direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×Dic10, D6.8D10, C20.27D6, Dic30⋊7C2, C12.15D10, C30.4C23, Dic5.1D6, C60.13C22, Dic3.7D10, Dic15.3C22, (C5×S3)⋊Q8, C5⋊2(S3×Q8), C15⋊Q8⋊2C2, C15⋊2(C2×Q8), C4.6(S3×D5), (C4×S3).1D5, C3⋊1(C2×Dic10), (S3×C20).1C2, C6.4(C22×D5), (C3×Dic10)⋊2C2, C10.4(C22×S3), (S3×Dic5).1C2, (S3×C10).6C22, (C5×Dic3).8C22, (C3×Dic5).1C22, C2.8(C2×S3×D5), SmallGroup(240,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Dic10
G = < a,b,c,d | a3=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 280 in 76 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, C15, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Dic6, C4×S3, C4×S3, C3×Q8, C5×S3, C30, Dic10, Dic10, C2×Dic5, C2×C20, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×Dic10, S3×Dic5, C15⋊Q8, C3×Dic10, S3×C20, Dic30, S3×Dic10
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, D10, C22×S3, Dic10, C22×D5, S3×Q8, S3×D5, C2×Dic10, C2×S3×D5, S3×Dic10
►On 120 points
Generators in S120
(1 60 69)(2 41 70)(3 42 71)(4 43 72)(5 44 73)(6 45 74)(7 46 75)(8 47 76)(9 48 77)(10 49 78)(11 50 79)(12 51 80)(13 52 61)(14 53 62)(15 54 63)(16 55 64)(17 56 65)(18 57 66)(19 58 67)(20 59 68)(21 106 84)(22 107 85)(23 108 86)(24 109 87)(25 110 88)(26 111 89)(27 112 90)(28 113 91)(29 114 92)(30 115 93)(31 116 94)(32 117 95)(33 118 96)(34 119 97)(35 120 98)(36 101 99)(37 102 100)(38 103 81)(39 104 82)(40 105 83)
(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(81 103)(82 104)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 101)(100 102)
(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 32 11 22)(2 31 12 21)(3 30 13 40)(4 29 14 39)(5 28 15 38)(6 27 16 37)(7 26 17 36)(8 25 18 35)(9 24 19 34)(10 23 20 33)(41 116 51 106)(42 115 52 105)(43 114 53 104)(44 113 54 103)(45 112 55 102)(46 111 56 101)(47 110 57 120)(48 109 58 119)(49 108 59 118)(50 107 60 117)(61 83 71 93)(62 82 72 92)(63 81 73 91)(64 100 74 90)(65 99 75 89)(66 98 76 88)(67 97 77 87)(68 96 78 86)(69 95 79 85)(70 94 80 84)
G:=sub<Sym(120)| (1,60,69)(2,41,70)(3,42,71)(4,43,72)(5,44,73)(6,45,74)(7,46,75)(8,47,76)(9,48,77)(10,49,78)(11,50,79)(12,51,80)(13,52,61)(14,53,62)(15,54,63)(16,55,64)(17,56,65)(18,57,66)(19,58,67)(20,59,68)(21,106,84)(22,107,85)(23,108,86)(24,109,87)(25,110,88)(26,111,89)(27,112,90)(28,113,91)(29,114,92)(30,115,93)(31,116,94)(32,117,95)(33,118,96)(34,119,97)(35,120,98)(36,101,99)(37,102,100)(38,103,81)(39,104,82)(40,105,83), (41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,103)(82,104)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,101)(100,102), (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), (1,32,11,22)(2,31,12,21)(3,30,13,40)(4,29,14,39)(5,28,15,38)(6,27,16,37)(7,26,17,36)(8,25,18,35)(9,24,19,34)(10,23,20,33)(41,116,51,106)(42,115,52,105)(43,114,53,104)(44,113,54,103)(45,112,55,102)(46,111,56,101)(47,110,57,120)(48,109,58,119)(49,108,59,118)(50,107,60,117)(61,83,71,93)(62,82,72,92)(63,81,73,91)(64,100,74,90)(65,99,75,89)(66,98,76,88)(67,97,77,87)(68,96,78,86)(69,95,79,85)(70,94,80,84)>;
G:=Group( (1,60,69)(2,41,70)(3,42,71)(4,43,72)(5,44,73)(6,45,74)(7,46,75)(8,47,76)(9,48,77)(10,49,78)(11,50,79)(12,51,80)(13,52,61)(14,53,62)(15,54,63)(16,55,64)(17,56,65)(18,57,66)(19,58,67)(20,59,68)(21,106,84)(22,107,85)(23,108,86)(24,109,87)(25,110,88)(26,111,89)(27,112,90)(28,113,91)(29,114,92)(30,115,93)(31,116,94)(32,117,95)(33,118,96)(34,119,97)(35,120,98)(36,101,99)(37,102,100)(38,103,81)(39,104,82)(40,105,83), (41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,103)(82,104)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,101)(100,102), (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), (1,32,11,22)(2,31,12,21)(3,30,13,40)(4,29,14,39)(5,28,15,38)(6,27,16,37)(7,26,17,36)(8,25,18,35)(9,24,19,34)(10,23,20,33)(41,116,51,106)(42,115,52,105)(43,114,53,104)(44,113,54,103)(45,112,55,102)(46,111,56,101)(47,110,57,120)(48,109,58,119)(49,108,59,118)(50,107,60,117)(61,83,71,93)(62,82,72,92)(63,81,73,91)(64,100,74,90)(65,99,75,89)(66,98,76,88)(67,97,77,87)(68,96,78,86)(69,95,79,85)(70,94,80,84) );
G=PermutationGroup([[(1,60,69),(2,41,70),(3,42,71),(4,43,72),(5,44,73),(6,45,74),(7,46,75),(8,47,76),(9,48,77),(10,49,78),(11,50,79),(12,51,80),(13,52,61),(14,53,62),(15,54,63),(16,55,64),(17,56,65),(18,57,66),(19,58,67),(20,59,68),(21,106,84),(22,107,85),(23,108,86),(24,109,87),(25,110,88),(26,111,89),(27,112,90),(28,113,91),(29,114,92),(30,115,93),(31,116,94),(32,117,95),(33,118,96),(34,119,97),(35,120,98),(36,101,99),(37,102,100),(38,103,81),(39,104,82),(40,105,83)], [(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(81,103),(82,104),(83,105),(84,106),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,101),(100,102)], [(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)], [(1,32,11,22),(2,31,12,21),(3,30,13,40),(4,29,14,39),(5,28,15,38),(6,27,16,37),(7,26,17,36),(8,25,18,35),(9,24,19,34),(10,23,20,33),(41,116,51,106),(42,115,52,105),(43,114,53,104),(44,113,54,103),(45,112,55,102),(46,111,56,101),(47,110,57,120),(48,109,58,119),(49,108,59,118),(50,107,60,117),(61,83,71,93),(62,82,72,92),(63,81,73,91),(64,100,74,90),(65,99,75,89),(66,98,76,88),(67,97,77,87),(68,96,78,86),(69,95,79,85),(70,94,80,84)]])
S3×Dic10 is a maximal subgroup of
Dic20⋊S3 C40.2D6 C60.10C23 Dic10.26D6 D20.39D6 C30.C24 C15⋊2- 1+4 D12.29D10 S3×Q8×D5
S3×Dic10 is a maximal quotient of
Dic3⋊5Dic10 Dic15⋊1Q8 Dic3⋊Dic10 Dic30⋊14C4 Dic3.Dic10 Dic3.2Dic10 D6⋊Dic10 C60.45D4 C60.46D4 Dic3.3Dic10 C60.48D4 D6⋊1Dic10 D6⋊2Dic10 D6⋊3Dic10 D6⋊4Dic10 C20⋊4Dic6
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | D10 | D10 | D10 | Dic10 | S3×Q8 | S3×D5 | C2×S3×D5 | S3×Dic10 |
| kernel | S3×Dic10 | S3×Dic5 | C15⋊Q8 | C3×Dic10 | S3×C20 | Dic30 | Dic10 | C5×S3 | C4×S3 | Dic5 | C20 | Dic3 | C12 | D6 | S3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of S3×Dic10 ►in GL4(𝔽61) generated by
| 60 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 32 |
| 0 | 0 | 2 | 34 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 27 | 31 |
| 0 | 0 | 4 | 34 |
G:=sub<GL(4,GF(61))| [60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[1,60,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,2,0,0,32,34],[60,0,0,0,0,60,0,0,0,0,27,4,0,0,31,34] >;
S3×Dic10 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{10} % in TeX
G:=Group("S3xDic10"); // GroupNames label
G:=SmallGroup(240,128);
// by ID
G=gap.SmallGroup(240,128);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,218,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^20=1,d^2=c^10,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