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G = D2017D6order 480 = 25·3·5

11st semidirect product of D20 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2017D6, D1217D10, D6013C22, C30.39C24, C60.63C23, C1572+ 1+4, D30.20C23, Dic15.42C23, (C4×D5)⋊6D6, (D5×D12)⋊7C2, (C4×S3)⋊6D10, (S3×D20)⋊7C2, C54(D4○D12), (C5×Q8)⋊16D6, Q811(S3×D5), C20⋊D67C2, Q82D58S3, Q83S35D5, (C3×Q8)⋊13D10, C15⋊Q819C22, Q83D157C2, C34(D48D10), (S3×C20)⋊9C22, (C4×D15)⋊9C22, (D5×C12)⋊9C22, C6.39(C23×D5), D6.D108C2, (C5×D12)⋊13C22, (C3×D20)⋊13C22, C15⋊D417C22, C3⋊D2016C22, C5⋊D1217C22, C10.39(S3×C23), C20.63(C22×S3), (Q8×C15)⋊11C22, (C6×D5).17C23, D6.18(C22×D5), C12.63(C22×D5), (S3×C10).20C23, D10.20(C22×S3), (C3×Dic5).50C23, (C5×Dic3).34C23, Dic3.29(C22×D5), Dic5.45(C22×S3), C4.63(C2×S3×D5), (C2×S3×D5)⋊7C22, C2.42(C22×S3×D5), (C5×Q83S3)⋊7C2, (C3×Q82D5)⋊7C2, SmallGroup(480,1111)

Series: Derived Chief Lower central Upper central

C1C30 — D2017D6
C1C5C15C30C6×D5C2×S3×D5D5×D12 — D2017D6
C15C30 — D2017D6
C1C2Q8

Generators and relations for D2017D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a8b, dcd=c-1 >

Subgroups: 2044 in 332 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×9], C3, C4 [×3], C4 [×3], C22 [×15], C5, S3 [×6], C6, C6 [×3], C2×C4 [×9], D4 [×18], Q8, Q8, C23 [×6], D5 [×6], C10, C10 [×3], Dic3, Dic3, C12 [×3], C12, D6 [×3], D6 [×9], C2×C6 [×3], C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5, C20 [×3], C20, D10 [×3], D10 [×9], C2×C10 [×3], Dic6, C4×S3 [×3], C4×S3 [×3], D12 [×3], D12 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C5×S3 [×3], C3×D5 [×3], D15 [×3], C30, 2+ 1+4, Dic10, C4×D5 [×3], C4×D5 [×3], D20 [×3], D20 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×6], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q83S3, Q83S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60 [×3], S3×D5 [×6], C6×D5 [×3], S3×C10 [×3], D30 [×3], C2×D20 [×3], C4○D20 [×3], D4×D5 [×6], Q82D5, Q82D5, C5×C4○D4, D4○D12, C15⋊D4 [×3], C3⋊D20 [×3], C5⋊D12 [×3], C15⋊Q8, D5×C12 [×3], C3×D20 [×3], S3×C20 [×3], C5×D12 [×3], C4×D15 [×3], D60 [×3], Q8×C15, C2×S3×D5 [×6], D48D10, D6.D10 [×3], D5×D12 [×3], S3×D20 [×3], C20⋊D6 [×3], C3×Q82D5, C5×Q83S3, Q83D15, D2017D6
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, D2017D6

Smallest permutation representation of D2017D6
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 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 120)
(1 95 30 78 56 115)(2 84 31 67 57 104)(3 93 32 76 58 113)(4 82 33 65 59 102)(5 91 34 74 60 111)(6 100 35 63 41 120)(7 89 36 72 42 109)(8 98 37 61 43 118)(9 87 38 70 44 107)(10 96 39 79 45 116)(11 85 40 68 46 105)(12 94 21 77 47 114)(13 83 22 66 48 103)(14 92 23 75 49 112)(15 81 24 64 50 101)(16 90 25 73 51 110)(17 99 26 62 52 119)(18 88 27 71 53 108)(19 97 28 80 54 117)(20 86 29 69 55 106)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 107)(62 106)(63 105)(64 104)(65 103)(66 102)(67 101)(68 120)(69 119)(70 118)(71 117)(72 116)(73 115)(74 114)(75 113)(76 112)(77 111)(78 110)(79 109)(80 108)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D10A10B10C···10H12A12B12C12D12E15A15B20A···20F20G20H20I20J30A30B60A···60F
order122222222223444444556666101010···101212121212151520···2020202020303060···60
size116661010103030302222610302222020202212···124441010444···46666448···8

57 irreducible representations

dim1111111122222222444448
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D5D6D6D6D10D10D102+ 1+4S3×D5D4○D12C2×S3×D5D48D10D2017D6
kernelD2017D6D6.D10D5×D12S3×D20C20⋊D6C3×Q82D5C5×Q83S3Q83D15Q82D5Q83S3C4×D5D20C5×Q8C4×S3D12C3×Q8C15Q8C5C4C3C1
# reps1333311112331662122642

Matrix representation of D2017D6 in GL8(𝔽61)

181000000
4260000000
00010000
0060170000
000043600
0000252700
00004365725
000025273634
,
6060000000
01000000
00010000
00100000
000043600
0000255700
000000436
0000002557
,
252746260000
383641350000
21216000000
12524410000
000060020
00001812559
00000010
0000004360
,
363415350000
232520260000
4040200000
49934590000
0000255700
0000343600
00002557364
000034362725

G:=sub<GL(8,GF(61))| [18,42,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,4,25,4,25,0,0,0,0,36,27,36,27,0,0,0,0,0,0,57,36,0,0,0,0,0,0,25,34],[60,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,36,57,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,36,57],[25,38,21,12,0,0,0,0,27,36,21,52,0,0,0,0,46,41,60,44,0,0,0,0,26,35,0,1,0,0,0,0,0,0,0,0,60,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,25,1,43,0,0,0,0,0,59,0,60],[36,23,40,49,0,0,0,0,34,25,40,9,0,0,0,0,15,20,2,34,0,0,0,0,35,26,0,59,0,0,0,0,0,0,0,0,25,34,25,34,0,0,0,0,57,36,57,36,0,0,0,0,0,0,36,27,0,0,0,0,0,0,4,25] >;

D2017D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{17}D_6
% in TeX

G:=Group("D20:17D6");
// GroupNames label

G:=SmallGroup(480,1111);
// by ID

G=gap.SmallGroup(480,1111);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations

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