metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊3S3, Dic6⋊3D5, D10.2D6, C20.14D6, C12.14D10, C30.3C23, C60.26C22, Dic3.2D10, D30.8C22, Dic15.10C22, (C3×D20)⋊5C2, (C4×D15)⋊5C2, C15⋊2(C4○D4), C3⋊D20⋊1C2, C4.19(S3×D5), C5⋊1(D4⋊2S3), (C5×Dic6)⋊5C2, (D5×Dic3)⋊2C2, C3⋊2(Q8⋊2D5), C6.3(C22×D5), C10.3(C22×S3), (C6×D5).2C22, (C5×Dic3).2C22, C2.7(C2×S3×D5), SmallGroup(240,127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊S3
G = < a,b,c,d | a20=b2=c3=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, dcd=c-1 >
Subgroups: 360 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C4○D4, Dic5, C20, C20, D10, D10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×D5, D15, C30, C4×D5, D20, D20, C5×Q8, D4⋊2S3, C5×Dic3, Dic15, C60, C6×D5, D30, Q8⋊2D5, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C4×D15, D20⋊S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, D4⋊2S3, S3×D5, Q8⋊2D5, C2×S3×D5, D20⋊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)
(1 101)(2 120)(3 119)(4 118)(5 117)(6 116)(7 115)(8 114)(9 113)(10 112)(11 111)(12 110)(13 109)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 84)(49 83)(50 82)(51 81)(52 100)(53 99)(54 98)(55 97)(56 96)(57 95)(58 94)(59 93)(60 92)
(1 72 93)(2 73 94)(3 74 95)(4 75 96)(5 76 97)(6 77 98)(7 78 99)(8 79 100)(9 80 81)(10 61 82)(11 62 83)(12 63 84)(13 64 85)(14 65 86)(15 66 87)(16 67 88)(17 68 89)(18 69 90)(19 70 91)(20 71 92)(21 41 103)(22 42 104)(23 43 105)(24 44 106)(25 45 107)(26 46 108)(27 47 109)(28 48 110)(29 49 111)(30 50 112)(31 51 113)(32 52 114)(33 53 115)(34 54 116)(35 55 117)(36 56 118)(37 57 119)(38 58 120)(39 59 101)(40 60 102)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 59)(22 48)(23 57)(24 46)(25 55)(26 44)(27 53)(28 42)(29 51)(30 60)(31 49)(32 58)(33 47)(34 56)(35 45)(36 54)(37 43)(38 52)(39 41)(40 50)(61 94)(62 83)(63 92)(64 81)(65 90)(66 99)(67 88)(68 97)(69 86)(70 95)(71 84)(72 93)(73 82)(74 91)(75 100)(76 89)(77 98)(78 87)(79 96)(80 85)(101 103)(102 112)(104 110)(105 119)(106 108)(107 117)(109 115)(111 113)(114 120)(116 118)
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), (1,101)(2,120)(3,119)(4,118)(5,117)(6,116)(7,115)(8,114)(9,113)(10,112)(11,111)(12,110)(13,109)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92), (1,72,93)(2,73,94)(3,74,95)(4,75,96)(5,76,97)(6,77,98)(7,78,99)(8,79,100)(9,80,81)(10,61,82)(11,62,83)(12,63,84)(13,64,85)(14,65,86)(15,66,87)(16,67,88)(17,68,89)(18,69,90)(19,70,91)(20,71,92)(21,41,103)(22,42,104)(23,43,105)(24,44,106)(25,45,107)(26,46,108)(27,47,109)(28,48,110)(29,49,111)(30,50,112)(31,51,113)(32,52,114)(33,53,115)(34,54,116)(35,55,117)(36,56,118)(37,57,119)(38,58,120)(39,59,101)(40,60,102), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,59)(22,48)(23,57)(24,46)(25,55)(26,44)(27,53)(28,42)(29,51)(30,60)(31,49)(32,58)(33,47)(34,56)(35,45)(36,54)(37,43)(38,52)(39,41)(40,50)(61,94)(62,83)(63,92)(64,81)(65,90)(66,99)(67,88)(68,97)(69,86)(70,95)(71,84)(72,93)(73,82)(74,91)(75,100)(76,89)(77,98)(78,87)(79,96)(80,85)(101,103)(102,112)(104,110)(105,119)(106,108)(107,117)(109,115)(111,113)(114,120)(116,118)>;
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), (1,101)(2,120)(3,119)(4,118)(5,117)(6,116)(7,115)(8,114)(9,113)(10,112)(11,111)(12,110)(13,109)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92), (1,72,93)(2,73,94)(3,74,95)(4,75,96)(5,76,97)(6,77,98)(7,78,99)(8,79,100)(9,80,81)(10,61,82)(11,62,83)(12,63,84)(13,64,85)(14,65,86)(15,66,87)(16,67,88)(17,68,89)(18,69,90)(19,70,91)(20,71,92)(21,41,103)(22,42,104)(23,43,105)(24,44,106)(25,45,107)(26,46,108)(27,47,109)(28,48,110)(29,49,111)(30,50,112)(31,51,113)(32,52,114)(33,53,115)(34,54,116)(35,55,117)(36,56,118)(37,57,119)(38,58,120)(39,59,101)(40,60,102), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,59)(22,48)(23,57)(24,46)(25,55)(26,44)(27,53)(28,42)(29,51)(30,60)(31,49)(32,58)(33,47)(34,56)(35,45)(36,54)(37,43)(38,52)(39,41)(40,50)(61,94)(62,83)(63,92)(64,81)(65,90)(66,99)(67,88)(68,97)(69,86)(70,95)(71,84)(72,93)(73,82)(74,91)(75,100)(76,89)(77,98)(78,87)(79,96)(80,85)(101,103)(102,112)(104,110)(105,119)(106,108)(107,117)(109,115)(111,113)(114,120)(116,118) );
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)], [(1,101),(2,120),(3,119),(4,118),(5,117),(6,116),(7,115),(8,114),(9,113),(10,112),(11,111),(12,110),(13,109),(14,108),(15,107),(16,106),(17,105),(18,104),(19,103),(20,102),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,84),(49,83),(50,82),(51,81),(52,100),(53,99),(54,98),(55,97),(56,96),(57,95),(58,94),(59,93),(60,92)], [(1,72,93),(2,73,94),(3,74,95),(4,75,96),(5,76,97),(6,77,98),(7,78,99),(8,79,100),(9,80,81),(10,61,82),(11,62,83),(12,63,84),(13,64,85),(14,65,86),(15,66,87),(16,67,88),(17,68,89),(18,69,90),(19,70,91),(20,71,92),(21,41,103),(22,42,104),(23,43,105),(24,44,106),(25,45,107),(26,46,108),(27,47,109),(28,48,110),(29,49,111),(30,50,112),(31,51,113),(32,52,114),(33,53,115),(34,54,116),(35,55,117),(36,56,118),(37,57,119),(38,58,120),(39,59,101),(40,60,102)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,59),(22,48),(23,57),(24,46),(25,55),(26,44),(27,53),(28,42),(29,51),(30,60),(31,49),(32,58),(33,47),(34,56),(35,45),(36,54),(37,43),(38,52),(39,41),(40,50),(61,94),(62,83),(63,92),(64,81),(65,90),(66,99),(67,88),(68,97),(69,86),(70,95),(71,84),(72,93),(73,82),(74,91),(75,100),(76,89),(77,98),(78,87),(79,96),(80,85),(101,103),(102,112),(104,110),(105,119),(106,108),(107,117),(109,115),(111,113),(114,120),(116,118)]])
D20⋊S3 is a maximal subgroup of
C40⋊8D6 Dic6.D10 D40⋊5S3 D30.3D4 D20.10D6 Dic6⋊D10 D20.16D6 D20.17D6 D20.38D6 D20⋊24D6 D20⋊25D6 D5×D4⋊2S3 D20⋊14D6 D20.29D6 S3×Q8⋊2D5
D20⋊S3 is a maximal quotient of
Dic15⋊7Q8 C4⋊Dic5⋊S3 Dic3.2Dic10 (C4×D15)⋊8C4 D30.34D4 (C2×C12).D10 (C4×Dic15)⋊C2 C60.88D4 D30⋊9Q8 C12.Dic10 D10.19(C4×S3) Dic15⋊13D4 (C6×D5).D4 Dic3⋊D20 D10.16D12 D10⋊1Dic6 D20⋊8Dic3 (C2×Dic6)⋊D5 C12⋊2D20
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 10 | 10 | 30 | 2 | 2 | 6 | 6 | 15 | 15 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D4⋊2S3 | S3×D5 | Q8⋊2D5 | C2×S3×D5 | D20⋊S3 |
| kernel | D20⋊S3 | D5×Dic3 | C3⋊D20 | C3×D20 | C5×Dic6 | C4×D15 | D20 | Dic6 | C20 | D10 | C15 | Dic3 | C12 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of D20⋊S3 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 50 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 59 | 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 | 1 | 41 |
| 0 | 0 | 0 | 0 | 52 | 59 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 20 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,17,0,0,0,0,0,0,50,50,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,52,0,0,0,0,41,59],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,20,1] >;
D20⋊S3 in GAP, Magma, Sage, TeX
D_{20}\rtimes S_3 % in TeX
G:=Group("D20:S3"); // GroupNames label
G:=SmallGroup(240,127);
// by ID
G=gap.SmallGroup(240,127);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations