direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×Dic15, D6.D15, C6.2D30, C30.22D6, C10.2S32, (S3×C6).D5, (S3×C10).S3, (C3×S3)⋊Dic5, C6.2(S3×D5), C5⋊4(S3×Dic3), (S3×C15)⋊3C4, C15⋊16(C4×S3), C3⋊3(S3×Dic5), C2.2(S3×D15), (C3×C6).2D10, (S3×C30).2C2, C3⋊Dic15⋊6C2, (C5×S3)⋊2Dic3, C15⋊5(C2×Dic3), C3⋊1(C2×Dic15), C32⋊2(C2×Dic5), (C3×Dic15)⋊5C2, (C3×C30).16C22, (C3×C15)⋊20(C2×C4), SmallGroup(360,78)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — S3×Dic15 |
Generators and relations for S3×Dic15
G = < a,b,c,d | a3=b2=c30=1, d2=c15, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 348 in 70 conjugacy classes, 32 normal (24 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, C2×C10, C4×S3, C2×Dic3, C5×S3, C30, C30, C3×Dic3, C3⋊Dic3, S3×C6, C2×Dic5, C3×C15, C3×Dic5, Dic15, Dic15, S3×C10, C2×C30, S3×Dic3, S3×C15, C3×C30, S3×Dic5, C2×Dic15, C3×Dic15, C3⋊Dic15, S3×C30, S3×Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, Dic5, D10, C4×S3, C2×Dic3, D15, S32, C2×Dic5, Dic15, S3×D5, D30, S3×Dic3, S3×Dic5, C2×Dic15, S3×D15, S3×Dic15
►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 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 101 111)(92 102 112)(93 103 113)(94 104 114)(95 105 115)(96 106 116)(97 107 117)(98 108 118)(99 109 119)(100 110 120)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)
(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 61 16 76)(2 90 17 75)(3 89 18 74)(4 88 19 73)(5 87 20 72)(6 86 21 71)(7 85 22 70)(8 84 23 69)(9 83 24 68)(10 82 25 67)(11 81 26 66)(12 80 27 65)(13 79 28 64)(14 78 29 63)(15 77 30 62)(31 107 46 92)(32 106 47 91)(33 105 48 120)(34 104 49 119)(35 103 50 118)(36 102 51 117)(37 101 52 116)(38 100 53 115)(39 99 54 114)(40 98 55 113)(41 97 56 112)(42 96 57 111)(43 95 58 110)(44 94 59 109)(45 93 60 108)
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)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100), (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), (1,61,16,76)(2,90,17,75)(3,89,18,74)(4,88,19,73)(5,87,20,72)(6,86,21,71)(7,85,22,70)(8,84,23,69)(9,83,24,68)(10,82,25,67)(11,81,26,66)(12,80,27,65)(13,79,28,64)(14,78,29,63)(15,77,30,62)(31,107,46,92)(32,106,47,91)(33,105,48,120)(34,104,49,119)(35,103,50,118)(36,102,51,117)(37,101,52,116)(38,100,53,115)(39,99,54,114)(40,98,55,113)(41,97,56,112)(42,96,57,111)(43,95,58,110)(44,94,59,109)(45,93,60,108)>;
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)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100), (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), (1,61,16,76)(2,90,17,75)(3,89,18,74)(4,88,19,73)(5,87,20,72)(6,86,21,71)(7,85,22,70)(8,84,23,69)(9,83,24,68)(10,82,25,67)(11,81,26,66)(12,80,27,65)(13,79,28,64)(14,78,29,63)(15,77,30,62)(31,107,46,92)(32,106,47,91)(33,105,48,120)(34,104,49,119)(35,103,50,118)(36,102,51,117)(37,101,52,116)(38,100,53,115)(39,99,54,114)(40,98,55,113)(41,97,56,112)(42,96,57,111)(43,95,58,110)(44,94,59,109)(45,93,60,108) );
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),(61,81,71),(62,82,72),(63,83,73),(64,84,74),(65,85,75),(66,86,76),(67,87,77),(68,88,78),(69,89,79),(70,90,80),(91,101,111),(92,102,112),(93,103,113),(94,104,114),(95,105,115),(96,106,116),(97,107,117),(98,108,118),(99,109,119),(100,110,120)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100)], [(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)], [(1,61,16,76),(2,90,17,75),(3,89,18,74),(4,88,19,73),(5,87,20,72),(6,86,21,71),(7,85,22,70),(8,84,23,69),(9,83,24,68),(10,82,25,67),(11,81,26,66),(12,80,27,65),(13,79,28,64),(14,78,29,63),(15,77,30,62),(31,107,46,92),(32,106,47,91),(33,105,48,120),(34,104,49,119),(35,103,50,118),(36,102,51,117),(37,101,52,116),(38,100,53,115),(39,99,54,114),(40,98,55,113),(41,97,56,112),(42,96,57,111),(43,95,58,110),(44,94,59,109),(45,93,60,108)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 15C | 15D | 15E | ··· | 15J | 30A | 30B | 30C | 30D | 30E | ··· | 30J | 30K | ··· | 30R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 3 | 3 | 2 | 2 | 4 | 15 | 15 | 45 | 45 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 6 | 6 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | Dic5 | D10 | C4×S3 | D15 | Dic15 | D30 | S32 | S3×D5 | S3×Dic3 | S3×Dic5 | S3×D15 | S3×Dic15 |
| kernel | S3×Dic15 | C3×Dic15 | C3⋊Dic15 | S3×C30 | S3×C15 | Dic15 | S3×C10 | S3×C6 | C5×S3 | C30 | C3×S3 | C3×C6 | C15 | D6 | S3 | C6 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 4 | 1 | 2 | 1 | 2 | 4 | 4 |
Matrix representation of S3×Dic15 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 44 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 22 | 47 | 0 | 0 | 0 | 0 |
| 39 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[44,1,0,0,0,0,60,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,39,0,0,0,0,47,39,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
S3×Dic15 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{15} % in TeX
G:=Group("S3xDic15"); // GroupNames label
G:=SmallGroup(360,78);
// by ID
G=gap.SmallGroup(360,78);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,201,1444,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^30=1,d^2=c^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations