metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊16D6, D12⋊16D10, C60.62C23, C30.38C24, D30.42C23, Dic30⋊11C22, Dic15.22C23, (C4×D5)⋊11D6, (Q8×D15)⋊7C2, (C5×Q8)⋊15D6, Q8⋊10(S3×D5), (C4×S3)⋊11D10, C20⋊D6⋊6C2, Q8⋊2D5⋊9S3, (C3×Q8)⋊12D10, Q8⋊3S3⋊6D5, D15⋊3(C4○D4), D20⋊5S3⋊7C2, D12⋊5D5⋊7C2, (S3×C20)⋊8C22, (D5×C12)⋊8C22, C15⋊D4⋊6C22, C6.38(C23×D5), (C5×D12)⋊12C22, (C3×D20)⋊12C22, C20.62(C22×S3), C10.38(S3×C23), (Q8×C15)⋊10C22, D6.17(C22×D5), (C6×D5).16C23, C12.62(C22×D5), (S3×C10).19C23, (S3×Dic5)⋊14C22, (D5×Dic3)⋊14C22, (C4×D15).23C22, D10.19(C22×S3), D30.C2.19C22, (C3×Dic5).49C23, Dic5.59(C22×S3), (C5×Dic3).33C23, Dic3.36(C22×D5), (C4×S3×D5)⋊7C2, C5⋊6(S3×C4○D4), C3⋊6(D5×C4○D4), C4.62(C2×S3×D5), C15⋊16(C2×C4○D4), C2.41(C22×S3×D5), (C3×Q8⋊2D5)⋊6C2, (C5×Q8⋊3S3)⋊6C2, (C2×S3×D5).13C22, SmallGroup(480,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1660 in 328 conjugacy classes, 110 normal (24 characteristic)
C1, C2, C2 [×8], C3, C4 [×3], C4 [×5], C22 [×13], C5, S3 [×5], C6, C6 [×3], C2×C4 [×16], D4 [×12], Q8, Q8 [×3], C23 [×3], D5 [×5], C10, C10 [×3], Dic3, Dic3 [×3], C12 [×3], C12, D6 [×3], D6 [×7], C2×C6 [×3], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5, Dic5 [×3], C20 [×3], C20, D10 [×3], D10 [×7], C2×C10 [×3], Dic6 [×3], C4×S3 [×3], C4×S3 [×7], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C5×S3 [×3], C3×D5 [×3], D15 [×2], C30, C2×C4○D4, Dic10 [×3], C4×D5 [×3], C4×D5 [×7], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15 [×3], C60 [×3], S3×D5 [×6], C6×D5 [×3], S3×C10 [×3], D30, C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C5×C4○D4, S3×C4○D4, D5×Dic3 [×3], S3×Dic5 [×3], D30.C2, C15⋊D4 [×6], D5×C12 [×3], C3×D20 [×3], S3×C20 [×3], C5×D12 [×3], Dic30 [×3], C4×D15 [×3], Q8×C15, C2×S3×D5 [×3], D5×C4○D4, D20⋊5S3 [×3], D12⋊5D5 [×3], C4×S3×D5 [×3], C20⋊D6 [×3], C3×Q8⋊2D5, C5×Q8⋊3S3, Q8×D15, D20⋊16D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D20⋊16D6
Generators and relations
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 >
►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 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 76 89 102 42 36)(2 65 90 111 43 25)(3 74 91 120 44 34)(4 63 92 109 45 23)(5 72 93 118 46 32)(6 61 94 107 47 21)(7 70 95 116 48 30)(8 79 96 105 49 39)(9 68 97 114 50 28)(10 77 98 103 51 37)(11 66 99 112 52 26)(12 75 100 101 53 35)(13 64 81 110 54 24)(14 73 82 119 55 33)(15 62 83 108 56 22)(16 71 84 117 57 31)(17 80 85 106 58 40)(18 69 86 115 59 29)(19 78 87 104 60 38)(20 67 88 113 41 27)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 21)(17 22)(18 23)(19 24)(20 25)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 61)(58 62)(59 63)(60 64)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 101)(99 102)(100 103)
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)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,76,89,102,42,36)(2,65,90,111,43,25)(3,74,91,120,44,34)(4,63,92,109,45,23)(5,72,93,118,46,32)(6,61,94,107,47,21)(7,70,95,116,48,30)(8,79,96,105,49,39)(9,68,97,114,50,28)(10,77,98,103,51,37)(11,66,99,112,52,26)(12,75,100,101,53,35)(13,64,81,110,54,24)(14,73,82,119,55,33)(15,62,83,108,56,22)(16,71,84,117,57,31)(17,80,85,106,58,40)(18,69,86,115,59,29)(19,78,87,104,60,38)(20,67,88,113,41,27), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,61)(58,62)(59,63)(60,64)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,101)(99,102)(100,103)>;
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)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,76,89,102,42,36)(2,65,90,111,43,25)(3,74,91,120,44,34)(4,63,92,109,45,23)(5,72,93,118,46,32)(6,61,94,107,47,21)(7,70,95,116,48,30)(8,79,96,105,49,39)(9,68,97,114,50,28)(10,77,98,103,51,37)(11,66,99,112,52,26)(12,75,100,101,53,35)(13,64,81,110,54,24)(14,73,82,119,55,33)(15,62,83,108,56,22)(16,71,84,117,57,31)(17,80,85,106,58,40)(18,69,86,115,59,29)(19,78,87,104,60,38)(20,67,88,113,41,27), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,61)(58,62)(59,63)(60,64)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,101)(99,102)(100,103) );
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),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)], [(1,76,89,102,42,36),(2,65,90,111,43,25),(3,74,91,120,44,34),(4,63,92,109,45,23),(5,72,93,118,46,32),(6,61,94,107,47,21),(7,70,95,116,48,30),(8,79,96,105,49,39),(9,68,97,114,50,28),(10,77,98,103,51,37),(11,66,99,112,52,26),(12,75,100,101,53,35),(13,64,81,110,54,24),(14,73,82,119,55,33),(15,62,83,108,56,22),(16,71,84,117,57,31),(17,80,85,106,58,40),(18,69,86,115,59,29),(19,78,87,104,60,38),(20,67,88,113,41,27)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,21),(17,22),(18,23),(19,24),(20,25),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,61),(58,62),(59,63),(60,64),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,101),(99,102),(100,103)])
Matrix representation ►G ⊆ 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] >;
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 |
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