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## G = C3×D30.C2order 360 = 23·32·5

### Direct product of C3 and D30.C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C3×D30.C2
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C32×Dic5 — C3×D30.C2
 Lower central C15 — C3×D30.C2
 Upper central C1 — C6

Generators and relations for C3×D30.C2
G = < a,b,c,d | a3=b30=c2=1, d2=b15, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b19, dcd-1=b18c >

Subgroups: 268 in 70 conjugacy classes, 32 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, C20, D10, C4×S3, C2×C12, C3×D5, D15, C30, C30, C3×Dic3, C3×C12, S3×C6, C4×D5, C3×C15, C5×Dic3, C3×Dic5, C3×Dic5, C60, C6×D5, D30, S3×C12, C3×D15, C3×C30, D30.C2, D5×C12, C32×Dic5, Dic3×C15, C6×D15, C3×D30.C2
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D5, C12, D6, C2×C6, C3×S3, D10, C4×S3, C2×C12, C3×D5, S3×C6, C4×D5, S3×D5, C6×D5, S3×C12, D30.C2, D5×C12, C3×S3×D5, C3×D30.C2

Smallest permutation representation of C3×D30.C2
On 120 points
Generators in S120
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 28)(9 19 29)(10 20 30)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 71 81)(62 72 82)(63 73 83)(64 74 84)(65 75 85)(66 76 86)(67 77 87)(68 78 88)(69 79 89)(70 80 90)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 91)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 90)(58 89)(59 88)(60 87)
(1 66 16 81)(2 85 17 70)(3 74 18 89)(4 63 19 78)(5 82 20 67)(6 71 21 86)(7 90 22 75)(8 79 23 64)(9 68 24 83)(10 87 25 72)(11 76 26 61)(12 65 27 80)(13 84 28 69)(14 73 29 88)(15 62 30 77)(31 112 46 97)(32 101 47 116)(33 120 48 105)(34 109 49 94)(35 98 50 113)(36 117 51 102)(37 106 52 91)(38 95 53 110)(39 114 54 99)(40 103 55 118)(41 92 56 107)(42 111 57 96)(43 100 58 115)(44 119 59 104)(45 108 60 93)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 10A 10B 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12N 15A 15B 15C 15D 15E ··· 15J 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30J 60A ··· 60H order 1 2 2 2 3 3 3 3 3 4 4 4 4 5 5 6 6 6 6 6 6 6 6 6 10 10 12 12 12 12 12 12 12 12 12 ··· 12 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 15 15 1 1 2 2 2 3 3 5 5 2 2 1 1 2 2 2 15 15 15 15 2 2 3 3 3 3 5 5 5 5 10 ··· 10 2 2 2 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D5 D6 C3×S3 D10 C4×S3 C3×D5 S3×C6 C4×D5 C6×D5 S3×C12 D5×C12 S3×D5 D30.C2 C3×S3×D5 C3×D30.C2 kernel C3×D30.C2 C32×Dic5 Dic3×C15 C6×D15 D30.C2 C3×D15 C5×Dic3 C3×Dic5 D30 D15 C3×Dic5 C3×Dic3 C30 Dic5 C3×C6 C15 Dic3 C10 C32 C6 C5 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 2 1 2 2 2 4 2 4 4 4 8 2 2 4 4

Matrix representation of C3×D30.C2 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 13 0 0 0 29 47 0 0 0 0 17 16 0 0 1 1
,
 32 27 0 0 57 29 0 0 0 0 36 22 0 0 16 25
,
 1 0 0 0 0 1 0 0 0 0 29 26 0 0 38 32
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[13,29,0,0,0,47,0,0,0,0,17,1,0,0,16,1],[32,57,0,0,27,29,0,0,0,0,36,16,0,0,22,25],[1,0,0,0,0,1,0,0,0,0,29,38,0,0,26,32] >;

C3×D30.C2 in GAP, Magma, Sage, TeX

C_3\times D_{30}.C_2
% in TeX

G:=Group("C3xD30.C2");
// GroupNames label

G:=SmallGroup(360,60);
// by ID

G=gap.SmallGroup(360,60);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,79,730,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^30=c^2=1,d^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^19,d*c*d^-1=b^18*c>;
// generators/relations

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