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G = D2029D6order 480 = 25·3·5

3rd semidirect product of D20 and D6 acting through Inn(D20)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2029D6, D1229D10, Dic1026D6, Dic626D10, D6041C22, C30.23C24, D30.9C23, C1532+ 1+4, C60.115C23, (C4×D5)⋊2D6, C5⋊D49D6, C4○D209S3, C4○D129D5, (C4×S3)⋊2D10, (C2×C20)⋊10D6, C52(D4○D12), C3⋊D49D10, (S3×D20)⋊12C2, (C2×D60)⋊19C2, (D5×D12)⋊12C2, (C2×C12)⋊10D10, D10⋊D61C2, C32(D48D10), (C2×C60)⋊11C22, C3⋊D202C22, C5⋊D122C22, D60⋊C212C2, C12.28D1012C2, D6.9(C22×D5), (C6×D5).9C23, C6.23(C23×D5), (S3×C20)⋊11C22, (D5×C12)⋊11C22, (C3×D20)⋊26C22, (C5×D12)⋊26C22, (S3×C10).9C23, C10.23(S3×C23), D30.C21C22, D10.9(C22×S3), (C2×C30).242C23, C20.132(C22×S3), (C5×Dic6)⋊23C22, C12.130(C22×D5), (C3×Dic10)⋊23C22, (C22×D15)⋊10C22, Dic3.11(C22×D5), (C3×Dic5).11C23, (C5×Dic3).11C23, Dic5.11(C22×S3), (C2×C4)⋊6(S3×D5), C4.89(C2×S3×D5), (C2×S3×D5)⋊3C22, (C3×C4○D20)⋊8C2, (C5×C4○D12)⋊8C2, C2.26(C22×S3×D5), C22.19(C2×S3×D5), (C5×C3⋊D4)⋊11C22, (C3×C5⋊D4)⋊11C22, (C2×C6).13(C22×D5), (C2×C10).13(C22×S3), SmallGroup(480,1095)

Series: Derived Chief Lower central Upper central

C1C30 — D2029D6
C1C5C15C30C6×D5C2×S3×D5D5×D12 — D2029D6
C15C30 — D2029D6
C1C2C2×C4

Generators and relations for D2029D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a10b, dbd=a8b, dcd=c-1 >

Subgroups: 2060 in 332 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], C5, S3 [×6], C6, C6 [×3], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×10], C2×C6, C2×C6 [×2], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×10], C2×C10, C2×C10 [×2], Dic6, C4×S3 [×2], C4×S3 [×4], D12, D12 [×8], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×6], C5×S3 [×2], C3×D5 [×2], D15 [×4], C30, C30, 2+ 1+4, Dic10, C4×D5 [×2], C4×D5 [×4], D20, D20 [×8], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5 [×6], C2×D12 [×3], C4○D12, C4○D12 [×2], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30 [×4], D30 [×2], C2×C30, C2×D20 [×3], C4○D20, C4○D20 [×2], D4×D5 [×6], Q82D5 [×2], C5×C4○D4, D4○D12, D30.C2 [×4], C3⋊D20 [×4], C5⋊D12 [×4], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], D60 [×4], C2×C60, C2×S3×D5 [×4], C22×D15 [×2], D48D10, D60⋊C2 [×2], C12.28D10 [×2], D5×D12 [×2], S3×D20 [×2], D10⋊D6 [×4], C3×C4○D20, C5×C4○D12, C2×D60, D2029D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5 [×3], D48D10, C22×S3×D5, D2029D6

Smallest permutation representation of D2029D6
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(116 120)(117 119)
(1 35 66 81 47 106)(2 36 67 82 48 107)(3 37 68 83 49 108)(4 38 69 84 50 109)(5 39 70 85 51 110)(6 40 71 86 52 111)(7 21 72 87 53 112)(8 22 73 88 54 113)(9 23 74 89 55 114)(10 24 75 90 56 115)(11 25 76 91 57 116)(12 26 77 92 58 117)(13 27 78 93 59 118)(14 28 79 94 60 119)(15 29 80 95 41 120)(16 30 61 96 42 101)(17 31 62 97 43 102)(18 32 63 98 44 103)(19 33 64 99 45 104)(20 34 65 100 46 105)
(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 80)(8 79)(9 78)(10 77)(11 76)(12 75)(13 74)(14 73)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(81 106)(82 105)(83 104)(84 103)(85 102)(86 101)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D10A10B10C10D10E10F10G10H12A12B12C12D12E15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order1222222222234444445566661010101010101010121212121215152020202020202020202030···3060···60
size1126610103030303022266101022242020224412121212224202044222244121212124···44···4

63 irreducible representations

dim1111111112222222222224444444
type++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6D10D10D10D10D102+ 1+4S3×D5D4○D12C2×S3×D5C2×S3×D5D48D10D2029D6
kernelD2029D6D60⋊C2C12.28D10D5×D12S3×D20D10⋊D6C3×C4○D20C5×C4○D12C2×D60C4○D20C4○D12Dic10C4×D5D20C5⋊D4C2×C20Dic6C4×S3D12C3⋊D4C2×C12C15C2×C4C5C4C22C3C1
# reps1222241111212121242421224248

Matrix representation of D2029D6 in GL8(𝔽61)

431000000
600000000
004310000
006000000
0000363200
000023400
0000510732
0000310292
,
143000000
060000000
001430000
000600000
00001000
0000196000
0000270060
0000270600
,
006000000
000600000
106000000
010600000
00004194444
0000328043
00003019942
00004664820
,
431818430000
60181430000
0018430000
001430000
000043100
0000431800
00004237600
0000319441

G:=sub<GL(8,GF(61))| [43,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,2,51,3,0,0,0,0,32,34,0,10,0,0,0,0,0,0,7,29,0,0,0,0,0,0,32,2],[1,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,1,19,27,27,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0,60,0,0,0,0,0,0,60,0,60,0,0,0,0,0,0,0,0,4,3,30,46,0,0,0,0,19,28,19,6,0,0,0,0,44,0,9,48,0,0,0,0,44,43,42,20],[43,60,0,0,0,0,0,0,18,18,0,0,0,0,0,0,18,1,18,1,0,0,0,0,43,43,43,43,0,0,0,0,0,0,0,0,43,43,42,31,0,0,0,0,1,18,37,9,0,0,0,0,0,0,60,44,0,0,0,0,0,0,0,1] >;

D2029D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{29}D_6
% in TeX

G:=Group("D20:29D6");
// GroupNames label

G:=SmallGroup(480,1095);
// by ID

G=gap.SmallGroup(480,1095);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,80,1356,18822]);
// 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^8*b,d*c*d=c^-1>;
// generators/relations

׿
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