metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊21D6, C60.59D4, Dic10⋊18D6, D12.35D10, C60.152C23, C4○D20⋊3S3, (C2×D12)⋊8D5, (C10×D12)⋊1C2, C15⋊D8⋊14C2, C5⋊4(D4⋊D6), C15⋊8(C8⋊C22), C30.74(C2×D4), (C2×C30).42D4, (C2×C20).86D6, C60.7C4⋊3C2, (C2×C12).87D10, C15⋊3C8⋊22C22, C20.D6⋊13C2, C3⋊4(D4.D10), (C3×D20)⋊28C22, C4.16(C15⋊D4), C12.82(C5⋊D4), C20.24(C3⋊D4), (C2×C60).24C22, C20.87(C22×S3), C12.87(C22×D5), (C5×D12).41C22, C22.9(C15⋊D4), (C3×Dic10)⋊24C22, (C2×C4).8(S3×D5), C4.125(C2×S3×D5), (C3×C4○D20)⋊1C2, C6.74(C2×C5⋊D4), C2.8(C2×C15⋊D4), C10.75(C2×C3⋊D4), (C2×C6).10(C5⋊D4), (C2×C10).52(C3⋊D4), SmallGroup(480,375)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊21D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, dbd=a5b, dcd=c-1 >
Subgroups: 620 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C12, C12, D6, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C22×C10, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×D12, C3×C4○D4, C3×Dic5, C60, C6×D5, S3×C10, C2×C30, C4.Dic5, D4⋊D5, D4.D5, C4○D20, D4×C10, D4⋊D6, C15⋊3C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×D12, C5×D12, C2×C60, S3×C2×C10, D4.D10, C15⋊D8, C20.D6, C60.7C4, C3×C4○D20, C10×D12, D20⋊21D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8⋊C22, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, D4⋊D6, C15⋊D4, C2×S3×D5, D4.D10, C2×C15⋊D4, D20⋊21D6
►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 96)(2 95)(3 94)(4 93)(5 92)(6 91)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 84)(14 83)(15 82)(16 81)(17 100)(18 99)(19 98)(20 97)(21 79)(22 78)(23 77)(24 76)(25 75)(26 74)(27 73)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 80)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 120)(53 119)(54 118)(55 117)(56 116)(57 115)(58 114)(59 113)(60 112)
(1 113 73)(2 114 74)(3 115 75)(4 116 76)(5 117 77)(6 118 78)(7 119 79)(8 120 80)(9 101 61)(10 102 62)(11 103 63)(12 104 64)(13 105 65)(14 106 66)(15 107 67)(16 108 68)(17 109 69)(18 110 70)(19 111 71)(20 112 72)(21 100 53 31 90 43)(22 81 54 32 91 44)(23 82 55 33 92 45)(24 83 56 34 93 46)(25 84 57 35 94 47)(26 85 58 36 95 48)(27 86 59 37 96 49)(28 87 60 38 97 50)(29 88 41 39 98 51)(30 89 42 40 99 52)
(1 73)(2 64)(3 75)(4 66)(5 77)(6 68)(7 79)(8 70)(9 61)(10 72)(11 63)(12 74)(13 65)(14 76)(15 67)(16 78)(17 69)(18 80)(19 71)(20 62)(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 56)(42 47)(43 58)(44 49)(45 60)(46 51)(48 53)(50 55)(52 57)(54 59)(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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,100)(18,99)(19,98)(20,97)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,80)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,120)(53,119)(54,118)(55,117)(56,116)(57,115)(58,114)(59,113)(60,112), (1,113,73)(2,114,74)(3,115,75)(4,116,76)(5,117,77)(6,118,78)(7,119,79)(8,120,80)(9,101,61)(10,102,62)(11,103,63)(12,104,64)(13,105,65)(14,106,66)(15,107,67)(16,108,68)(17,109,69)(18,110,70)(19,111,71)(20,112,72)(21,100,53,31,90,43)(22,81,54,32,91,44)(23,82,55,33,92,45)(24,83,56,34,93,46)(25,84,57,35,94,47)(26,85,58,36,95,48)(27,86,59,37,96,49)(28,87,60,38,97,50)(29,88,41,39,98,51)(30,89,42,40,99,52), (1,73)(2,64)(3,75)(4,66)(5,77)(6,68)(7,79)(8,70)(9,61)(10,72)(11,63)(12,74)(13,65)(14,76)(15,67)(16,78)(17,69)(18,80)(19,71)(20,62)(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)(41,56)(42,47)(43,58)(44,49)(45,60)(46,51)(48,53)(50,55)(52,57)(54,59)(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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,100)(18,99)(19,98)(20,97)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,80)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,120)(53,119)(54,118)(55,117)(56,116)(57,115)(58,114)(59,113)(60,112), (1,113,73)(2,114,74)(3,115,75)(4,116,76)(5,117,77)(6,118,78)(7,119,79)(8,120,80)(9,101,61)(10,102,62)(11,103,63)(12,104,64)(13,105,65)(14,106,66)(15,107,67)(16,108,68)(17,109,69)(18,110,70)(19,111,71)(20,112,72)(21,100,53,31,90,43)(22,81,54,32,91,44)(23,82,55,33,92,45)(24,83,56,34,93,46)(25,84,57,35,94,47)(26,85,58,36,95,48)(27,86,59,37,96,49)(28,87,60,38,97,50)(29,88,41,39,98,51)(30,89,42,40,99,52), (1,73)(2,64)(3,75)(4,66)(5,77)(6,68)(7,79)(8,70)(9,61)(10,72)(11,63)(12,74)(13,65)(14,76)(15,67)(16,78)(17,69)(18,80)(19,71)(20,62)(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)(41,56)(42,47)(43,58)(44,49)(45,60)(46,51)(48,53)(50,55)(52,57)(54,59)(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)], [(1,96),(2,95),(3,94),(4,93),(5,92),(6,91),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,84),(14,83),(15,82),(16,81),(17,100),(18,99),(19,98),(20,97),(21,79),(22,78),(23,77),(24,76),(25,75),(26,74),(27,73),(28,72),(29,71),(30,70),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64),(37,63),(38,62),(39,61),(40,80),(41,111),(42,110),(43,109),(44,108),(45,107),(46,106),(47,105),(48,104),(49,103),(50,102),(51,101),(52,120),(53,119),(54,118),(55,117),(56,116),(57,115),(58,114),(59,113),(60,112)], [(1,113,73),(2,114,74),(3,115,75),(4,116,76),(5,117,77),(6,118,78),(7,119,79),(8,120,80),(9,101,61),(10,102,62),(11,103,63),(12,104,64),(13,105,65),(14,106,66),(15,107,67),(16,108,68),(17,109,69),(18,110,70),(19,111,71),(20,112,72),(21,100,53,31,90,43),(22,81,54,32,91,44),(23,82,55,33,92,45),(24,83,56,34,93,46),(25,84,57,35,94,47),(26,85,58,36,95,48),(27,86,59,37,96,49),(28,87,60,38,97,50),(29,88,41,39,98,51),(30,89,42,40,99,52)], [(1,73),(2,64),(3,75),(4,66),(5,77),(6,68),(7,79),(8,70),(9,61),(10,72),(11,63),(12,74),(13,65),(14,76),(15,67),(16,78),(17,69),(18,80),(19,71),(20,62),(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),(41,56),(42,47),(43,58),(44,49),(45,60),(46,51),(48,53),(50,55),(52,57),(54,59),(102,112),(104,114),(106,116),(108,118),(110,120)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 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 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 12 | 20 | 2 | 2 | 2 | 20 | 2 | 2 | 2 | 4 | 20 | 20 | 60 | 60 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | C5⋊D4 | C5⋊D4 | C8⋊C22 | S3×D5 | D4⋊D6 | C15⋊D4 | C2×S3×D5 | C15⋊D4 | D4.D10 | D20⋊21D6 |
| kernel | D20⋊21D6 | C15⋊D8 | C20.D6 | C60.7C4 | C3×C4○D20 | C10×D12 | C4○D20 | C60 | C2×C30 | C2×D12 | Dic10 | D20 | C2×C20 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D20⋊21D6 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 91 | 0 | 0 |
| 0 | 0 | 150 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 143 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 143 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 91 | 0 | 0 |
| 0 | 0 | 150 | 0 | 0 | 0 |
| 240 | 28 | 0 | 0 | 0 | 0 |
| 129 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 2 | 213 | 0 | 0 | 0 | 0 |
| 112 | 239 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,150,0,0,0,0,91,0,0,0,0,0,0,0,0,98,0,0,0,0,143,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,150,0,0,0,0,91,0,0,0,0,98,0,0,0,0,143,0,0,0],[240,129,0,0,0,0,28,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[2,112,0,0,0,0,213,239,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D20⋊21D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{21}D_6 % in TeX
G:=Group("D20:21D6"); // GroupNames label
G:=SmallGroup(480,375);
// by ID
G=gap.SmallGroup(480,375);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,100,675,346,80,1356,18822]);
// 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,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations