metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊1Dic3, D4⋊1(C3⋊F5), (C3×D4)⋊1F5, (D4×C15)⋊1C4, (C3×D20)⋊1C4, C5⋊(D4⋊Dic3), C60⋊C4⋊5C2, (C3×D5).6D8, (D4×D5).2S3, C12.9(C2×F5), C60.33(C2×C4), C3⋊3(D20⋊C4), (C5×D4)⋊1Dic3, (C6×D5).76D4, (C4×D5).27D6, C60.C4⋊6C2, C15⋊9(D4⋊C4), D5.3(D4⋊S3), (C3×D5).8SD16, C20.1(C2×Dic3), D5.3(D4.S3), (C3×Dic5).37D4, C6.20(C22⋊F5), D10.33(C3⋊D4), C30.20(C22⋊C4), Dic5.6(C3⋊D4), (D5×C12).68C22, C10.5(C6.D4), C2.6(D10.D6), C4.1(C2×C3⋊F5), (C3×D4×D5).2C2, SmallGroup(480,312)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊Dic3
G = < a,b,c,d | a20=b2=c6=1, d2=c3, bab=a-1, cac-1=a9, dad-1=a7, cbc-1=a8b, dbd-1=ab, dcd-1=c-1 >
Subgroups: 604 in 100 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, D4⋊C4, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C2×C3⋊C8, C4⋊Dic3, C6×D4, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, D5⋊C8, C4⋊F5, D4×D5, D4⋊Dic3, C15⋊C8, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, C2×C3⋊F5, D5×C2×C6, D20⋊C4, C60.C4, C60⋊C4, C3×D4×D5, D20⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, F5, C2×Dic3, C3⋊D4, D4⋊C4, C2×F5, D4⋊S3, D4.S3, C6.D4, C3⋊F5, C22⋊F5, D4⋊Dic3, C2×C3⋊F5, D20⋊C4, D10.D6, D20⋊Dic3
►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 44)(2 43)(3 42)(4 41)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)
(1 107 21)(2 116 22 10 108 30)(3 105 23 19 109 39)(4 114 24 8 110 28)(5 103 25 17 111 37)(6 112 26)(7 101 27 15 113 35)(9 119 29 13 115 33)(11 117 31)(12 106 32 20 118 40)(14 104 34 18 120 38)(16 102 36)(41 99 65 49 91 73)(42 88 66 58 92 62)(43 97 67 47 93 71)(44 86 68 56 94 80)(45 95 69)(46 84 70 54 96 78)(48 82 72 52 98 76)(50 100 74)(51 89 75 59 81 63)(53 87 77 57 83 61)(55 85 79)(60 90 64)
(2 4 10 8)(3 7 19 15)(5 13 17 9)(6 16)(12 14 20 18)(21 107)(22 110 30 114)(23 113 39 101)(24 116 28 108)(25 119 37 115)(26 102)(27 105 35 109)(29 111 33 103)(31 117)(32 120 40 104)(34 106 38 118)(36 112)(41 58 49 42)(43 44 47 56)(45 50)(46 53 54 57)(48 59 52 51)(55 60)(61 96 77 84)(62 99 66 91)(63 82 75 98)(64 85)(65 88 73 92)(67 94 71 86)(68 97 80 93)(69 100)(70 83 78 87)(72 89 76 81)(74 95)(79 90)
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), (1,44)(2,43)(3,42)(4,41)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101), (1,107,21)(2,116,22,10,108,30)(3,105,23,19,109,39)(4,114,24,8,110,28)(5,103,25,17,111,37)(6,112,26)(7,101,27,15,113,35)(9,119,29,13,115,33)(11,117,31)(12,106,32,20,118,40)(14,104,34,18,120,38)(16,102,36)(41,99,65,49,91,73)(42,88,66,58,92,62)(43,97,67,47,93,71)(44,86,68,56,94,80)(45,95,69)(46,84,70,54,96,78)(48,82,72,52,98,76)(50,100,74)(51,89,75,59,81,63)(53,87,77,57,83,61)(55,85,79)(60,90,64), (2,4,10,8)(3,7,19,15)(5,13,17,9)(6,16)(12,14,20,18)(21,107)(22,110,30,114)(23,113,39,101)(24,116,28,108)(25,119,37,115)(26,102)(27,105,35,109)(29,111,33,103)(31,117)(32,120,40,104)(34,106,38,118)(36,112)(41,58,49,42)(43,44,47,56)(45,50)(46,53,54,57)(48,59,52,51)(55,60)(61,96,77,84)(62,99,66,91)(63,82,75,98)(64,85)(65,88,73,92)(67,94,71,86)(68,97,80,93)(69,100)(70,83,78,87)(72,89,76,81)(74,95)(79,90)>;
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), (1,44)(2,43)(3,42)(4,41)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101), (1,107,21)(2,116,22,10,108,30)(3,105,23,19,109,39)(4,114,24,8,110,28)(5,103,25,17,111,37)(6,112,26)(7,101,27,15,113,35)(9,119,29,13,115,33)(11,117,31)(12,106,32,20,118,40)(14,104,34,18,120,38)(16,102,36)(41,99,65,49,91,73)(42,88,66,58,92,62)(43,97,67,47,93,71)(44,86,68,56,94,80)(45,95,69)(46,84,70,54,96,78)(48,82,72,52,98,76)(50,100,74)(51,89,75,59,81,63)(53,87,77,57,83,61)(55,85,79)(60,90,64), (2,4,10,8)(3,7,19,15)(5,13,17,9)(6,16)(12,14,20,18)(21,107)(22,110,30,114)(23,113,39,101)(24,116,28,108)(25,119,37,115)(26,102)(27,105,35,109)(29,111,33,103)(31,117)(32,120,40,104)(34,106,38,118)(36,112)(41,58,49,42)(43,44,47,56)(45,50)(46,53,54,57)(48,59,52,51)(55,60)(61,96,77,84)(62,99,66,91)(63,82,75,98)(64,85)(65,88,73,92)(67,94,71,86)(68,97,80,93)(69,100)(70,83,78,87)(72,89,76,81)(74,95)(79,90) );
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)], [(1,44),(2,43),(3,42),(4,41),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,68),(22,67),(23,66),(24,65),(25,64),(26,63),(27,62),(28,61),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,72),(38,71),(39,70),(40,69),(81,120),(82,119),(83,118),(84,117),(85,116),(86,115),(87,114),(88,113),(89,112),(90,111),(91,110),(92,109),(93,108),(94,107),(95,106),(96,105),(97,104),(98,103),(99,102),(100,101)], [(1,107,21),(2,116,22,10,108,30),(3,105,23,19,109,39),(4,114,24,8,110,28),(5,103,25,17,111,37),(6,112,26),(7,101,27,15,113,35),(9,119,29,13,115,33),(11,117,31),(12,106,32,20,118,40),(14,104,34,18,120,38),(16,102,36),(41,99,65,49,91,73),(42,88,66,58,92,62),(43,97,67,47,93,71),(44,86,68,56,94,80),(45,95,69),(46,84,70,54,96,78),(48,82,72,52,98,76),(50,100,74),(51,89,75,59,81,63),(53,87,77,57,83,61),(55,85,79),(60,90,64)], [(2,4,10,8),(3,7,19,15),(5,13,17,9),(6,16),(12,14,20,18),(21,107),(22,110,30,114),(23,113,39,101),(24,116,28,108),(25,119,37,115),(26,102),(27,105,35,109),(29,111,33,103),(31,117),(32,120,40,104),(34,106,38,118),(36,112),(41,58,49,42),(43,44,47,56),(45,50),(46,53,54,57),(48,59,52,51),(55,60),(61,96,77,84),(62,99,66,91),(63,82,75,98),(64,85),(65,88,73,92),(67,94,71,86),(68,97,80,93),(69,100),(70,83,78,87),(72,89,76,81),(74,95),(79,90)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 12A | 12B | 15A | 15B | 20 | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 5 | 5 | 20 | 2 | 2 | 10 | 60 | 60 | 4 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 30 | 30 | 30 | 30 | 4 | 8 | 8 | 4 | 20 | 4 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 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 | 8 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | - | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | Dic3 | Dic3 | D8 | SD16 | C3⋊D4 | C3⋊D4 | F5 | C2×F5 | D4⋊S3 | D4.S3 | C3⋊F5 | C22⋊F5 | C2×C3⋊F5 | D10.D6 | D20⋊C4 | D20⋊Dic3 |
| kernel | D20⋊Dic3 | C60.C4 | C60⋊C4 | C3×D4×D5 | C3×D20 | D4×C15 | D4×D5 | C3×Dic5 | C6×D5 | C4×D5 | D20 | C5×D4 | C3×D5 | C3×D5 | Dic5 | D10 | C3×D4 | C12 | D5 | D5 | D4 | C6 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 |
Matrix representation of D20⋊Dic3 ►in GL6(𝔽241)
| 240 | 96 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 22 | 149 | 0 | 0 | 0 | 0 |
| 186 | 219 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 117 | 124 | 234 |
| 0 | 0 | 124 | 0 | 117 | 234 |
| 0 | 0 | 7 | 234 | 117 | 0 |
| 0 | 0 | 0 | 234 | 124 | 117 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 114 | 127 | 229 | 0 |
| 0 | 0 | 0 | 115 | 229 | 12 |
| 0 | 0 | 229 | 115 | 0 | 126 |
| 0 | 0 | 229 | 127 | 114 | 12 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 162 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 240 |
G:=sub<GL(6,GF(241))| [240,5,0,0,0,0,96,1,0,0,0,0,0,0,1,1,1,1,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0],[22,186,0,0,0,0,149,219,0,0,0,0,0,0,7,124,7,0,0,0,117,0,234,234,0,0,124,117,117,124,0,0,234,234,0,117],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,114,0,229,229,0,0,127,115,115,127,0,0,229,229,0,114,0,0,0,12,126,12],[64,162,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,240,240,240,240] >;
D20⋊Dic3 in GAP, Magma, Sage, TeX
D_{20}\rtimes {\rm Dic}_3 % in TeX
G:=Group("D20:Dic3"); // GroupNames label
G:=SmallGroup(480,312);
// by ID
G=gap.SmallGroup(480,312);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^7,c*b*c^-1=a^8*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations