metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊16D6, D12⋊16D10, C30.38C24, C60.62C23, D30.42C23, Dic30⋊11C22, Dic15.22C23, (C4×D5)⋊11D6, Q8⋊10(S3×D5), (C5×Q8)⋊15D6, (Q8×D15)⋊7C2, (C4×S3)⋊11D10, C20⋊D6⋊6C2, Q8⋊2D5⋊9S3, Q8⋊3S3⋊6D5, (C3×Q8)⋊12D10, D15⋊3(C4○D4), D12⋊5D5⋊7C2, D20⋊5S3⋊7C2, (S3×C20)⋊8C22, (D5×C12)⋊8C22, C15⋊D4⋊6C22, C6.38(C23×D5), (C5×D12)⋊12C22, (C3×D20)⋊12C22, C10.38(S3×C23), C20.62(C22×S3), (Q8×C15)⋊10C22, D6.17(C22×D5), (C6×D5).16C23, C12.62(C22×D5), (S3×C10).19C23, (S3×Dic5)⋊14C22, (D5×Dic3)⋊14C22, D10.19(C22×S3), (C4×D15).23C22, D30.C2.19C22, Dic3.36(C22×D5), (C5×Dic3).33C23, Dic5.59(C22×S3), (C3×Dic5).49C23, (C4×S3×D5)⋊7C2, C3⋊6(D5×C4○D4), C5⋊6(S3×C4○D4), C4.62(C2×S3×D5), C15⋊16(C2×C4○D4), C2.41(C22×S3×D5), (C5×Q8⋊3S3)⋊6C2, (C3×Q8⋊2D5)⋊6C2, (C2×S3×D5).13C22, SmallGroup(480,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊16D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=a9, ad=da, cbc-1=a18b, dbd=a10b, dcd=c-1 >
Subgroups: 1660 in 328 conjugacy classes, 110 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, S3×C4○D4, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, D5×C12, C3×D20, S3×C20, C5×D12, Dic30, C4×D15, Q8×C15, C2×S3×D5, D5×C4○D4, D20⋊5S3, D12⋊5D5, C4×S3×D5, C20⋊D6, C3×Q8⋊2D5, C5×Q8⋊3S3, Q8×D15, D20⋊16D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5, D5×C4○D4, C22×S3×D5, D20⋊16D6
►On 120 points
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 22)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 32 52 72 111 92)(2 21 53 61 112 81)(3 30 54 70 113 90)(4 39 55 79 114 99)(5 28 56 68 115 88)(6 37 57 77 116 97)(7 26 58 66 117 86)(8 35 59 75 118 95)(9 24 60 64 119 84)(10 33 41 73 120 93)(11 22 42 62 101 82)(12 31 43 71 102 91)(13 40 44 80 103 100)(14 29 45 69 104 89)(15 38 46 78 105 98)(16 27 47 67 106 87)(17 36 48 76 107 96)(18 25 49 65 108 85)(19 34 50 74 109 94)(20 23 51 63 110 83)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 81)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116), (1,32,52,72,111,92)(2,21,53,61,112,81)(3,30,54,70,113,90)(4,39,55,79,114,99)(5,28,56,68,115,88)(6,37,57,77,116,97)(7,26,58,66,117,86)(8,35,59,75,118,95)(9,24,60,64,119,84)(10,33,41,73,120,93)(11,22,42,62,101,82)(12,31,43,71,102,91)(13,40,44,80,103,100)(14,29,45,69,104,89)(15,38,46,78,105,98)(16,27,47,67,106,87)(17,36,48,76,107,96)(18,25,49,65,108,85)(19,34,50,74,109,94)(20,23,51,63,110,83), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)>;
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116), (1,32,52,72,111,92)(2,21,53,61,112,81)(3,30,54,70,113,90)(4,39,55,79,114,99)(5,28,56,68,115,88)(6,37,57,77,116,97)(7,26,58,66,117,86)(8,35,59,75,118,95)(9,24,60,64,119,84)(10,33,41,73,120,93)(11,22,42,62,101,82)(12,31,43,71,102,91)(13,40,44,80,103,100)(14,29,45,69,104,89)(15,38,46,78,105,98)(16,27,47,67,106,87)(17,36,48,76,107,96)(18,25,49,65,108,85)(19,34,50,74,109,94)(20,23,51,63,110,83), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70) );
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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,22),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,82),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116)], [(1,32,52,72,111,92),(2,21,53,61,112,81),(3,30,54,70,113,90),(4,39,55,79,114,99),(5,28,56,68,115,88),(6,37,57,77,116,97),(7,26,58,66,117,86),(8,35,59,75,118,95),(9,24,60,64,119,84),(10,33,41,73,120,93),(11,22,42,62,101,82),(12,31,43,71,102,91),(13,40,44,80,103,100),(14,29,45,69,104,89),(15,38,46,78,105,98),(16,27,47,67,106,87),(17,36,48,76,107,96),(18,25,49,65,108,85),(19,34,50,74,109,94),(20,23,51,63,110,83)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,81),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20F | 20G | 20H | 20I | 20J | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 6 | 6 | 10 | 10 | 10 | 15 | 15 | 2 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 30 | 30 | 30 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | S3×D5 | S3×C4○D4 | C2×S3×D5 | D5×C4○D4 | D20⋊16D6 |
| kernel | D20⋊16D6 | D20⋊5S3 | D12⋊5D5 | C4×S3×D5 | C20⋊D6 | C3×Q8⋊2D5 | C5×Q8⋊3S3 | Q8×D15 | Q8⋊2D5 | Q8⋊3S3 | C4×D5 | D20 | C5×Q8 | D15 | C4×S3 | D12 | C3×Q8 | Q8 | C5 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 4 | 6 | 6 | 2 | 2 | 2 | 6 | 4 | 2 |
Matrix representation of D20⋊16D6 ►in GL6(𝔽61)
| 43 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 1 | 43 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 18 | 43 | 0 | 0 | 0 | 0 |
| 1 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 |
G:=sub<GL(6,GF(61))| [43,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,11,0,0,0,0,11,0],[1,0,0,0,0,0,43,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,11,0,0,0,0,50,0],[18,1,0,0,0,0,43,43,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,60,0] >;
D20⋊16D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{16}D_6 % in TeX
G:=Group("D20:16D6"); // GroupNames label
G:=SmallGroup(480,1110);
// by ID
G=gap.SmallGroup(480,1110);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,185,80,1356,18822]);
// 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=a^9,a*d=d*a,c*b*c^-1=a^18*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations