metabelian, supersoluble, monomial, A-group
Aliases: D30.S3, C30.17D6, D15⋊2Dic3, Dic15⋊2S3, C10.8S32, C15⋊8(C4×S3), C5⋊3(S3×Dic3), C32⋊4(C4×D5), (C3×D15)⋊5C4, C3⋊Dic3⋊2D5, C3⋊2(D5×Dic3), (C3×C6).8D10, C6.23(S3×D5), (C6×D15).3C2, C15⋊6(C2×Dic3), C3⋊3(D30.C2), C2.1(D15⋊S3), (C3×Dic15)⋊8C2, (C3×C30).22C22, (C3×C15)⋊22(C2×C4), (C5×C3⋊Dic3)⋊4C2, SmallGroup(360,84)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — D30.S3 |
Generators and relations for D30.S3
G = < a,b,c,d | a30=b2=c3=1, d2=a15, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 356 in 70 conjugacy classes, 27 normal (23 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C22, C5, S3 [×2], C6 [×2], C6 [×3], C2×C4, C32, D5 [×2], C10, Dic3 [×4], C12, D6, C2×C6, C15 [×2], C15, C3×S3 [×2], C3×C6, Dic5, C20, D10, C4×S3, C2×Dic3, C3×D5 [×2], D15 [×2], C30 [×2], C30, C3×Dic3, C3⋊Dic3, S3×C6, C4×D5, C3×C15, C5×Dic3 [×3], C3×Dic5, Dic15, C6×D5, D30, S3×Dic3, C3×D15 [×2], C3×C30, D5×Dic3, D30.C2, C3×Dic15, C5×C3⋊Dic3, C6×D15, D30.S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D5, Dic3 [×2], D6 [×2], D10, C4×S3, C2×Dic3, S32, C4×D5, S3×D5 [×2], S3×Dic3, D5×Dic3, D30.C2, D15⋊S3, D30.S3
►On 120 points
Generators in S120
(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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 120)(39 119)(40 118)(41 117)(42 116)(43 115)(44 114)(45 113)(46 112)(47 111)(48 110)(49 109)(50 108)(51 107)(52 106)(53 105)(54 104)(55 103)(56 102)(57 101)(58 100)(59 99)(60 98)
(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 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)(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)(91 111 101)(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 56 16 41)(2 37 17 52)(3 48 18 33)(4 59 19 44)(5 40 20 55)(6 51 21 36)(7 32 22 47)(8 43 23 58)(9 54 24 39)(10 35 25 50)(11 46 26 31)(12 57 27 42)(13 38 28 53)(14 49 29 34)(15 60 30 45)(61 103 76 118)(62 114 77 99)(63 95 78 110)(64 106 79 91)(65 117 80 102)(66 98 81 113)(67 109 82 94)(68 120 83 105)(69 101 84 116)(70 112 85 97)(71 93 86 108)(72 104 87 119)(73 115 88 100)(74 96 89 111)(75 107 90 92)
G:=sub<Sym(120)| (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,113)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(57,101)(58,100)(59,99)(60,98), (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)(31,41,51)(32,42,52)(33,43,53)(34,44,54)(35,45,55)(36,46,56)(37,47,57)(38,48,58)(39,49,59)(40,50,60)(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)(91,111,101)(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), (1,56,16,41)(2,37,17,52)(3,48,18,33)(4,59,19,44)(5,40,20,55)(6,51,21,36)(7,32,22,47)(8,43,23,58)(9,54,24,39)(10,35,25,50)(11,46,26,31)(12,57,27,42)(13,38,28,53)(14,49,29,34)(15,60,30,45)(61,103,76,118)(62,114,77,99)(63,95,78,110)(64,106,79,91)(65,117,80,102)(66,98,81,113)(67,109,82,94)(68,120,83,105)(69,101,84,116)(70,112,85,97)(71,93,86,108)(72,104,87,119)(73,115,88,100)(74,96,89,111)(75,107,90,92)>;
G:=Group( (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,113)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(57,101)(58,100)(59,99)(60,98), (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)(31,41,51)(32,42,52)(33,43,53)(34,44,54)(35,45,55)(36,46,56)(37,47,57)(38,48,58)(39,49,59)(40,50,60)(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)(91,111,101)(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), (1,56,16,41)(2,37,17,52)(3,48,18,33)(4,59,19,44)(5,40,20,55)(6,51,21,36)(7,32,22,47)(8,43,23,58)(9,54,24,39)(10,35,25,50)(11,46,26,31)(12,57,27,42)(13,38,28,53)(14,49,29,34)(15,60,30,45)(61,103,76,118)(62,114,77,99)(63,95,78,110)(64,106,79,91)(65,117,80,102)(66,98,81,113)(67,109,82,94)(68,120,83,105)(69,101,84,116)(70,112,85,97)(71,93,86,108)(72,104,87,119)(73,115,88,100)(74,96,89,111)(75,107,90,92) );
G=PermutationGroup([(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)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,84),(8,83),(9,82),(10,81),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,97),(32,96),(33,95),(34,94),(35,93),(36,92),(37,91),(38,120),(39,119),(40,118),(41,117),(42,116),(43,115),(44,114),(45,113),(46,112),(47,111),(48,110),(49,109),(50,108),(51,107),(52,106),(53,105),(54,104),(55,103),(56,102),(57,101),(58,100),(59,99),(60,98)], [(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),(31,41,51),(32,42,52),(33,43,53),(34,44,54),(35,45,55),(36,46,56),(37,47,57),(38,48,58),(39,49,59),(40,50,60),(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),(91,111,101),(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)], [(1,56,16,41),(2,37,17,52),(3,48,18,33),(4,59,19,44),(5,40,20,55),(6,51,21,36),(7,32,22,47),(8,43,23,58),(9,54,24,39),(10,35,25,50),(11,46,26,31),(12,57,27,42),(13,38,28,53),(14,49,29,34),(15,60,30,45),(61,103,76,118),(62,114,77,99),(63,95,78,110),(64,106,79,91),(65,117,80,102),(66,98,81,113),(67,109,82,94),(68,120,83,105),(69,101,84,116),(70,112,85,97),(71,93,86,108),(72,104,87,119),(73,115,88,100),(74,96,89,111),(75,107,90,92)])
42 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 | ··· | 15H | 20A | 20B | 20C | 20D | 30A | ··· | 30H |
| 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 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 15 | 15 | 2 | 2 | 4 | 9 | 9 | 15 | 15 | 2 | 2 | 2 | 2 | 4 | 30 | 30 | 2 | 2 | 30 | 30 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D5 | Dic3 | D6 | D10 | C4×S3 | C4×D5 | S32 | S3×D5 | S3×Dic3 | D5×Dic3 | D30.C2 | D15⋊S3 | D30.S3 |
| kernel | D30.S3 | C3×Dic15 | C5×C3⋊Dic3 | C6×D15 | C3×D15 | Dic15 | D30 | C3⋊Dic3 | D15 | C30 | C3×C6 | C15 | C32 | C10 | C6 | C5 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D30.S3 ►in GL6(𝔽61)
| 44 | 60 | 0 | 0 | 0 | 0 |
| 45 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 47 | 0 | 0 | 0 | 0 |
| 32 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [44,45,0,0,0,0,60,60,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,32,0,0,0,0,47,49,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D30.S3 in GAP, Magma, Sage, TeX
D_{30}.S_3 % in TeX
G:=Group("D30.S3"); // GroupNames label
G:=SmallGroup(360,84);
// by ID
G=gap.SmallGroup(360,84);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,31,387,201,730,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^3=1,d^2=a^15,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations