direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×S3×Dic5, C30.29D6, Dic15⋊3C6, C5⋊4(S3×C12), D6.(C3×D5), (S3×C10).C6, C6.2(C6×D5), (C5×S3)⋊2C12, (S3×C15)⋊4C4, C15⋊17(C4×S3), C15⋊5(C2×C12), C10.2(S3×C6), C30.2(C2×C6), (S3×C6).2D5, C3⋊1(C6×Dic5), C6.29(S3×D5), (S3×C30).1C2, (C3×Dic5)⋊1C6, (C3×C6).14D10, C32⋊4(C2×Dic5), (C3×Dic15)⋊3C2, (C3×C30).2C22, (C32×Dic5)⋊1C2, C2.2(C3×S3×D5), (C3×C15)⋊14(C2×C4), SmallGroup(360,59)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C3×S3×Dic5 |
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
►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