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G = D2019D6order 480 = 25·3·5

2nd semidirect product of D20 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2019D6, C60.60D4, C12.51D20, Dic1016D6, C60.102C23, D60.43C22, C3⋊C83D10, C4○D204S3, (C2×C6).5D20, (C2×D60)⋊17C2, C51(D4⋊D6), C3⋊D4013C2, C35(C8⋊D10), C159(C8⋊C22), C6.47(C2×D20), (C2×C30).44D4, C30.76(C2×D4), (C2×C20).87D6, C4.Dic37D5, (C2×C12).89D10, C15⋊SD1613C2, (C3×D20)⋊21C22, C4.16(C3⋊D20), C20.25(C3⋊D4), (C2×C60).93C22, C12.89(C22×D5), C20.152(C22×S3), C22.9(C3⋊D20), (C3×Dic10)⋊18C22, (C2×C4).9(S3×D5), C4.101(C2×S3×D5), (C3×C4○D20)⋊6C2, (C5×C3⋊C8)⋊17C22, C2.6(C2×C3⋊D20), C10.2(C2×C3⋊D4), (C5×C4.Dic3)⋊8C2, (C2×C10).10(C3⋊D4), SmallGroup(480,377)

Series: Derived Chief Lower central Upper central

C1C60 — D2019D6
C1C5C15C30C60C3×D20C3⋊D40 — D2019D6
C15C30C60 — D2019D6
C1C2C2×C4

Generators and relations for D2019D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a10b, dbd=a13b, dcd=c-1 >

Subgroups: 1004 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×3], C10, C10, C12 [×2], C12, D6 [×4], C2×C6, C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10 [×5], C2×C10, C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×D4 [×2], C3×Q8, C22×S3, C3×D5, D15 [×2], C30, C30, C8⋊C22, C40 [×2], Dic10, C4×D5, D20, D20 [×3], C5⋊D4, C2×C20, C22×D5, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, C3×Dic5, C60 [×2], C6×D5, D30 [×4], C2×C30, C40⋊C2 [×2], D40 [×2], C5×M4(2), C2×D20, C4○D20, D4⋊D6, C5×C3⋊C8 [×2], C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, D60 [×2], D60, C2×C60, C22×D15, C8⋊D10, C3⋊D40 [×2], C15⋊SD16 [×2], C5×C4.Dic3, C3×C4○D20, C2×D60, D2019D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, D20 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×D20, D4⋊D6, C3⋊D20 [×2], C2×S3×D5, C8⋊D10, C2×C3⋊D20, D2019D6

Smallest permutation representation of D2019D6
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)
(1 109)(2 108)(3 107)(4 106)(5 105)(6 104)(7 103)(8 102)(9 101)(10 120)(11 119)(12 118)(13 117)(14 116)(15 115)(16 114)(17 113)(18 112)(19 111)(20 110)(21 41)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 100)(77 99)(78 98)(79 97)(80 96)
(1 71 22)(2 72 23)(3 73 24)(4 74 25)(5 75 26)(6 76 27)(7 77 28)(8 78 29)(9 79 30)(10 80 31)(11 61 32)(12 62 33)(13 63 34)(14 64 35)(15 65 36)(16 66 37)(17 67 38)(18 68 39)(19 69 40)(20 70 21)(41 120 86 51 110 96)(42 101 87 52 111 97)(43 102 88 53 112 98)(44 103 89 54 113 99)(45 104 90 55 114 100)(46 105 91 56 115 81)(47 106 92 57 116 82)(48 107 93 58 117 83)(49 108 94 59 118 84)(50 109 95 60 119 85)
(1 22)(2 21)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 120)(57 119)(58 118)(59 117)(60 116)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D8A8B10A10B10C10D12A12B12C12D12E15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order122222344455666688101010101212121212151520202020202030···3040···4060···60
size1122060602222022242020121222442242020442222444···412···124···4

57 irreducible representations

dim111111222222222222244444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D6D10D10C3⋊D4C3⋊D4D20D20C8⋊C22S3×D5D4⋊D6C3⋊D20C2×S3×D5C3⋊D20C8⋊D10D2019D6
kernelD2019D6C3⋊D40C15⋊SD16C5×C4.Dic3C3×C4○D20C2×D60C4○D20C60C2×C30C4.Dic3Dic10D20C2×C20C3⋊C8C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps122111111211142224412222248

Matrix representation of D2019D6 in GL4(𝔽241) generated by

34400
1977800
00344
0019778
,
00344
00197238
34400
19723800
,
1478400
1579300
0014884
0015794
,
1478400
1109400
006119
00184235
G:=sub<GL(4,GF(241))| [3,197,0,0,44,78,0,0,0,0,3,197,0,0,44,78],[0,0,3,197,0,0,44,238,3,197,0,0,44,238,0,0],[147,157,0,0,84,93,0,0,0,0,148,157,0,0,84,94],[147,110,0,0,84,94,0,0,0,0,6,184,0,0,119,235] >;

D2019D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{19}D_6
% in TeX

G:=Group("D20:19D6");
// GroupNames label

G:=SmallGroup(480,377);
// by ID

G=gap.SmallGroup(480,377);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,219,100,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^10*b,d*b*d=a^13*b,d*c*d=c^-1>;
// generators/relations

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