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## G = C3×S3×Dic5order 360 = 23·32·5

### Direct product of C3, S3 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C3×S3×Dic5
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C32×Dic5 — C3×S3×Dic5
 Lower central C15 — C3×S3×Dic5
 Upper central C1 — C6

Generators and relations for C3×S3×Dic5
G = < a,b,c,d,e | a3=b3=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 204 in 70 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, Dic5, C2×C10, C4×S3, C2×C12, C5×S3, C30, C30, C3×Dic3, C3×C12, S3×C6, C2×Dic5, C3×C15, C3×Dic5, C3×Dic5, Dic15, S3×C10, C2×C30, S3×C12, S3×C15, C3×C30, S3×Dic5, C6×Dic5, C32×Dic5, C3×Dic15, S3×C30, C3×S3×Dic5
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D5, C12, D6, C2×C6, C3×S3, Dic5, D10, C4×S3, C2×C12, C3×D5, S3×C6, C2×Dic5, C3×Dic5, S3×D5, C6×D5, S3×C12, S3×Dic5, C6×Dic5, C3×S3×D5, C3×S3×Dic5

Smallest permutation representation of C3×S3×Dic5
On 120 points
Generators in S120
(1 25 33)(2 26 34)(3 27 35)(4 28 36)(5 29 37)(6 30 38)(7 21 39)(8 22 40)(9 23 31)(10 24 32)(11 111 107)(12 112 108)(13 113 109)(14 114 110)(15 115 101)(16 116 102)(17 117 103)(18 118 104)(19 119 105)(20 120 106)(41 63 52)(42 64 53)(43 65 54)(44 66 55)(45 67 56)(46 68 57)(47 69 58)(48 70 59)(49 61 60)(50 62 51)(71 96 85)(72 97 86)(73 98 87)(74 99 88)(75 100 89)(76 91 90)(77 92 81)(78 93 82)(79 94 83)(80 95 84)
(1 25 33)(2 26 34)(3 27 35)(4 28 36)(5 29 37)(6 30 38)(7 21 39)(8 22 40)(9 23 31)(10 24 32)(11 107 111)(12 108 112)(13 109 113)(14 110 114)(15 101 115)(16 102 116)(17 103 117)(18 104 118)(19 105 119)(20 106 120)(41 52 63)(42 53 64)(43 54 65)(44 55 66)(45 56 67)(46 57 68)(47 58 69)(48 59 70)(49 60 61)(50 51 62)(71 96 85)(72 97 86)(73 98 87)(74 99 88)(75 100 89)(76 91 90)(77 92 81)(78 93 82)(79 94 83)(80 95 84)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 51)(9 52)(10 53)(11 97)(12 98)(13 99)(14 100)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 49)(22 50)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 61)(40 62)(71 106)(72 107)(73 108)(74 109)(75 110)(76 101)(77 102)(78 103)(79 104)(80 105)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)
(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 86 6 81)(2 85 7 90)(3 84 8 89)(4 83 9 88)(5 82 10 87)(11 70 16 65)(12 69 17 64)(13 68 18 63)(14 67 19 62)(15 66 20 61)(21 76 26 71)(22 75 27 80)(23 74 28 79)(24 73 29 78)(25 72 30 77)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 109 46 104)(42 108 47 103)(43 107 48 102)(44 106 49 101)(45 105 50 110)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)

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

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

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

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 10C 10D 10E 10F 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 15A 15B 15C 15D 15E ··· 15J 30A 30B 30C 30D 30E ··· 30J 30K ··· 30R 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 10 10 10 10 12 12 12 12 12 ··· 12 12 12 12 12 15 15 15 15 15 ··· 15 30 30 30 30 30 ··· 30 30 ··· 30 size 1 1 3 3 1 1 2 2 2 5 5 15 15 2 2 1 1 2 2 2 3 3 3 3 2 2 6 6 6 6 5 5 5 5 10 ··· 10 15 15 15 15 2 2 2 2 4 ··· 4 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 Dic5 D10 C4×S3 C3×D5 S3×C6 C3×Dic5 C6×D5 S3×C12 S3×D5 S3×Dic5 C3×S3×D5 C3×S3×Dic5 kernel C3×S3×Dic5 C32×Dic5 C3×Dic15 S3×C30 S3×Dic5 S3×C15 C3×Dic5 Dic15 S3×C10 C5×S3 C3×Dic5 S3×C6 C30 Dic5 C3×S3 C3×C6 C15 D6 C10 S3 C6 C5 C6 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 2 1 2 4 2 2 4 2 8 4 4 2 2 4 4

Matrix representation of C3×S3×Dic5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 47 0 0 0 0 47
,
 1 0 0 0 0 1 0 0 0 0 47 0 0 0 13 13
,
 1 0 0 0 0 1 0 0 0 0 1 49 0 0 0 60
,
 0 60 0 0 1 44 0 0 0 0 60 0 0 0 0 60
,
 57 57 0 0 50 4 0 0 0 0 11 0 0 0 0 11
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[1,0,0,0,0,1,0,0,0,0,47,13,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,49,60],[0,1,0,0,60,44,0,0,0,0,60,0,0,0,0,60],[57,50,0,0,57,4,0,0,0,0,11,0,0,0,0,11] >;

C3×S3×Dic5 in GAP, Magma, Sage, TeX

C_3\times S_3\times {\rm Dic}_5
% in TeX

G:=Group("C3xS3xDic5");
// GroupNames label

G:=SmallGroup(360,59);
// by ID

G=gap.SmallGroup(360,59);
# by ID

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

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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