direct product, metabelian, supersoluble, monomial
Aliases: C3×C5⋊D12, D30⋊4C6, C15⋊8D12, C30.33D6, (C3×C15)⋊6D4, C5⋊2(C3×D12), (S3×C6)⋊2D5, D6⋊2(C3×D5), C15⋊3(C3×D4), Dic5⋊(C3×S3), C6.6(C6×D5), (S3×C10)⋊2C6, (S3×C30)⋊2C2, (C6×D15)⋊4C2, C10.6(S3×C6), C30.6(C2×C6), C6.33(S3×D5), (C3×Dic5)⋊3C6, (C3×Dic5)⋊4S3, (C3×C6).18D10, C32⋊5(C5⋊D4), (C3×C30).6C22, (C32×Dic5)⋊3C2, C2.6(C3×S3×D5), C3⋊1(C3×C5⋊D4), SmallGroup(360,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C5⋊D12
G = < a,b,c,d | a3=b5=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 300 in 70 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, C12, D6, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, D10, C2×C10, D12, C3×D4, C5×S3, C3×D5, D15, C30, C30, C3×C12, S3×C6, S3×C6, C5⋊D4, C3×C15, C3×Dic5, C3×Dic5, C6×D5, S3×C10, D30, C2×C30, C3×D12, S3×C15, C3×D15, C3×C30, C5⋊D12, C3×C5⋊D4, C32×Dic5, S3×C30, C6×D15, C3×C5⋊D12
Quotients: C1, C2, C3, C22, S3, C6, D4, D5, D6, C2×C6, C3×S3, D10, D12, C3×D4, C3×D5, S3×C6, C5⋊D4, S3×D5, C6×D5, C3×D12, C5⋊D12, C3×C5⋊D4, C3×S3×D5, C3×C5⋊D12
►On 120 points
Generators in S120
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)(97 105 101)(98 106 102)(99 107 103)(100 108 104)(109 117 113)(110 118 114)(111 119 115)(112 120 116)
(1 19 26 110 98)(2 99 111 27 20)(3 21 28 112 100)(4 101 113 29 22)(5 23 30 114 102)(6 103 115 31 24)(7 13 32 116 104)(8 105 117 33 14)(9 15 34 118 106)(10 107 119 35 16)(11 17 36 120 108)(12 97 109 25 18)(37 49 73 89 69)(38 70 90 74 50)(39 51 75 91 71)(40 72 92 76 52)(41 53 77 93 61)(42 62 94 78 54)(43 55 79 95 63)(44 64 96 80 56)(45 57 81 85 65)(46 66 86 82 58)(47 59 83 87 67)(48 68 88 84 60)
(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 49)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)(73 98)(74 97)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)
G:=sub<Sym(120)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96)(97,105,101)(98,106,102)(99,107,103)(100,108,104)(109,117,113)(110,118,114)(111,119,115)(112,120,116), (1,19,26,110,98)(2,99,111,27,20)(3,21,28,112,100)(4,101,113,29,22)(5,23,30,114,102)(6,103,115,31,24)(7,13,32,116,104)(8,105,117,33,14)(9,15,34,118,106)(10,107,119,35,16)(11,17,36,120,108)(12,97,109,25,18)(37,49,73,89,69)(38,70,90,74,50)(39,51,75,91,71)(40,72,92,76,52)(41,53,77,93,61)(42,62,94,78,54)(43,55,79,95,63)(44,64,96,80,56)(45,57,81,85,65)(46,66,86,82,58)(47,59,83,87,67)(48,68,88,84,60), (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), (1,49)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(73,98)(74,97)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96)(97,105,101)(98,106,102)(99,107,103)(100,108,104)(109,117,113)(110,118,114)(111,119,115)(112,120,116), (1,19,26,110,98)(2,99,111,27,20)(3,21,28,112,100)(4,101,113,29,22)(5,23,30,114,102)(6,103,115,31,24)(7,13,32,116,104)(8,105,117,33,14)(9,15,34,118,106)(10,107,119,35,16)(11,17,36,120,108)(12,97,109,25,18)(37,49,73,89,69)(38,70,90,74,50)(39,51,75,91,71)(40,72,92,76,52)(41,53,77,93,61)(42,62,94,78,54)(43,55,79,95,63)(44,64,96,80,56)(45,57,81,85,65)(46,66,86,82,58)(47,59,83,87,67)(48,68,88,84,60), (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), (1,49)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(73,98)(74,97)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,89,93),(86,90,94),(87,91,95),(88,92,96),(97,105,101),(98,106,102),(99,107,103),(100,108,104),(109,117,113),(110,118,114),(111,119,115),(112,120,116)], [(1,19,26,110,98),(2,99,111,27,20),(3,21,28,112,100),(4,101,113,29,22),(5,23,30,114,102),(6,103,115,31,24),(7,13,32,116,104),(8,105,117,33,14),(9,15,34,118,106),(10,107,119,35,16),(11,17,36,120,108),(12,97,109,25,18),(37,49,73,89,69),(38,70,90,74,50),(39,51,75,91,71),(40,72,92,76,52),(41,53,77,93,61),(42,62,94,78,54),(43,55,79,95,63),(44,64,96,80,56),(45,57,81,85,65),(46,66,86,82,58),(47,59,83,87,67),(48,68,88,84,60)], [(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)], [(1,49),(2,60),(3,59),(4,58),(5,57),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,48),(21,47),(22,46),(23,45),(24,44),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,72),(36,71),(73,98),(74,97),(75,108),(76,107),(77,106),(78,105),(79,104),(80,103),(81,102),(82,101),(83,100),(84,99),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 10A | 10B | 10C | 10D | 10E | 10F | 12A | ··· | 12H | 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 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 30 | ··· | 30 |
| size | 1 | 1 | 6 | 30 | 1 | 1 | 2 | 2 | 2 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 30 | 30 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D5 | D6 | C3×S3 | D10 | D12 | C3×D4 | C3×D5 | S3×C6 | C5⋊D4 | C6×D5 | C3×D12 | C3×C5⋊D4 | S3×D5 | C5⋊D12 | C3×S3×D5 | C3×C5⋊D12 |
| kernel | C3×C5⋊D12 | C32×Dic5 | S3×C30 | C6×D15 | C5⋊D12 | C3×Dic5 | S3×C10 | D30 | C3×Dic5 | C3×C15 | S3×C6 | C30 | Dic5 | C3×C6 | C15 | C15 | D6 | C10 | C32 | C6 | C5 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C5⋊D12 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 60 |
| 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 |
| 0 | 0 | 2 | 59 |
| 0 | 0 | 32 | 59 |
| 0 | 21 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 2 | 59 |
| 0 | 0 | 32 | 59 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,17,1,0,0,60,0],[32,0,0,0,0,21,0,0,0,0,2,32,0,0,59,59],[0,32,0,0,21,0,0,0,0,0,2,32,0,0,59,59] >;
C3×C5⋊D12 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes D_{12} % in TeX
G:=Group("C3xC5:D12"); // GroupNames label
G:=SmallGroup(360,63);
// by ID
G=gap.SmallGroup(360,63);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,169,730,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations