metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.6D6, D30.8D4, D12.6D10, C60.6C23, Dic15.40D4, Dic30⋊1C22, C3⋊C8⋊6D10, D4⋊D5⋊3S3, D4⋊S3⋊3D5, (C5×D4)⋊4D6, D4⋊6(S3×D5), C5⋊2C8⋊6D6, (C3×D4)⋊4D10, C5⋊3(D8⋊S3), C6.68(D4×D5), C20⋊D6⋊2C2, C3⋊3(D8⋊D5), C10.69(S3×D4), D4⋊2D15⋊1C2, C15⋊15(C8⋊C22), D12.D5⋊1C2, C6.D20⋊1C2, C30.168(C2×D4), (D4×C15)⋊6C22, C20.6(C22×S3), C12.6(C22×D5), D30.5C4⋊1C2, (C4×D15).2C22, (C5×D12).2C22, (C3×D20).2C22, C2.21(D10⋊D6), C4.6(C2×S3×D5), (C5×D4⋊S3)⋊4C2, (C3×D4⋊D5)⋊4C2, (C5×C3⋊C8)⋊4C22, (C3×C5⋊2C8)⋊4C22, SmallGroup(480,558)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.6D6
G = < a,b,c,d | a20=b2=c6=1, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a15b, dcd-1=a5c-1 >
Subgroups: 940 in 136 conjugacy classes, 38 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 [×2], C10, C10 [×2], Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, Dic15, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D8⋊S3, C5×C3⋊C8, C3×C5⋊2C8, C15⋊D4, C3×D20, C5×D12, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, C2×S3×D5, D8⋊D5, D30.5C4, C6.D20, D12.D5, C3×D4⋊D5, C5×D4⋊S3, C20⋊D6, D4⋊2D15, D20.6D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D8⋊D5, D10⋊D6, D20.6D6
►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 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 22 86 47 102 70)(2 33 87 58 103 61)(3 24 88 49 104 72)(4 35 89 60 105 63)(5 26 90 51 106 74)(6 37 91 42 107 65)(7 28 92 53 108 76)(8 39 93 44 109 67)(9 30 94 55 110 78)(10 21 95 46 111 69)(11 32 96 57 112 80)(12 23 97 48 113 71)(13 34 98 59 114 62)(14 25 99 50 115 73)(15 36 100 41 116 64)(16 27 81 52 117 75)(17 38 82 43 118 66)(18 29 83 54 119 77)(19 40 84 45 120 68)(20 31 85 56 101 79)
(1 70 6 75 11 80 16 65)(2 71 7 76 12 61 17 66)(3 72 8 77 13 62 18 67)(4 73 9 78 14 63 19 68)(5 74 10 79 15 64 20 69)(21 106 26 111 31 116 36 101)(22 107 27 112 32 117 37 102)(23 108 28 113 33 118 38 103)(24 109 29 114 34 119 39 104)(25 110 30 115 35 120 40 105)(41 85 46 90 51 95 56 100)(42 86 47 91 52 96 57 81)(43 87 48 92 53 97 58 82)(44 88 49 93 54 98 59 83)(45 89 50 94 55 99 60 84)
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)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,22,86,47,102,70)(2,33,87,58,103,61)(3,24,88,49,104,72)(4,35,89,60,105,63)(5,26,90,51,106,74)(6,37,91,42,107,65)(7,28,92,53,108,76)(8,39,93,44,109,67)(9,30,94,55,110,78)(10,21,95,46,111,69)(11,32,96,57,112,80)(12,23,97,48,113,71)(13,34,98,59,114,62)(14,25,99,50,115,73)(15,36,100,41,116,64)(16,27,81,52,117,75)(17,38,82,43,118,66)(18,29,83,54,119,77)(19,40,84,45,120,68)(20,31,85,56,101,79), (1,70,6,75,11,80,16,65)(2,71,7,76,12,61,17,66)(3,72,8,77,13,62,18,67)(4,73,9,78,14,63,19,68)(5,74,10,79,15,64,20,69)(21,106,26,111,31,116,36,101)(22,107,27,112,32,117,37,102)(23,108,28,113,33,118,38,103)(24,109,29,114,34,119,39,104)(25,110,30,115,35,120,40,105)(41,85,46,90,51,95,56,100)(42,86,47,91,52,96,57,81)(43,87,48,92,53,97,58,82)(44,88,49,93,54,98,59,83)(45,89,50,94,55,99,60,84)>;
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)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,22,86,47,102,70)(2,33,87,58,103,61)(3,24,88,49,104,72)(4,35,89,60,105,63)(5,26,90,51,106,74)(6,37,91,42,107,65)(7,28,92,53,108,76)(8,39,93,44,109,67)(9,30,94,55,110,78)(10,21,95,46,111,69)(11,32,96,57,112,80)(12,23,97,48,113,71)(13,34,98,59,114,62)(14,25,99,50,115,73)(15,36,100,41,116,64)(16,27,81,52,117,75)(17,38,82,43,118,66)(18,29,83,54,119,77)(19,40,84,45,120,68)(20,31,85,56,101,79), (1,70,6,75,11,80,16,65)(2,71,7,76,12,61,17,66)(3,72,8,77,13,62,18,67)(4,73,9,78,14,63,19,68)(5,74,10,79,15,64,20,69)(21,106,26,111,31,116,36,101)(22,107,27,112,32,117,37,102)(23,108,28,113,33,118,38,103)(24,109,29,114,34,119,39,104)(25,110,30,115,35,120,40,105)(41,85,46,90,51,95,56,100)(42,86,47,91,52,96,57,81)(43,87,48,92,53,97,58,82)(44,88,49,93,54,98,59,83)(45,89,50,94,55,99,60,84) );
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),(21,27),(22,26),(23,25),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)], [(1,22,86,47,102,70),(2,33,87,58,103,61),(3,24,88,49,104,72),(4,35,89,60,105,63),(5,26,90,51,106,74),(6,37,91,42,107,65),(7,28,92,53,108,76),(8,39,93,44,109,67),(9,30,94,55,110,78),(10,21,95,46,111,69),(11,32,96,57,112,80),(12,23,97,48,113,71),(13,34,98,59,114,62),(14,25,99,50,115,73),(15,36,100,41,116,64),(16,27,81,52,117,75),(17,38,82,43,118,66),(18,29,83,54,119,77),(19,40,84,45,120,68),(20,31,85,56,101,79)], [(1,70,6,75,11,80,16,65),(2,71,7,76,12,61,17,66),(3,72,8,77,13,62,18,67),(4,73,9,78,14,63,19,68),(5,74,10,79,15,64,20,69),(21,106,26,111,31,116,36,101),(22,107,27,112,32,117,37,102),(23,108,28,113,33,118,38,103),(24,109,29,114,34,119,39,104),(25,110,30,115,35,120,40,105),(41,85,46,90,51,95,56,100),(42,86,47,91,52,96,57,81),(43,87,48,92,53,97,58,82),(44,88,49,93,54,98,59,83),(45,89,50,94,55,99,60,84)])
42 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 | 12 | 15A | 15B | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 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 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 12 | 20 | 30 | 2 | 2 | 30 | 60 | 2 | 2 | 2 | 8 | 40 | 12 | 20 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
42 irreducible representations
| dim | 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 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | D8⋊S3 | C2×S3×D5 | D8⋊D5 | D10⋊D6 | D20.6D6 |
| kernel | D20.6D6 | D30.5C4 | C6.D20 | D12.D5 | C3×D4⋊D5 | C5×D4⋊S3 | C20⋊D6 | D4⋊2D15 | D4⋊D5 | Dic15 | D30 | D4⋊S3 | C5⋊2C8 | D20 | C5×D4 | C3⋊C8 | D12 | C3×D4 | C15 | C10 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.6D6 ►in GL6(𝔽241)
| 189 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 189 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 194 | 47 | 194 | 47 |
| 0 | 0 | 194 | 147 | 194 | 147 |
| 0 | 0 | 194 | 47 | 47 | 194 |
| 0 | 0 | 194 | 147 | 47 | 94 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 194 | 194 | 47 |
| 0 | 0 | 147 | 194 | 94 | 47 |
| 0 | 0 | 47 | 194 | 47 | 194 |
| 0 | 0 | 147 | 194 | 147 | 194 |
G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,189,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,194,194,194,194,0,0,47,147,47,147,0,0,194,194,47,47,0,0,47,147,194,94],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,147,47,147,0,0,194,194,194,194,0,0,194,94,47,147,0,0,47,47,194,194] >;
D20.6D6 in GAP, Magma, Sage, TeX
D_{20}._6D_6 % in TeX
G:=Group("D20.6D6"); // GroupNames label
G:=SmallGroup(480,558);
// by ID
G=gap.SmallGroup(480,558);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations