metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊19D6, C60.60D4, C12.51D20, Dic10⋊16D6, C60.102C23, D60.43C22, C3⋊C8⋊3D10, C4○D20⋊4S3, (C2×C6).5D20, (C2×D60)⋊17C2, C5⋊1(D4⋊D6), C3⋊D40⋊13C2, C3⋊5(C8⋊D10), C15⋊9(C8⋊C22), C6.47(C2×D20), (C2×C30).44D4, C30.76(C2×D4), (C2×C20).87D6, C4.Dic3⋊7D5, (C2×C12).89D10, C15⋊SD16⋊13C2, (C3×D20)⋊21C22, C4.16(C3⋊D20), C20.25(C3⋊D4), (C2×C60).93C22, C12.89(C22×D5), C20.152(C22×S3), C22.9(C3⋊D20), (C3×Dic10)⋊18C22, (C2×C4).9(S3×D5), C4.101(C2×S3×D5), (C3×C4○D20)⋊6C2, (C5×C3⋊C8)⋊17C22, C2.6(C2×C3⋊D20), C10.2(C2×C3⋊D4), (C5×C4.Dic3)⋊8C2, (C2×C10).10(C3⋊D4), SmallGroup(480,377)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊19D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a10b, dbd=a13b, dcd=c-1 >
Subgroups: 1004 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, C3⋊C8, D12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C3×D5, D15, C30, C30, C8⋊C22, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C22×D5, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×D12, C3×C4○D4, C3×Dic5, C60, C6×D5, D30, C2×C30, C40⋊C2, D40, C5×M4(2), C2×D20, C4○D20, D4⋊D6, C5×C3⋊C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, D60, D60, C2×C60, C22×D15, C8⋊D10, C3⋊D40, C15⋊SD16, C5×C4.Dic3, C3×C4○D20, C2×D60, D20⋊19D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8⋊C22, D20, C22×D5, C2×C3⋊D4, S3×D5, C2×D20, D4⋊D6, C3⋊D20, C2×S3×D5, C8⋊D10, C2×C3⋊D20, D20⋊19D6
►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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 60)(17 59)(18 58)(19 57)(20 56)(21 63)(22 62)(23 61)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 72)(33 71)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)
(1 77 97)(2 78 98)(3 79 99)(4 80 100)(5 61 81)(6 62 82)(7 63 83)(8 64 84)(9 65 85)(10 66 86)(11 67 87)(12 68 88)(13 69 89)(14 70 90)(15 71 91)(16 72 92)(17 73 93)(18 74 94)(19 75 95)(20 76 96)(21 118 49 31 108 59)(22 119 50 32 109 60)(23 120 51 33 110 41)(24 101 52 34 111 42)(25 102 53 35 112 43)(26 103 54 36 113 44)(27 104 55 37 114 45)(28 105 56 38 115 46)(29 106 57 39 116 47)(30 107 58 40 117 48)
(1 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 100)(19 99)(20 98)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 120)(57 119)(58 118)(59 117)(60 116)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)
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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,60)(17,59)(18,58)(19,57)(20,56)(21,63)(22,62)(23,61)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111), (1,77,97)(2,78,98)(3,79,99)(4,80,100)(5,61,81)(6,62,82)(7,63,83)(8,64,84)(9,65,85)(10,66,86)(11,67,87)(12,68,88)(13,69,89)(14,70,90)(15,71,91)(16,72,92)(17,73,93)(18,74,94)(19,75,95)(20,76,96)(21,118,49,31,108,59)(22,119,50,32,109,60)(23,120,51,33,110,41)(24,101,52,34,111,42)(25,102,53,35,112,43)(26,103,54,36,113,44)(27,104,55,37,114,45)(28,105,56,38,115,46)(29,106,57,39,116,47)(30,107,58,40,117,48), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,100)(19,99)(20,98)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,120)(57,119)(58,118)(59,117)(60,116)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)>;
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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,60)(17,59)(18,58)(19,57)(20,56)(21,63)(22,62)(23,61)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111), (1,77,97)(2,78,98)(3,79,99)(4,80,100)(5,61,81)(6,62,82)(7,63,83)(8,64,84)(9,65,85)(10,66,86)(11,67,87)(12,68,88)(13,69,89)(14,70,90)(15,71,91)(16,72,92)(17,73,93)(18,74,94)(19,75,95)(20,76,96)(21,118,49,31,108,59)(22,119,50,32,109,60)(23,120,51,33,110,41)(24,101,52,34,111,42)(25,102,53,35,112,43)(26,103,54,36,113,44)(27,104,55,37,114,45)(28,105,56,38,115,46)(29,106,57,39,116,47)(30,107,58,40,117,48), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,100)(19,99)(20,98)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,120)(57,119)(58,118)(59,117)(60,116)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78) );
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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,60),(17,59),(18,58),(19,57),(20,56),(21,63),(22,62),(23,61),(24,80),(25,79),(26,78),(27,77),(28,76),(29,75),(30,74),(31,73),(32,72),(33,71),(34,70),(35,69),(36,68),(37,67),(38,66),(39,65),(40,64),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111)], [(1,77,97),(2,78,98),(3,79,99),(4,80,100),(5,61,81),(6,62,82),(7,63,83),(8,64,84),(9,65,85),(10,66,86),(11,67,87),(12,68,88),(13,69,89),(14,70,90),(15,71,91),(16,72,92),(17,73,93),(18,74,94),(19,75,95),(20,76,96),(21,118,49,31,108,59),(22,119,50,32,109,60),(23,120,51,33,110,41),(24,101,52,34,111,42),(25,102,53,35,112,43),(26,103,54,36,113,44),(27,104,55,37,114,45),(28,105,56,38,115,46),(29,106,57,39,116,47),(30,107,58,40,117,48)], [(1,97),(2,96),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,88),(11,87),(12,86),(13,85),(14,84),(15,83),(16,82),(17,81),(18,100),(19,99),(20,98),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,115),(42,114),(43,113),(44,112),(45,111),(46,110),(47,109),(48,108),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,120),(57,119),(58,118),(59,117),(60,116),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 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 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 60 | 60 | 2 | 2 | 2 | 20 | 2 | 2 | 2 | 4 | 20 | 20 | 12 | 12 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 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 | D20 | D20 | C8⋊C22 | S3×D5 | D4⋊D6 | C3⋊D20 | C2×S3×D5 | C3⋊D20 | C8⋊D10 | D20⋊19D6 |
| kernel | D20⋊19D6 | C3⋊D40 | C15⋊SD16 | C5×C4.Dic3 | C3×C4○D20 | C2×D60 | C4○D20 | C60 | C2×C30 | C4.Dic3 | Dic10 | D20 | C2×C20 | C3⋊C8 | 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⋊19D6 ►in GL4(𝔽241) generated by
| 3 | 44 | 0 | 0 |
| 197 | 78 | 0 | 0 |
| 0 | 0 | 3 | 44 |
| 0 | 0 | 197 | 78 |
| 0 | 0 | 3 | 44 |
| 0 | 0 | 197 | 238 |
| 3 | 44 | 0 | 0 |
| 197 | 238 | 0 | 0 |
| 147 | 84 | 0 | 0 |
| 157 | 93 | 0 | 0 |
| 0 | 0 | 148 | 84 |
| 0 | 0 | 157 | 94 |
| 147 | 84 | 0 | 0 |
| 110 | 94 | 0 | 0 |
| 0 | 0 | 6 | 119 |
| 0 | 0 | 184 | 235 |
G:=sub<GL(4,GF(241))| [3,197,0,0,44,78,0,0,0,0,3,197,0,0,44,78],[0,0,3,197,0,0,44,238,3,197,0,0,44,238,0,0],[147,157,0,0,84,93,0,0,0,0,148,157,0,0,84,94],[147,110,0,0,84,94,0,0,0,0,6,184,0,0,119,235] >;
D20⋊19D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{19}D_6 % in TeX
G:=Group("D20:19D6"); // GroupNames label
G:=SmallGroup(480,377);
// by ID
G=gap.SmallGroup(480,377);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,219,100,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^10*b,d*b*d=a^13*b,d*c*d=c^-1>;
// generators/relations