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G = D2016D6order 480 = 25·3·5

10th semidirect product of D20 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2016D6, D1216D10, C30.38C24, C60.62C23, D30.42C23, Dic3011C22, Dic15.22C23, (C4×D5)⋊11D6, Q810(S3×D5), (C5×Q8)⋊15D6, (Q8×D15)⋊7C2, (C4×S3)⋊11D10, C20⋊D66C2, Q82D59S3, Q83S36D5, (C3×Q8)⋊12D10, D153(C4○D4), D125D57C2, D205S37C2, (S3×C20)⋊8C22, (D5×C12)⋊8C22, C15⋊D46C22, 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, C36(D5×C4○D4), C56(S3×C4○D4), C4.62(C2×S3×D5), C1516(C2×C4○D4), C2.41(C22×S3×D5), (C5×Q83S3)⋊6C2, (C3×Q82D5)⋊6C2, (C2×S3×D5).13C22, SmallGroup(480,1110)

Series: Derived Chief Lower central Upper central

C1C30 — D2016D6
C1C5C15C30C6×D5C2×S3×D5C4×S3×D5 — D2016D6
C15C30 — D2016D6
C1C2Q8

Generators and relations for D2016D6
 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 [×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], D42S3 [×3], S3×Q8, Q83S3, 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], D42D5 [×3], Q8×D5, Q82D5, 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, D205S3 [×3], D125D5 [×3], C4×S3×D5 [×3], C20⋊D6 [×3], C3×Q82D5, C5×Q83S3, Q8×D15, D2016D6
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, D2016D6

Smallest permutation representation of D2016D6
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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 58 98 40 101 62)(2 47 99 29 102 71)(3 56 100 38 103 80)(4 45 81 27 104 69)(5 54 82 36 105 78)(6 43 83 25 106 67)(7 52 84 34 107 76)(8 41 85 23 108 65)(9 50 86 32 109 74)(10 59 87 21 110 63)(11 48 88 30 111 72)(12 57 89 39 112 61)(13 46 90 28 113 70)(14 55 91 37 114 79)(15 44 92 26 115 68)(16 53 93 35 116 77)(17 42 94 24 117 66)(18 51 95 33 118 75)(19 60 96 22 119 64)(20 49 97 31 120 73)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)

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)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116), (1,58,98,40,101,62)(2,47,99,29,102,71)(3,56,100,38,103,80)(4,45,81,27,104,69)(5,54,82,36,105,78)(6,43,83,25,106,67)(7,52,84,34,107,76)(8,41,85,23,108,65)(9,50,86,32,109,74)(10,59,87,21,110,63)(11,48,88,30,111,72)(12,57,89,39,112,61)(13,46,90,28,113,70)(14,55,91,37,114,79)(15,44,92,26,115,68)(16,53,93,35,116,77)(17,42,94,24,117,66)(18,51,95,33,118,75)(19,60,96,22,119,64)(20,49,97,31,120,73), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)>;

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)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116), (1,58,98,40,101,62)(2,47,99,29,102,71)(3,56,100,38,103,80)(4,45,81,27,104,69)(5,54,82,36,105,78)(6,43,83,25,106,67)(7,52,84,34,107,76)(8,41,85,23,108,65)(9,50,86,32,109,74)(10,59,87,21,110,63)(11,48,88,30,111,72)(12,57,89,39,112,61)(13,46,90,28,113,70)(14,55,91,37,114,79)(15,44,92,26,115,68)(16,53,93,35,116,77)(17,42,94,24,117,66)(18,51,95,33,118,75)(19,60,96,22,119,64)(20,49,97,31,120,73), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113) );

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),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116)], [(1,58,98,40,101,62),(2,47,99,29,102,71),(3,56,100,38,103,80),(4,45,81,27,104,69),(5,54,82,36,105,78),(6,43,83,25,106,67),(7,52,84,34,107,76),(8,41,85,23,108,65),(9,50,86,32,109,74),(10,59,87,21,110,63),(11,48,88,30,111,72),(12,57,89,39,112,61),(13,46,90,28,113,70),(14,55,91,37,114,79),(15,44,92,26,115,68),(16,53,93,35,116,77),(17,42,94,24,117,66),(18,51,95,33,118,75),(19,60,96,22,119,64),(20,49,97,31,120,73)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109),(57,110),(58,111),(59,112),(60,113)])

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D10A10B10C···10H12A12B12C12D12E15A15B20A···20F20G20H20I20J30A30B60A···60F
order122222222234444444444556666101010···101212121212151520···2020202020303060···60
size116661010101515222233553030302222020202212···124441010444···46666448···8

60 irreducible representations

dim1111111122222222244448
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10D10S3×D5S3×C4○D4C2×S3×D5D5×C4○D4D2016D6
kernelD2016D6D205S3D125D5C4×S3×D5C20⋊D6C3×Q82D5C5×Q83S3Q8×D15Q82D5Q83S3C4×D5D20C5×Q8D15C4×S3D12C3×Q8Q8C5C4C3C1
# reps1333311112331466222642

Matrix representation of D2016D6 in GL6(𝔽61)

4310000
6000000
0060000
0006000
0000011
0000110
,
1430000
0600000
0060000
0006000
0000050
0000110
,
18430000
1430000
001100
0060000
000001
000010
,
6000000
0600000
001100
0006000
0000060
0000600

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] >;

D2016D6 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

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