direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×D15, C6.1D30, D30.2S3, C30.16D6, C10.1S32, C15⋊7(C4×S3), C3⋊3(C4×D15), C6.1(S3×D5), C5⋊2(S3×Dic3), C32⋊2(C4×D5), (C3×D15)⋊4C4, C3⋊1(D5×Dic3), (C3×C6).1D10, C2.1(S3×D15), C3⋊Dic15⋊5C2, (C5×Dic3)⋊1S3, (C3×Dic3)⋊2D5, (C6×D15).2C2, C15⋊7(C2×Dic3), (Dic3×C15)⋊2C2, (C3×C30).15C22, (C3×C15)⋊19(C2×C4), SmallGroup(360,77)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — Dic3×D15 |
Generators and relations for Dic3×D15
G = < a,b,c,d | a6=c15=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 412 in 70 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, C20, D10, C4×S3, C2×Dic3, C3×D5, D15, C30, C30, C3×Dic3, C3⋊Dic3, S3×C6, C4×D5, C3×C15, C5×Dic3, Dic15, C60, C6×D5, D30, S3×Dic3, C3×D15, C3×C30, D5×Dic3, C4×D15, Dic3×C15, C3⋊Dic15, C6×D15, Dic3×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, D10, C4×S3, C2×Dic3, D15, S32, C4×D5, S3×D5, D30, S3×Dic3, D5×Dic3, C4×D15, S3×D15, Dic3×D15
►On 120 points
Generators in S120
(1 25 11 20 6 30)(2 26 12 21 7 16)(3 27 13 22 8 17)(4 28 14 23 9 18)(5 29 15 24 10 19)(31 55 36 60 41 50)(32 56 37 46 42 51)(33 57 38 47 43 52)(34 58 39 48 44 53)(35 59 40 49 45 54)(61 79 66 84 71 89)(62 80 67 85 72 90)(63 81 68 86 73 76)(64 82 69 87 74 77)(65 83 70 88 75 78)(91 117 101 112 96 107)(92 118 102 113 97 108)(93 119 103 114 98 109)(94 120 104 115 99 110)(95 106 105 116 100 111)
(1 46 20 32)(2 47 21 33)(3 48 22 34)(4 49 23 35)(5 50 24 36)(6 51 25 37)(7 52 26 38)(8 53 27 39)(9 54 28 40)(10 55 29 41)(11 56 30 42)(12 57 16 43)(13 58 17 44)(14 59 18 45)(15 60 19 31)(61 116 84 95)(62 117 85 96)(63 118 86 97)(64 119 87 98)(65 120 88 99)(66 106 89 100)(67 107 90 101)(68 108 76 102)(69 109 77 103)(70 110 78 104)(71 111 79 105)(72 112 80 91)(73 113 81 92)(74 114 82 93)(75 115 83 94)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 80)(17 79)(18 78)(19 77)(20 76)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 105)(45 104)(46 108)(47 107)(48 106)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)
G:=sub<Sym(120)| (1,25,11,20,6,30)(2,26,12,21,7,16)(3,27,13,22,8,17)(4,28,14,23,9,18)(5,29,15,24,10,19)(31,55,36,60,41,50)(32,56,37,46,42,51)(33,57,38,47,43,52)(34,58,39,48,44,53)(35,59,40,49,45,54)(61,79,66,84,71,89)(62,80,67,85,72,90)(63,81,68,86,73,76)(64,82,69,87,74,77)(65,83,70,88,75,78)(91,117,101,112,96,107)(92,118,102,113,97,108)(93,119,103,114,98,109)(94,120,104,115,99,110)(95,106,105,116,100,111), (1,46,20,32)(2,47,21,33)(3,48,22,34)(4,49,23,35)(5,50,24,36)(6,51,25,37)(7,52,26,38)(8,53,27,39)(9,54,28,40)(10,55,29,41)(11,56,30,42)(12,57,16,43)(13,58,17,44)(14,59,18,45)(15,60,19,31)(61,116,84,95)(62,117,85,96)(63,118,86,97)(64,119,87,98)(65,120,88,99)(66,106,89,100)(67,107,90,101)(68,108,76,102)(69,109,77,103)(70,110,78,104)(71,111,79,105)(72,112,80,91)(73,113,81,92)(74,114,82,93)(75,115,83,94), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,80)(17,79)(18,78)(19,77)(20,76)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,105)(45,104)(46,108)(47,107)(48,106)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)>;
G:=Group( (1,25,11,20,6,30)(2,26,12,21,7,16)(3,27,13,22,8,17)(4,28,14,23,9,18)(5,29,15,24,10,19)(31,55,36,60,41,50)(32,56,37,46,42,51)(33,57,38,47,43,52)(34,58,39,48,44,53)(35,59,40,49,45,54)(61,79,66,84,71,89)(62,80,67,85,72,90)(63,81,68,86,73,76)(64,82,69,87,74,77)(65,83,70,88,75,78)(91,117,101,112,96,107)(92,118,102,113,97,108)(93,119,103,114,98,109)(94,120,104,115,99,110)(95,106,105,116,100,111), (1,46,20,32)(2,47,21,33)(3,48,22,34)(4,49,23,35)(5,50,24,36)(6,51,25,37)(7,52,26,38)(8,53,27,39)(9,54,28,40)(10,55,29,41)(11,56,30,42)(12,57,16,43)(13,58,17,44)(14,59,18,45)(15,60,19,31)(61,116,84,95)(62,117,85,96)(63,118,86,97)(64,119,87,98)(65,120,88,99)(66,106,89,100)(67,107,90,101)(68,108,76,102)(69,109,77,103)(70,110,78,104)(71,111,79,105)(72,112,80,91)(73,113,81,92)(74,114,82,93)(75,115,83,94), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,80)(17,79)(18,78)(19,77)(20,76)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,105)(45,104)(46,108)(47,107)(48,106)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109) );
G=PermutationGroup([[(1,25,11,20,6,30),(2,26,12,21,7,16),(3,27,13,22,8,17),(4,28,14,23,9,18),(5,29,15,24,10,19),(31,55,36,60,41,50),(32,56,37,46,42,51),(33,57,38,47,43,52),(34,58,39,48,44,53),(35,59,40,49,45,54),(61,79,66,84,71,89),(62,80,67,85,72,90),(63,81,68,86,73,76),(64,82,69,87,74,77),(65,83,70,88,75,78),(91,117,101,112,96,107),(92,118,102,113,97,108),(93,119,103,114,98,109),(94,120,104,115,99,110),(95,106,105,116,100,111)], [(1,46,20,32),(2,47,21,33),(3,48,22,34),(4,49,23,35),(5,50,24,36),(6,51,25,37),(7,52,26,38),(8,53,27,39),(9,54,28,40),(10,55,29,41),(11,56,30,42),(12,57,16,43),(13,58,17,44),(14,59,18,45),(15,60,19,31),(61,116,84,95),(62,117,85,96),(63,118,86,97),(64,119,87,98),(65,120,88,99),(66,106,89,100),(67,107,90,101),(68,108,76,102),(69,109,77,103),(70,110,78,104),(71,111,79,105),(72,112,80,91),(73,113,81,92),(74,114,82,93),(75,115,83,94)], [(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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,75),(10,74),(11,73),(12,72),(13,71),(14,70),(15,69),(16,80),(17,79),(18,78),(19,77),(20,76),(21,90),(22,89),(23,88),(24,87),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,103),(32,102),(33,101),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,105),(45,104),(46,108),(47,107),(48,106),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,112),(58,111),(59,110),(60,109)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 12A | 12B | 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 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 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 | 2 | 2 | 4 | 3 | 3 | 45 | 45 | 2 | 2 | 2 | 2 | 4 | 30 | 30 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D5 | Dic3 | D6 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 | S32 | S3×D5 | S3×Dic3 | D5×Dic3 | S3×D15 | Dic3×D15 |
| kernel | Dic3×D15 | Dic3×C15 | C3⋊Dic15 | C6×D15 | C3×D15 | C5×Dic3 | D30 | C3×Dic3 | D15 | C30 | C3×C6 | C15 | Dic3 | C32 | C6 | C3 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 4 |
Matrix representation of Dic3×D15 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 44 | 17 | 0 | 0 | 0 | 0 |
| 44 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 44 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[44,44,0,0,0,0,17,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,44,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic3×D15 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times D_{15} % in TeX
G:=Group("Dic3xD15"); // GroupNames label
G:=SmallGroup(360,77);
// by ID
G=gap.SmallGroup(360,77);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,31,201,1444,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^15=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations