metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5S3, D10.1D6, D6.5D10, C20.26D6, Dic30⋊6C2, C12.13D10, C30.2C23, C60.12C22, Dic3.10D10, Dic15.2C22, (C4×S3)⋊1D5, C4.5(S3×D5), (S3×C20)⋊1C2, (C3×D20)⋊2C2, C15⋊1(C4○D4), C3⋊3(C4○D20), C15⋊D4⋊1C2, C5⋊2(D4⋊2S3), (D5×Dic3)⋊1C2, C6.2(C22×D5), C10.2(C22×S3), (C6×D5).1C22, (S3×C10).5C22, (C5×Dic3).7C22, C2.6(C2×S3×D5), SmallGroup(240,126)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊5S3
G = < a,b,c,d | a20=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 328 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C4○D4, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C5×S3, C3×D5, C30, Dic10, C4×D5, D20, C5⋊D4, C2×C20, D4⋊2S3, C5×Dic3, Dic15, C60, C6×D5, S3×C10, C4○D20, D5×Dic3, C15⋊D4, C3×D20, S3×C20, Dic30, D20⋊5S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, D4⋊2S3, S3×D5, C4○D20, C2×S3×D5, D20⋊5S3
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)
(1 47 24)(2 48 25)(3 49 26)(4 50 27)(5 51 28)(6 52 29)(7 53 30)(8 54 31)(9 55 32)(10 56 33)(11 57 34)(12 58 35)(13 59 36)(14 60 37)(15 41 38)(16 42 39)(17 43 40)(18 44 21)(19 45 22)(20 46 23)(61 119 94)(62 120 95)(63 101 96)(64 102 97)(65 103 98)(66 104 99)(67 105 100)(68 106 81)(69 107 82)(70 108 83)(71 109 84)(72 110 85)(73 111 86)(74 112 87)(75 113 88)(76 114 89)(77 115 90)(78 116 91)(79 117 92)(80 118 93)
(1 80)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,89)(82,88)(83,87)(84,86)(90,100)(91,99)(92,98)(93,97)(94,96)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111), (1,47,24)(2,48,25)(3,49,26)(4,50,27)(5,51,28)(6,52,29)(7,53,30)(8,54,31)(9,55,32)(10,56,33)(11,57,34)(12,58,35)(13,59,36)(14,60,37)(15,41,38)(16,42,39)(17,43,40)(18,44,21)(19,45,22)(20,46,23)(61,119,94)(62,120,95)(63,101,96)(64,102,97)(65,103,98)(66,104,99)(67,105,100)(68,106,81)(69,107,82)(70,108,83)(71,109,84)(72,110,85)(73,111,86)(74,112,87)(75,113,88)(76,114,89)(77,115,90)(78,116,91)(79,117,92)(80,118,93), (1,80)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,89)(82,88)(83,87)(84,86)(90,100)(91,99)(92,98)(93,97)(94,96)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111), (1,47,24)(2,48,25)(3,49,26)(4,50,27)(5,51,28)(6,52,29)(7,53,30)(8,54,31)(9,55,32)(10,56,33)(11,57,34)(12,58,35)(13,59,36)(14,60,37)(15,41,38)(16,42,39)(17,43,40)(18,44,21)(19,45,22)(20,46,23)(61,119,94)(62,120,95)(63,101,96)(64,102,97)(65,103,98)(66,104,99)(67,105,100)(68,106,81)(69,107,82)(70,108,83)(71,109,84)(72,110,85)(73,111,86)(74,112,87)(75,113,88)(76,114,89)(77,115,90)(78,116,91)(79,117,92)(80,118,93), (1,80)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,89),(82,88),(83,87),(84,86),(90,100),(91,99),(92,98),(93,97),(94,96),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111)], [(1,47,24),(2,48,25),(3,49,26),(4,50,27),(5,51,28),(6,52,29),(7,53,30),(8,54,31),(9,55,32),(10,56,33),(11,57,34),(12,58,35),(13,59,36),(14,60,37),(15,41,38),(16,42,39),(17,43,40),(18,44,21),(19,45,22),(20,46,23),(61,119,94),(62,120,95),(63,101,96),(64,102,97),(65,103,98),(66,104,99),(67,105,100),(68,106,81),(69,107,82),(70,108,83),(71,109,84),(72,110,85),(73,111,86),(74,112,87),(75,113,88),(76,114,89),(77,115,90),(78,116,91),(79,117,92),(80,118,93)], [(1,80),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,99),(54,100),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86)]])
D20⋊5S3 is a maximal subgroup of
D40⋊S3 D6.1D20 D40⋊7S3 C40.2D6 D20.24D6 D20⋊10D6 D20.27D6 D20.28D6 D20.39D6 S3×C4○D20 D20⋊25D6 D5×D4⋊2S3 D20⋊13D6 D20.29D6 D20⋊16D6
D20⋊5S3 is a maximal quotient of
(S3×C20)⋊5C4 Dic15.2Q8 Dic30⋊14C4 D6⋊C4.D5 (C2×C60).C22 (C4×Dic3)⋊D5 C60.44D4 C60.45D4 C60.6Q8 (D5×Dic3)⋊C4 (C6×D5).D4 Dic15⋊D4 D10.17D12 Dic3×D20 D10⋊4Dic6 C15⋊17(C4×D4) D10⋊C4⋊S3 C60⋊4D4 Dic15.10D4
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 6 | 10 | 10 | 2 | 2 | 3 | 3 | 30 | 30 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 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 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D20 | D4⋊2S3 | S3×D5 | C2×S3×D5 | D20⋊5S3 |
| kernel | D20⋊5S3 | D5×Dic3 | C15⋊D4 | C3×D20 | S3×C20 | Dic30 | D20 | C4×S3 | C20 | D10 | C15 | Dic3 | C12 | D6 | C3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of D20⋊5S3 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 4 | 36 |
| 0 | 0 | 25 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 60 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 44 |
| 0 | 0 | 17 | 31 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,4,25,0,0,36,27],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,30,17,0,0,44,31] >;
D20⋊5S3 in GAP, Magma, Sage, TeX
D_{20}\rtimes_5S_3 % in TeX
G:=Group("D20:5S3"); // GroupNames label
G:=SmallGroup(240,126);
// by ID
G=gap.SmallGroup(240,126);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,50,490,6917]);
// 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,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations