C6×D5 D30 C3×Dic6 C3×D15 C3×Dic10 Dic30 C6×D15 C3×Dic30 kernel C3×Dic30 C3×Dic15 C3×C60 Dic30 Dic15 C60 C60 C3×C15 C3×C12 C30 C20 C3×C6 C15 C15 C12 C12 C10 C32 C6 C6 C5 C4 C3 C3 C2 C1 # reps 1 2 1 2 4 2 1 1 2 1 2 2 2 2 4 4 2 4 4 4 4 8 8 8 8 16

Matrix representation of C3×Dic30 in GL2(𝔽61) generated by

 47 0 0 47
,
 18 0 34 17
,
 11 23 0 50
G:=sub<GL(2,GF(61))| [47,0,0,47],[18,34,0,17],[11,0,23,50] >;

C3×Dic30 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{30}
% in TeX

G:=Group("C3xDic30");
// GroupNames label

G:=SmallGroup(360,100);
// by ID

G=gap.SmallGroup(360,100);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,169,79,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^60=1,c^2=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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