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G = D2010D6order 480 = 25·3·5

4th semidirect product of D20 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2010D6, D12.8D10, C60.18C23, Dic305C22, D4⋊D54S3, (S3×D4)⋊2D5, C52C87D6, C15⋊D84C2, C56(D8⋊S3), D4.4(S3×D5), D4.D158C2, (C3×D4).6D10, (C4×S3).7D10, (C5×D4).21D6, D205S32C2, C1518(C8⋊C22), C153C89C22, (S3×C10).33D4, C10.144(S3×D4), C30.180(C2×D4), D12.D53C2, (C3×D20)⋊4C22, D6.Dic53C2, D6.13(C5⋊D4), C32(D4.D10), (S3×C20).6C22, C20.18(C22×S3), (C5×Dic3).13D4, (C5×D12).5C22, C12.18(C22×D5), (D4×C15).12C22, Dic3.10(C5⋊D4), (C5×S3×D4)⋊2C2, C4.18(C2×S3×D5), (C3×D4⋊D5)⋊6C2, C2.25(S3×C5⋊D4), C6.47(C2×C5⋊D4), (C3×C52C8)⋊7C22, SmallGroup(480,570)

Series: Derived Chief Lower central Upper central

C1C60 — D2010D6
C1C5C15C30C60C3×D20D205S3 — D2010D6
C15C30C60 — D2010D6
C1C2C4D4

Generators and relations for D2010D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=a15b, dbd=a5b, dcd=c-1 >

Subgroups: 684 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5, C10, C10 [×3], Dic3, Dic3, C12, D6, D6 [×3], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10 [×5], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C5×S3 [×2], C3×D5, C30, C30, C8⋊C22, C52C8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4 [×2], C22×C10, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, Dic15, C60, C6×D5, S3×C10, S3×C10 [×3], C2×C30, C4.Dic5, D4⋊D5, D4⋊D5, D4.D5 [×2], C4○D20, D4×C10, D8⋊S3, C3×C52C8, C153C8, D5×Dic3, C15⋊D4, C3×D20, S3×C20, C5×D12, C5×C3⋊D4, Dic30, D4×C15, S3×C2×C10, D4.D10, D6.Dic5, C15⋊D8, D12.D5, C3×D4⋊D5, D4.D15, D205S3, C5×S3×D4, D2010D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D8⋊S3, C2×S3×D5, D4.D10, S3×C5⋊D4, D2010D6

Smallest permutation representation of D2010D6
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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 104)(13 103)(14 102)(15 101)(16 120)(17 119)(18 118)(19 117)(20 116)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 100)(38 99)(39 98)(40 97)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 80)
(1 53 28 16 58 23)(2 44 29 7 59 34)(3 55 30 18 60 25)(4 46 31 9 41 36)(5 57 32 20 42 27)(6 48 33 11 43 38)(8 50 35 13 45 40)(10 52 37 15 47 22)(12 54 39 17 49 24)(14 56 21 19 51 26)(61 88 114)(62 99 115 72 89 105)(63 90 116)(64 81 117 74 91 107)(65 92 118)(66 83 119 76 93 109)(67 94 120)(68 85 101 78 95 111)(69 96 102)(70 87 103 80 97 113)(71 98 104)(73 100 106)(75 82 108)(77 84 110)(79 86 112)
(1 23)(2 34)(3 25)(4 36)(5 27)(6 38)(7 29)(8 40)(9 31)(10 22)(11 33)(12 24)(13 35)(14 26)(15 37)(16 28)(17 39)(18 30)(19 21)(20 32)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(61 71)(63 73)(65 75)(67 77)(69 79)(81 107)(82 118)(83 109)(84 120)(85 111)(86 102)(87 113)(88 104)(89 115)(90 106)(91 117)(92 108)(93 119)(94 110)(95 101)(96 112)(97 103)(98 114)(99 105)(100 116)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C8A8B10A10B10C10D10E10F10G10H10I10J10K10L10M10N 12 15A15B20A20B20C20D24A24B30A30B30C30D30E30F60A60B
order1222223444556668810101010101010101010101010101215152020202024243030303030306060
size11461220226602228402060224444666612121212444441212202044888888

48 irreducible representations

dim1111111122222222222244444448
type++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C5⋊D4C5⋊D4C8⋊C22S3×D4S3×D5D8⋊S3C2×S3×D5D4.D10S3×C5⋊D4D2010D6
kernelD2010D6D6.Dic5C15⋊D8D12.D5C3×D4⋊D5D4.D15D205S3C5×S3×D4D4⋊D5C5×Dic3S3×C10S3×D4C52C8D20C5×D4C4×S3D12C3×D4Dic3D6C15C10D4C5C4C3C2C1
# reps1111111111121112224411222442

Matrix representation of D2010D6 in GL6(𝔽241)

100000
010000
00872000
0018315400
0066196036
001451962050
,
100000
010000
00200108
00790205205
00145870239
008700239
,
110000
24000000
001300
00024000
00017410
001161740240
,
110000
02400000
001300
00024000
001161742400
00017401

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,87,183,66,145,0,0,20,154,196,196,0,0,0,0,0,205,0,0,0,0,36,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,79,145,87,0,0,0,0,87,0,0,0,0,205,0,0,0,0,108,205,239,239],[1,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,116,0,0,3,240,174,174,0,0,0,0,1,0,0,0,0,0,0,240],[1,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0,116,0,0,0,3,240,174,174,0,0,0,0,240,0,0,0,0,0,0,1] >;

D2010D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{10}D_6
% in TeX

G:=Group("D20:10D6");
// GroupNames label

G:=SmallGroup(480,570);
// by ID

G=gap.SmallGroup(480,570);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,675,185,80,1356,18822]);
// 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=d*a*d=a^11,c*b*c^-1=a^15*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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