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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, 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, C2×C30, C4.Dic5, D4⋊D5, D4⋊D5, D4.D5, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C5⋊D4, 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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 108)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 81)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)
(1 47 75 16 52 70)(2 58 76 7 53 61)(3 49 77 18 54 72)(4 60 78 9 55 63)(5 51 79 20 56 74)(6 42 80 11 57 65)(8 44 62 13 59 67)(10 46 64 15 41 69)(12 48 66 17 43 71)(14 50 68 19 45 73)(21 112 85 31 102 95)(22 103 86)(23 114 87 33 104 97)(24 105 88)(25 116 89 35 106 99)(26 107 90)(27 118 91 37 108 81)(28 109 92)(29 120 93 39 110 83)(30 111 94)(32 113 96)(34 115 98)(36 117 100)(38 119 82)(40 101 84)
(1 70)(2 61)(3 72)(4 63)(5 74)(6 65)(7 76)(8 67)(9 78)(10 69)(11 80)(12 71)(13 62)(14 73)(15 64)(16 75)(17 66)(18 77)(19 68)(20 79)(21 85)(22 96)(23 87)(24 98)(25 89)(26 100)(27 91)(28 82)(29 93)(30 84)(31 95)(32 86)(33 97)(34 88)(35 99)(36 90)(37 81)(38 92)(39 83)(40 94)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(101 111)(103 113)(105 115)(107 117)(109 119)
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94), (1,47,75,16,52,70)(2,58,76,7,53,61)(3,49,77,18,54,72)(4,60,78,9,55,63)(5,51,79,20,56,74)(6,42,80,11,57,65)(8,44,62,13,59,67)(10,46,64,15,41,69)(12,48,66,17,43,71)(14,50,68,19,45,73)(21,112,85,31,102,95)(22,103,86)(23,114,87,33,104,97)(24,105,88)(25,116,89,35,106,99)(26,107,90)(27,118,91,37,108,81)(28,109,92)(29,120,93,39,110,83)(30,111,94)(32,113,96)(34,115,98)(36,117,100)(38,119,82)(40,101,84), (1,70)(2,61)(3,72)(4,63)(5,74)(6,65)(7,76)(8,67)(9,78)(10,69)(11,80)(12,71)(13,62)(14,73)(15,64)(16,75)(17,66)(18,77)(19,68)(20,79)(21,85)(22,96)(23,87)(24,98)(25,89)(26,100)(27,91)(28,82)(29,93)(30,84)(31,95)(32,86)(33,97)(34,88)(35,99)(36,90)(37,81)(38,92)(39,83)(40,94)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(101,111)(103,113)(105,115)(107,117)(109,119)>;
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94), (1,47,75,16,52,70)(2,58,76,7,53,61)(3,49,77,18,54,72)(4,60,78,9,55,63)(5,51,79,20,56,74)(6,42,80,11,57,65)(8,44,62,13,59,67)(10,46,64,15,41,69)(12,48,66,17,43,71)(14,50,68,19,45,73)(21,112,85,31,102,95)(22,103,86)(23,114,87,33,104,97)(24,105,88)(25,116,89,35,106,99)(26,107,90)(27,118,91,37,108,81)(28,109,92)(29,120,93,39,110,83)(30,111,94)(32,113,96)(34,115,98)(36,117,100)(38,119,82)(40,101,84), (1,70)(2,61)(3,72)(4,63)(5,74)(6,65)(7,76)(8,67)(9,78)(10,69)(11,80)(12,71)(13,62)(14,73)(15,64)(16,75)(17,66)(18,77)(19,68)(20,79)(21,85)(22,96)(23,87)(24,98)(25,89)(26,100)(27,91)(28,82)(29,93)(30,84)(31,95)(32,86)(33,97)(34,88)(35,99)(36,90)(37,81)(38,92)(39,83)(40,94)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(101,111)(103,113)(105,115)(107,117)(109,119) );
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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,40),(17,39),(18,38),(19,37),(20,36),(41,107),(42,106),(43,105),(44,104),(45,103),(46,102),(47,101),(48,120),(49,119),(50,118),(51,117),(52,116),(53,115),(54,114),(55,113),(56,112),(57,111),(58,110),(59,109),(60,108),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82),(73,81),(74,100),(75,99),(76,98),(77,97),(78,96),(79,95),(80,94)], [(1,47,75,16,52,70),(2,58,76,7,53,61),(3,49,77,18,54,72),(4,60,78,9,55,63),(5,51,79,20,56,74),(6,42,80,11,57,65),(8,44,62,13,59,67),(10,46,64,15,41,69),(12,48,66,17,43,71),(14,50,68,19,45,73),(21,112,85,31,102,95),(22,103,86),(23,114,87,33,104,97),(24,105,88),(25,116,89,35,106,99),(26,107,90),(27,118,91,37,108,81),(28,109,92),(29,120,93,39,110,83),(30,111,94),(32,113,96),(34,115,98),(36,117,100),(38,119,82),(40,101,84)], [(1,70),(2,61),(3,72),(4,63),(5,74),(6,65),(7,76),(8,67),(9,78),(10,69),(11,80),(12,71),(13,62),(14,73),(15,64),(16,75),(17,66),(18,77),(19,68),(20,79),(21,85),(22,96),(23,87),(24,98),(25,89),(26,100),(27,91),(28,82),(29,93),(30,84),(31,95),(32,86),(33,97),(34,88),(35,99),(36,90),(37,81),(38,92),(39,83),(40,94),(41,46),(42,57),(43,48),(44,59),(45,50),(47,52),(49,54),(51,56),(53,58),(55,60),(101,111),(103,113),(105,115),(107,117),(109,119)]])
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