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G = D2013D6order 480 = 25·3·5

7th semidirect product of D20 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2013D6, D1213D10, C30.29C24, C60.53C23, C1542+ 1+4, Dic309C22, D30.14C23, Dic15.16C23, (D4×D5)⋊4S3, (S3×D4)⋊4D5, (C4×D5)⋊3D6, C5⋊D45D6, (C5×D4)⋊13D6, D410(S3×D5), (C4×S3)⋊3D10, C3⋊D45D10, (C3×D4)⋊13D10, C20⋊D65C2, C52(D46D6), (C22×D5)⋊6D6, C15⋊Q812C22, D42D157C2, D125D55C2, D205S35C2, C32(D46D10), (S3×C20)⋊4C22, (D5×C12)⋊4C22, (C3×D20)⋊9C22, (C22×S3)⋊5D10, (C4×D15)⋊4C22, (C5×D12)⋊9C22, C157D45C22, (C2×C30).5C23, C6.29(C23×D5), C30.C235C2, D6.D102C2, (D4×C15)⋊11C22, C5⋊D1214C22, C3⋊D2014C22, C15⋊D414C22, C10.29(S3×C23), C20.53(C22×S3), (D5×Dic3)⋊2C22, (S3×Dic5)⋊2C22, (C6×D5).45C23, D6.26(C22×D5), C12.53(C22×D5), (S3×C10).14C23, D10.14(C22×S3), (C2×Dic15)⋊18C22, Dic3.15(C22×D5), Dic5.15(C22×S3), (C3×Dic5).14C23, (C5×Dic3).16C23, (C5×S3×D4)⋊6C2, (C3×D4×D5)⋊6C2, C4.53(C2×S3×D5), (D5×C3⋊D4)⋊4C2, (S3×C5⋊D4)⋊4C2, (D5×C2×C6)⋊8C22, (C2×S3×D5)⋊4C22, C22.5(C2×S3×D5), (S3×C2×C10)⋊8C22, (C2×C15⋊D4)⋊20C2, C2.32(C22×S3×D5), (C5×C3⋊D4)⋊5C22, (C3×C5⋊D4)⋊5C22, (C2×C6).5(C22×D5), (C2×C10).5(C22×S3), SmallGroup(480,1101)

Series: Derived Chief Lower central Upper central

C1C30 — D2013D6
C1C5C15C30C6×D5C2×S3×D5D5×C3⋊D4 — D2013D6
C15C30 — D2013D6
C1C2D4

Generators and relations for D2013D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 1740 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×5], C22 [×2], C22 [×13], C5, S3 [×4], C6, C6 [×5], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], D5 [×4], C10, C10 [×5], Dic3, Dic3 [×3], C12, C12, D6, D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×5], C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5 [×3], C20, C20, D10, D10 [×2], D10 [×5], C2×C10 [×2], C2×C10 [×5], Dic6 [×2], C4×S3, C4×S3 [×3], D12, D12, C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12, C3×D4, C3×D4 [×3], C22×S3 [×2], C22×S3 [×2], C22×C6 [×2], C5×S3 [×3], C3×D5 [×3], D15, C30, C30 [×2], 2+ 1+4, Dic10 [×2], C4×D5, C4×D5 [×3], D20, D20, C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4, C5×D4 [×3], C22×D5 [×2], C22×D5 [×2], C22×C10 [×2], C4○D12 [×2], S3×D4, S3×D4 [×3], D42S3 [×4], C2×C3⋊D4 [×4], C6×D4, C5×Dic3, C3×Dic5, Dic15, Dic15 [×2], C60, S3×D5 [×2], C6×D5, C6×D5 [×2], C6×D5 [×2], S3×C10, S3×C10 [×2], S3×C10 [×2], D30, C2×C30 [×2], C4○D20 [×2], D4×D5, D4×D5 [×3], D42D5 [×4], C2×C5⋊D4 [×4], D4×C10, D46D6, D5×Dic3 [×2], S3×Dic5 [×2], C15⋊D4, C15⋊D4 [×6], C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4 [×2], S3×C20, C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15, C2×Dic15 [×2], C157D4 [×2], D4×C15, C2×S3×D5 [×2], D5×C2×C6 [×2], S3×C2×C10 [×2], D46D10, D205S3, D6.D10, D125D5, C20⋊D6, C30.C23 [×2], C2×C15⋊D4 [×2], D5×C3⋊D4 [×2], S3×C5⋊D4 [×2], C3×D4×D5, C5×S3×D4, D42D15, D2013D6
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, D46D6, C2×S3×D5 [×3], D46D10, C22×S3×D5, D2013D6

Smallest permutation representation of D2013D6
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 22)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 120)
(1 86 76 6 81 61)(2 97 77 17 82 72)(3 88 78 8 83 63)(4 99 79 19 84 74)(5 90 80 10 85 65)(7 92 62 12 87 67)(9 94 64 14 89 69)(11 96 66 16 91 71)(13 98 68 18 93 73)(15 100 70 20 95 75)(21 60 119 26 55 104)(22 51 120 37 56 115)(23 42 101 28 57 106)(24 53 102 39 58 117)(25 44 103 30 59 108)(27 46 105 32 41 110)(29 48 107 34 43 112)(31 50 109 36 45 114)(33 52 111 38 47 116)(35 54 113 40 49 118)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 80)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D30A30B30C30D30E30F60A60B
order122222222223444444556666666101010101010101010101010101012121515202020203030303030306060
size1122666101010302261030303022244101020202244446666121212124204444121244888888

57 irreducible representations

dim1111111111112222222222224444448
type++++++++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6D10D10D10D10D102+ 1+4S3×D5D46D6C2×S3×D5C2×S3×D5D46D10D2013D6
kernelD2013D6D205S3D6.D10D125D5C20⋊D6C30.C23C2×C15⋊D4D5×C3⋊D4S3×C5⋊D4C3×D4×D5C5×S3×D4D42D15D4×D5S3×D4C4×D5D20C5⋊D4C5×D4C22×D5C4×S3D12C3⋊D4C3×D4C22×S3C15D4C5C4C22C3C1
# reps1111122221111211212224241222442

Matrix representation of D2013D6 in GL6(𝔽61)

6000000
0600000
0000060
0000118
000100
00604300
,
100000
010000
0000060
0000600
0006000
0060000
,
4700000
0130000
000010
000001
001000
000100
,
0570000
1500000
00304400
00173100
00003044
00001731

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,1,43,0,0,0,1,0,0,0,0,60,18,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0],[47,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,15,0,0,0,0,57,0,0,0,0,0,0,0,30,17,0,0,0,0,44,31,0,0,0,0,0,0,30,17,0,0,0,0,44,31] >;

D2013D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{13}D_6
% in TeX

G:=Group("D20:13D6");
// GroupNames label

G:=SmallGroup(480,1101);
// by ID

G=gap.SmallGroup(480,1101);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,185,1356,18822]);
// 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^11,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽