direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×D30.C2, D30.C6, D15⋊2C12, C30.30D6, C5⋊2(S3×C12), C3⋊1(D5×C12), C6.3(C6×D5), C15⋊11(C4×S3), C32⋊5(C4×D5), C15⋊6(C2×C12), (C3×D15)⋊3C4, C10.3(S3×C6), C30.3(C2×C6), C6.30(S3×D5), (C3×Dic5)⋊2C6, (C3×Dic3)⋊5D5, (C3×Dic5)⋊5S3, (C5×Dic3)⋊2C6, Dic3⋊2(C3×D5), Dic5⋊2(C3×S3), (C6×D15).1C2, (C3×C6).15D10, (Dic3×C15)⋊7C2, (C3×C30).3C22, (C32×Dic5)⋊2C2, C2.3(C3×S3×D5), (C3×C15)⋊15(C2×C4), SmallGroup(360,60)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C3×D30.C2 |
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
►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