metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊17D6, D12⋊17D10, D60⋊13C22, C30.39C24, C60.63C23, C15⋊72+ 1+4, D30.20C23, Dic15.42C23, (C4×D5)⋊6D6, (D5×D12)⋊7C2, (C4×S3)⋊6D10, (S3×D20)⋊7C2, C5⋊4(D4○D12), (C5×Q8)⋊16D6, Q8⋊11(S3×D5), C20⋊D6⋊7C2, Q8⋊2D5⋊8S3, Q8⋊3S3⋊5D5, (C3×Q8)⋊13D10, C15⋊Q8⋊19C22, Q8⋊3D15⋊7C2, C3⋊4(D4⋊8D10), (S3×C20)⋊9C22, (C4×D15)⋊9C22, (D5×C12)⋊9C22, C6.39(C23×D5), D6.D10⋊8C2, (C5×D12)⋊13C22, (C3×D20)⋊13C22, C15⋊D4⋊17C22, C3⋊D20⋊16C22, C5⋊D12⋊17C22, C10.39(S3×C23), C20.63(C22×S3), (Q8×C15)⋊11C22, (C6×D5).17C23, D6.18(C22×D5), C12.63(C22×D5), (S3×C10).20C23, D10.20(C22×S3), (C3×Dic5).50C23, (C5×Dic3).34C23, Dic3.29(C22×D5), Dic5.45(C22×S3), C4.63(C2×S3×D5), (C2×S3×D5)⋊7C22, C2.42(C22×S3×D5), (C5×Q8⋊3S3)⋊7C2, (C3×Q8⋊2D5)⋊7C2, SmallGroup(480,1111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊17D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a8b, dcd=c-1 >
Subgroups: 2044 in 332 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, 2+ 1+4, Dic10, C4×D5, C4×D5, D20, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×D12, C4○D12, S3×D4, Q8⋊3S3, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C2×D20, C4○D20, D4×D5, Q8⋊2D5, Q8⋊2D5, C5×C4○D4, D4○D12, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C3×D20, S3×C20, C5×D12, C4×D15, D60, Q8×C15, C2×S3×D5, D4⋊8D10, D6.D10, D5×D12, S3×D20, C20⋊D6, C3×Q8⋊2D5, C5×Q8⋊3S3, Q8⋊3D15, D20⋊17D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2+ 1+4, C22×D5, S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5, D4⋊8D10, C22×S3×D5, D20⋊17D6
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)
(1 94 60 78 119 30)(2 83 41 67 120 39)(3 92 42 76 101 28)(4 81 43 65 102 37)(5 90 44 74 103 26)(6 99 45 63 104 35)(7 88 46 72 105 24)(8 97 47 61 106 33)(9 86 48 70 107 22)(10 95 49 79 108 31)(11 84 50 68 109 40)(12 93 51 77 110 29)(13 82 52 66 111 38)(14 91 53 75 112 27)(15 100 54 64 113 36)(16 89 55 73 114 25)(17 98 56 62 115 34)(18 87 57 71 116 23)(19 96 58 80 117 32)(20 85 59 69 118 21)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 60)(17 59)(18 58)(19 57)(20 56)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,100)(98,99)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119), (1,94,60,78,119,30)(2,83,41,67,120,39)(3,92,42,76,101,28)(4,81,43,65,102,37)(5,90,44,74,103,26)(6,99,45,63,104,35)(7,88,46,72,105,24)(8,97,47,61,106,33)(9,86,48,70,107,22)(10,95,49,79,108,31)(11,84,50,68,109,40)(12,93,51,77,110,29)(13,82,52,66,111,38)(14,91,53,75,112,27)(15,100,54,64,113,36)(16,89,55,73,114,25)(17,98,56,62,115,34)(18,87,57,71,116,23)(19,96,58,80,117,32)(20,85,59,69,118,21), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,60)(17,59)(18,58)(19,57)(20,56)(21,62)(22,61)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,100)(98,99)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119), (1,94,60,78,119,30)(2,83,41,67,120,39)(3,92,42,76,101,28)(4,81,43,65,102,37)(5,90,44,74,103,26)(6,99,45,63,104,35)(7,88,46,72,105,24)(8,97,47,61,106,33)(9,86,48,70,107,22)(10,95,49,79,108,31)(11,84,50,68,109,40)(12,93,51,77,110,29)(13,82,52,66,111,38)(14,91,53,75,112,27)(15,100,54,64,113,36)(16,89,55,73,114,25)(17,98,56,62,115,34)(18,87,57,71,116,23)(19,96,58,80,117,32)(20,85,59,69,118,21), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,60)(17,59)(18,58)(19,57)(20,56)(21,62)(22,61)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,64),(62,63),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89),(97,100),(98,99),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109),(117,120),(118,119)], [(1,94,60,78,119,30),(2,83,41,67,120,39),(3,92,42,76,101,28),(4,81,43,65,102,37),(5,90,44,74,103,26),(6,99,45,63,104,35),(7,88,46,72,105,24),(8,97,47,61,106,33),(9,86,48,70,107,22),(10,95,49,79,108,31),(11,84,50,68,109,40),(12,93,51,77,110,29),(13,82,52,66,111,38),(14,91,53,75,112,27),(15,100,54,64,113,36),(16,89,55,73,114,25),(17,98,56,62,115,34),(18,87,57,71,116,23),(19,96,58,80,117,32),(20,85,59,69,118,21)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,60),(17,59),(18,58),(19,57),(20,56),(21,62),(22,61),(23,80),(24,79),(25,78),(26,77),(27,76),(28,75),(29,74),(30,73),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63),(81,82),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,120),(114,119),(115,118),(116,117)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20F | 20G | 20H | 20I | 20J | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 6 | 6 | 10 | 10 | 10 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 6 | 10 | 30 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | 2+ 1+4 | S3×D5 | D4○D12 | C2×S3×D5 | D4⋊8D10 | D20⋊17D6 |
| kernel | D20⋊17D6 | D6.D10 | D5×D12 | S3×D20 | C20⋊D6 | C3×Q8⋊2D5 | C5×Q8⋊3S3 | Q8⋊3D15 | Q8⋊2D5 | Q8⋊3S3 | C4×D5 | D20 | C5×Q8 | C4×S3 | D12 | C3×Q8 | C15 | Q8 | C5 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 1 | 2 | 2 | 6 | 4 | 2 |
Matrix representation of D20⋊17D6 ►in GL8(𝔽61)
| 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 42 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 36 | 57 | 25 |
| 0 | 0 | 0 | 0 | 25 | 27 | 36 | 34 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 36 |
| 0 | 0 | 0 | 0 | 0 | 0 | 25 | 57 |
| 25 | 27 | 46 | 26 | 0 | 0 | 0 | 0 |
| 38 | 36 | 41 | 35 | 0 | 0 | 0 | 0 |
| 21 | 21 | 60 | 0 | 0 | 0 | 0 | 0 |
| 12 | 52 | 44 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 | 25 | 59 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 43 | 60 |
| 36 | 34 | 15 | 35 | 0 | 0 | 0 | 0 |
| 23 | 25 | 20 | 26 | 0 | 0 | 0 | 0 |
| 40 | 40 | 2 | 0 | 0 | 0 | 0 | 0 |
| 49 | 9 | 34 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 57 | 36 | 4 |
| 0 | 0 | 0 | 0 | 34 | 36 | 27 | 25 |
G:=sub<GL(8,GF(61))| [18,42,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,4,25,4,25,0,0,0,0,36,27,36,27,0,0,0,0,0,0,57,36,0,0,0,0,0,0,25,34],[60,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,36,57,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,36,57],[25,38,21,12,0,0,0,0,27,36,21,52,0,0,0,0,46,41,60,44,0,0,0,0,26,35,0,1,0,0,0,0,0,0,0,0,60,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,25,1,43,0,0,0,0,0,59,0,60],[36,23,40,49,0,0,0,0,34,25,40,9,0,0,0,0,15,20,2,34,0,0,0,0,35,26,0,59,0,0,0,0,0,0,0,0,25,34,25,34,0,0,0,0,57,36,57,36,0,0,0,0,0,0,36,27,0,0,0,0,0,0,4,25] >;
D20⋊17D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{17}D_6 % in TeX
G:=Group("D20:17D6"); // GroupNames label
G:=SmallGroup(480,1111);
// by ID
G=gap.SmallGroup(480,1111);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations