metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊26D6, D12⋊28D10, Dic10⋊23D6, D60⋊40C22, C30.22C24, D30.8C23, C15⋊22+ 1+4, C60.165C23, Dic30⋊37C22, Dic15.11C23, (C4×D5)⋊1D6, (C2×C20)⋊9D6, C4○D20⋊8S3, (C2×C12)⋊9D10, C5⋊1(D4○D12), C5⋊D4⋊14D6, (C10×D12)⋊3C2, (D5×D12)⋊11C2, (C2×D12)⋊14D5, (C2×C60)⋊3C22, C20⋊D6⋊12C2, C3⋊1(D4⋊6D10), (C22×S3)⋊4D10, C5⋊D12⋊1C22, C15⋊D4⋊2C22, D12⋊D5⋊12C2, D12⋊5D5⋊12C2, D60⋊11C2⋊4C2, D6.8(C22×D5), (C6×D5).8C23, C6.22(C23×D5), (C3×D20)⋊32C22, (C4×D15)⋊13C22, (C5×D12)⋊25C22, (D5×C12)⋊10C22, C15⋊7D4⋊16C22, (S3×C10).8C23, C10.22(S3×C23), (S3×Dic5)⋊1C22, D10.8(C22×S3), (C2×C30).241C23, C20.131(C22×S3), C12.162(C22×D5), (C3×Dic10)⋊29C22, Dic5.10(C22×S3), (C3×Dic5).10C23, (C2×C4)⋊5(S3×D5), C4.88(C2×S3×D5), (S3×C5⋊D4)⋊1C2, (C2×S3×D5)⋊2C22, (C3×C4○D20)⋊3C2, (S3×C2×C10)⋊6C22, C2.25(C22×S3×D5), C22.15(C2×S3×D5), (C3×C5⋊D4)⋊10C22, (C2×C6).12(C22×D5), (C2×C10).248(C22×S3), SmallGroup(480,1094)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊26D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, bd=db, dcd=c-1 >
Subgroups: 1868 in 332 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C5×S3, C3×D5, D15, C30, C30, 2+ 1+4, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C2×D12, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, S3×C10, D30, C2×C30, C4○D20, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, D4×C10, D4○D12, S3×Dic5, C15⋊D4, C5⋊D12, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×D12, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C2×S3×D5, S3×C2×C10, D4⋊6D10, D12⋊D5, D12⋊5D5, D5×D12, C20⋊D6, S3×C5⋊D4, C3×C4○D20, C10×D12, D60⋊11C2, D20⋊26D6
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⋊6D10, C22×S3×D5, D20⋊26D6
►On 120 points
(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 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(81 102)(82 101)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)
(1 103 30)(2 104 31)(3 105 32)(4 106 33)(5 107 34)(6 108 35)(7 109 36)(8 110 37)(9 111 38)(10 112 39)(11 113 40)(12 114 21)(13 115 22)(14 116 23)(15 117 24)(16 118 25)(17 119 26)(18 120 27)(19 101 28)(20 102 29)(41 68 84 51 78 94)(42 69 85 52 79 95)(43 70 86 53 80 96)(44 71 87 54 61 97)(45 72 88 55 62 98)(46 73 89 56 63 99)(47 74 90 57 64 100)(48 75 91 58 65 81)(49 76 92 59 66 82)(50 77 93 60 67 83)
(1 30)(2 21)(3 32)(4 23)(5 34)(6 25)(7 36)(8 27)(9 38)(10 29)(11 40)(12 31)(13 22)(14 33)(15 24)(16 35)(17 26)(18 37)(19 28)(20 39)(41 78)(42 69)(43 80)(44 71)(45 62)(46 73)(47 64)(48 75)(49 66)(50 77)(51 68)(52 79)(53 70)(54 61)(55 72)(56 63)(57 74)(58 65)(59 76)(60 67)(81 91)(83 93)(85 95)(87 97)(89 99)(102 112)(104 114)(106 116)(108 118)(110 120)
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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103), (1,103,30)(2,104,31)(3,105,32)(4,106,33)(5,107,34)(6,108,35)(7,109,36)(8,110,37)(9,111,38)(10,112,39)(11,113,40)(12,114,21)(13,115,22)(14,116,23)(15,117,24)(16,118,25)(17,119,26)(18,120,27)(19,101,28)(20,102,29)(41,68,84,51,78,94)(42,69,85,52,79,95)(43,70,86,53,80,96)(44,71,87,54,61,97)(45,72,88,55,62,98)(46,73,89,56,63,99)(47,74,90,57,64,100)(48,75,91,58,65,81)(49,76,92,59,66,82)(50,77,93,60,67,83), (1,30)(2,21)(3,32)(4,23)(5,34)(6,25)(7,36)(8,27)(9,38)(10,29)(11,40)(12,31)(13,22)(14,33)(15,24)(16,35)(17,26)(18,37)(19,28)(20,39)(41,78)(42,69)(43,80)(44,71)(45,62)(46,73)(47,64)(48,75)(49,66)(50,77)(51,68)(52,79)(53,70)(54,61)(55,72)(56,63)(57,74)(58,65)(59,76)(60,67)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120)>;
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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103), (1,103,30)(2,104,31)(3,105,32)(4,106,33)(5,107,34)(6,108,35)(7,109,36)(8,110,37)(9,111,38)(10,112,39)(11,113,40)(12,114,21)(13,115,22)(14,116,23)(15,117,24)(16,118,25)(17,119,26)(18,120,27)(19,101,28)(20,102,29)(41,68,84,51,78,94)(42,69,85,52,79,95)(43,70,86,53,80,96)(44,71,87,54,61,97)(45,72,88,55,62,98)(46,73,89,56,63,99)(47,74,90,57,64,100)(48,75,91,58,65,81)(49,76,92,59,66,82)(50,77,93,60,67,83), (1,30)(2,21)(3,32)(4,23)(5,34)(6,25)(7,36)(8,27)(9,38)(10,29)(11,40)(12,31)(13,22)(14,33)(15,24)(16,35)(17,26)(18,37)(19,28)(20,39)(41,78)(42,69)(43,80)(44,71)(45,62)(46,73)(47,64)(48,75)(49,66)(50,77)(51,68)(52,79)(53,70)(54,61)(55,72)(56,63)(57,74)(58,65)(59,76)(60,67)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120) );
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)], [(1,74),(2,73),(3,72),(4,71),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,63),(13,62),(14,61),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,60),(28,59),(29,58),(30,57),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(81,102),(82,101),(83,120),(84,119),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103)], [(1,103,30),(2,104,31),(3,105,32),(4,106,33),(5,107,34),(6,108,35),(7,109,36),(8,110,37),(9,111,38),(10,112,39),(11,113,40),(12,114,21),(13,115,22),(14,116,23),(15,117,24),(16,118,25),(17,119,26),(18,120,27),(19,101,28),(20,102,29),(41,68,84,51,78,94),(42,69,85,52,79,95),(43,70,86,53,80,96),(44,71,87,54,61,97),(45,72,88,55,62,98),(46,73,89,56,63,99),(47,74,90,57,64,100),(48,75,91,58,65,81),(49,76,92,59,66,82),(50,77,93,60,67,83)], [(1,30),(2,21),(3,32),(4,23),(5,34),(6,25),(7,36),(8,27),(9,38),(10,29),(11,40),(12,31),(13,22),(14,33),(15,24),(16,35),(17,26),(18,37),(19,28),(20,39),(41,78),(42,69),(43,80),(44,71),(45,62),(46,73),(47,64),(48,75),(49,66),(50,77),(51,68),(52,79),(53,70),(54,61),(55,72),(56,63),(57,74),(58,65),(59,76),(60,67),(81,91),(83,93),(85,95),(87,97),(89,99),(102,112),(104,114),(106,116),(108,118),(110,120)]])
63 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 | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | 2+ 1+4 | S3×D5 | D4○D12 | C2×S3×D5 | C2×S3×D5 | D4⋊6D10 | D20⋊26D6 |
| kernel | D20⋊26D6 | D12⋊D5 | D12⋊5D5 | D5×D12 | C20⋊D6 | S3×C5⋊D4 | C3×C4○D20 | C10×D12 | D60⋊11C2 | C4○D20 | C2×D12 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | D12 | C2×C12 | C22×S3 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 4 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of D20⋊26D6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 0 | 20 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,3,0,0,0,0,58,0,0,0,0,0,0,0,0,20,0,0,0,0,41,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,3,0,0,0,0,58,0,0,0,0,20,0,0,0,0,41,0,0,0],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1] >;
D20⋊26D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{26}D_6 % in TeX
G:=Group("D20:26D6"); // GroupNames label
G:=SmallGroup(480,1094);
// by ID
G=gap.SmallGroup(480,1094);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations