metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊10D6, D12.8D10, C60.18C23, Dic30⋊5C22, D4⋊D5⋊4S3, (S3×D4)⋊2D5, C5⋊2C8⋊7D6, C15⋊D8⋊4C2, C5⋊6(D8⋊S3), D4.4(S3×D5), D4.D15⋊8C2, (C3×D4).6D10, (C4×S3).7D10, (C5×D4).21D6, D20⋊5S3⋊2C2, C15⋊18(C8⋊C22), C15⋊3C8⋊9C22, (S3×C10).33D4, C10.144(S3×D4), C30.180(C2×D4), D12.D5⋊3C2, (C3×D20)⋊4C22, D6.Dic5⋊3C2, D6.13(C5⋊D4), C3⋊2(D4.D10), (S3×C20).6C22, C20.18(C22×S3), (C5×Dic3).13D4, (C5×D12).5C22, C12.18(C22×D5), (D4×C15).12C22, Dic3.10(C5⋊D4), (C5×S3×D4)⋊2C2, C4.18(C2×S3×D5), (C3×D4⋊D5)⋊6C2, C2.25(S3×C5⋊D4), C6.47(C2×C5⋊D4), (C3×C5⋊2C8)⋊7C22, SmallGroup(480,570)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊10D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=a15b, dbd=a5b, dcd=c-1 >
Subgroups: 684 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5, C10, C10 [×3], Dic3, Dic3, C12, D6, D6 [×3], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10 [×5], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C5×S3 [×2], C3×D5, C30, C30, C8⋊C22, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4 [×2], C22×C10, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C5×Dic3, Dic15, C60, C6×D5, S3×C10, S3×C10 [×3], C2×C30, C4.Dic5, D4⋊D5, D4⋊D5, D4.D5 [×2], C4○D20, D4×C10, D8⋊S3, C3×C5⋊2C8, C15⋊3C8, D5×Dic3, C15⋊D4, C3×D20, S3×C20, C5×D12, C5×C3⋊D4, Dic30, D4×C15, S3×C2×C10, D4.D10, D6.Dic5, C15⋊D8, D12.D5, C3×D4⋊D5, D4.D15, D20⋊5S3, C5×S3×D4, D20⋊10D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D8⋊S3, C2×S3×D5, D4.D10, S3×C5⋊D4, D20⋊10D6
►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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 104)(13 103)(14 102)(15 101)(16 120)(17 119)(18 118)(19 117)(20 116)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 100)(38 99)(39 98)(40 97)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 80)
(1 53 28 16 58 23)(2 44 29 7 59 34)(3 55 30 18 60 25)(4 46 31 9 41 36)(5 57 32 20 42 27)(6 48 33 11 43 38)(8 50 35 13 45 40)(10 52 37 15 47 22)(12 54 39 17 49 24)(14 56 21 19 51 26)(61 88 114)(62 99 115 72 89 105)(63 90 116)(64 81 117 74 91 107)(65 92 118)(66 83 119 76 93 109)(67 94 120)(68 85 101 78 95 111)(69 96 102)(70 87 103 80 97 113)(71 98 104)(73 100 106)(75 82 108)(77 84 110)(79 86 112)
(1 23)(2 34)(3 25)(4 36)(5 27)(6 38)(7 29)(8 40)(9 31)(10 22)(11 33)(12 24)(13 35)(14 26)(15 37)(16 28)(17 39)(18 30)(19 21)(20 32)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(61 71)(63 73)(65 75)(67 77)(69 79)(81 107)(82 118)(83 109)(84 120)(85 111)(86 102)(87 113)(88 104)(89 115)(90 106)(91 117)(92 108)(93 119)(94 110)(95 101)(96 112)(97 103)(98 114)(99 105)(100 116)
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), (1,115)(2,114)(3,113)(4,112)(5,111)(6,110)(7,109)(8,108)(9,107)(10,106)(11,105)(12,104)(13,103)(14,102)(15,101)(16,120)(17,119)(18,118)(19,117)(20,116)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,100)(38,99)(39,98)(40,97)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,80), (1,53,28,16,58,23)(2,44,29,7,59,34)(3,55,30,18,60,25)(4,46,31,9,41,36)(5,57,32,20,42,27)(6,48,33,11,43,38)(8,50,35,13,45,40)(10,52,37,15,47,22)(12,54,39,17,49,24)(14,56,21,19,51,26)(61,88,114)(62,99,115,72,89,105)(63,90,116)(64,81,117,74,91,107)(65,92,118)(66,83,119,76,93,109)(67,94,120)(68,85,101,78,95,111)(69,96,102)(70,87,103,80,97,113)(71,98,104)(73,100,106)(75,82,108)(77,84,110)(79,86,112), (1,23)(2,34)(3,25)(4,36)(5,27)(6,38)(7,29)(8,40)(9,31)(10,22)(11,33)(12,24)(13,35)(14,26)(15,37)(16,28)(17,39)(18,30)(19,21)(20,32)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(61,71)(63,73)(65,75)(67,77)(69,79)(81,107)(82,118)(83,109)(84,120)(85,111)(86,102)(87,113)(88,104)(89,115)(90,106)(91,117)(92,108)(93,119)(94,110)(95,101)(96,112)(97,103)(98,114)(99,105)(100,116)>;
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), (1,115)(2,114)(3,113)(4,112)(5,111)(6,110)(7,109)(8,108)(9,107)(10,106)(11,105)(12,104)(13,103)(14,102)(15,101)(16,120)(17,119)(18,118)(19,117)(20,116)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,100)(38,99)(39,98)(40,97)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,80), (1,53,28,16,58,23)(2,44,29,7,59,34)(3,55,30,18,60,25)(4,46,31,9,41,36)(5,57,32,20,42,27)(6,48,33,11,43,38)(8,50,35,13,45,40)(10,52,37,15,47,22)(12,54,39,17,49,24)(14,56,21,19,51,26)(61,88,114)(62,99,115,72,89,105)(63,90,116)(64,81,117,74,91,107)(65,92,118)(66,83,119,76,93,109)(67,94,120)(68,85,101,78,95,111)(69,96,102)(70,87,103,80,97,113)(71,98,104)(73,100,106)(75,82,108)(77,84,110)(79,86,112), (1,23)(2,34)(3,25)(4,36)(5,27)(6,38)(7,29)(8,40)(9,31)(10,22)(11,33)(12,24)(13,35)(14,26)(15,37)(16,28)(17,39)(18,30)(19,21)(20,32)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(61,71)(63,73)(65,75)(67,77)(69,79)(81,107)(82,118)(83,109)(84,120)(85,111)(86,102)(87,113)(88,104)(89,115)(90,106)(91,117)(92,108)(93,119)(94,110)(95,101)(96,112)(97,103)(98,114)(99,105)(100,116) );
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)], [(1,115),(2,114),(3,113),(4,112),(5,111),(6,110),(7,109),(8,108),(9,107),(10,106),(11,105),(12,104),(13,103),(14,102),(15,101),(16,120),(17,119),(18,118),(19,117),(20,116),(21,96),(22,95),(23,94),(24,93),(25,92),(26,91),(27,90),(28,89),(29,88),(30,87),(31,86),(32,85),(33,84),(34,83),(35,82),(36,81),(37,100),(38,99),(39,98),(40,97),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,80)], [(1,53,28,16,58,23),(2,44,29,7,59,34),(3,55,30,18,60,25),(4,46,31,9,41,36),(5,57,32,20,42,27),(6,48,33,11,43,38),(8,50,35,13,45,40),(10,52,37,15,47,22),(12,54,39,17,49,24),(14,56,21,19,51,26),(61,88,114),(62,99,115,72,89,105),(63,90,116),(64,81,117,74,91,107),(65,92,118),(66,83,119,76,93,109),(67,94,120),(68,85,101,78,95,111),(69,96,102),(70,87,103,80,97,113),(71,98,104),(73,100,106),(75,82,108),(77,84,110),(79,86,112)], [(1,23),(2,34),(3,25),(4,36),(5,27),(6,38),(7,29),(8,40),(9,31),(10,22),(11,33),(12,24),(13,35),(14,26),(15,37),(16,28),(17,39),(18,30),(19,21),(20,32),(41,46),(42,57),(43,48),(44,59),(45,50),(47,52),(49,54),(51,56),(53,58),(55,60),(61,71),(63,73),(65,75),(67,77),(69,79),(81,107),(82,118),(83,109),(84,120),(85,111),(86,102),(87,113),(88,104),(89,115),(90,106),(91,117),(92,108),(93,119),(94,110),(95,101),(96,112),(97,103),(98,114),(99,105),(100,116)])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 6 | 12 | 20 | 2 | 2 | 6 | 60 | 2 | 2 | 2 | 8 | 40 | 20 | 60 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | S3×D4 | S3×D5 | D8⋊S3 | C2×S3×D5 | D4.D10 | S3×C5⋊D4 | D20⋊10D6 |
| kernel | D20⋊10D6 | D6.Dic5 | C15⋊D8 | D12.D5 | C3×D4⋊D5 | D4.D15 | D20⋊5S3 | C5×S3×D4 | D4⋊D5 | C5×Dic3 | S3×C10 | S3×D4 | C5⋊2C8 | D20 | C5×D4 | C4×S3 | D12 | C3×D4 | Dic3 | D6 | C15 | C10 | D4 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20⋊10D6 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 87 | 20 | 0 | 0 |
| 0 | 0 | 183 | 154 | 0 | 0 |
| 0 | 0 | 66 | 196 | 0 | 36 |
| 0 | 0 | 145 | 196 | 205 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 108 |
| 0 | 0 | 79 | 0 | 205 | 205 |
| 0 | 0 | 145 | 87 | 0 | 239 |
| 0 | 0 | 87 | 0 | 0 | 239 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 174 | 1 | 0 |
| 0 | 0 | 116 | 174 | 0 | 240 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 116 | 174 | 240 | 0 |
| 0 | 0 | 0 | 174 | 0 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,87,183,66,145,0,0,20,154,196,196,0,0,0,0,0,205,0,0,0,0,36,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,79,145,87,0,0,0,0,87,0,0,0,0,205,0,0,0,0,108,205,239,239],[1,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,116,0,0,3,240,174,174,0,0,0,0,1,0,0,0,0,0,0,240],[1,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0,116,0,0,0,3,240,174,174,0,0,0,0,240,0,0,0,0,0,0,1] >;
D20⋊10D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{10}D_6 % in TeX
G:=Group("D20:10D6"); // GroupNames label
G:=SmallGroup(480,570);
// by ID
G=gap.SmallGroup(480,570);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=a^15*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations