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## G = D20⋊17D6order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20⋊17D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — D5×D12 — D20⋊17D6
 Lower central C15 — C30 — D20⋊17D6
 Upper central C1 — C2 — Q8

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, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, 2+ 1+4, Dic10, C4×D5, C4×D5, D20, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×D12, C4○D12, S3×D4, Q83S3, Q83S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C2×D20, C4○D20, D4×D5, Q82D5, Q82D5, C5×C4○D4, D4○D12, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C3×D20, S3×C20, C5×D12, C4×D15, D60, Q8×C15, C2×S3×D5, D48D10, D6.D10, D5×D12, S3×D20, C20⋊D6, C3×Q82D5, C5×Q83S3, Q83D15, D2017D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2+ 1+4, C22×D5, S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5, 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 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)
(1 94 60 78 119 30)(2 83 41 67 120 39)(3 92 42 76 101 28)(4 81 43 65 102 37)(5 90 44 74 103 26)(6 99 45 63 104 35)(7 88 46 72 105 24)(8 97 47 61 106 33)(9 86 48 70 107 22)(10 95 49 79 108 31)(11 84 50 68 109 40)(12 93 51 77 110 29)(13 82 52 66 111 38)(14 91 53 75 112 27)(15 100 54 64 113 36)(16 89 55 73 114 25)(17 98 56 62 115 34)(18 87 57 71 116 23)(19 96 58 80 117 32)(20 85 59 69 118 21)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 60)(17 59)(18 58)(19 57)(20 56)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 10A 10B 10C ··· 10H 12A 12B 12C 12D 12E 15A 15B 20A ··· 20F 20G 20H 20I 20J 30A 30B 60A ··· 60F order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 10 10 10 ··· 10 12 12 12 12 12 15 15 20 ··· 20 20 20 20 20 30 30 60 ··· 60 size 1 1 6 6 6 10 10 10 30 30 30 2 2 2 2 6 10 30 2 2 2 20 20 20 2 2 12 ··· 12 4 4 4 10 10 4 4 4 ··· 4 6 6 6 6 4 4 8 ··· 8

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D10 D10 D10 2+ 1+4 S3×D5 D4○D12 C2×S3×D5 D4⋊8D10 D20⋊17D6 kernel D20⋊17D6 D6.D10 D5×D12 S3×D20 C20⋊D6 C3×Q8⋊2D5 C5×Q8⋊3S3 Q8⋊3D15 Q8⋊2D5 Q8⋊3S3 C4×D5 D20 C5×Q8 C4×S3 D12 C3×Q8 C15 Q8 C5 C4 C3 C1 # reps 1 3 3 3 3 1 1 1 1 2 3 3 1 6 6 2 1 2 2 6 4 2

Matrix representation of D2017D6 in GL8(𝔽61)

 18 1 0 0 0 0 0 0 42 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 60 17 0 0 0 0 0 0 0 0 4 36 0 0 0 0 0 0 25 27 0 0 0 0 0 0 4 36 57 25 0 0 0 0 25 27 36 34
,
 60 60 0 0 0 0 0 0 0 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 36 0 0 0 0 0 0 25 57 0 0 0 0 0 0 0 0 4 36 0 0 0 0 0 0 25 57
,
 25 27 46 26 0 0 0 0 38 36 41 35 0 0 0 0 21 21 60 0 0 0 0 0 12 52 44 1 0 0 0 0 0 0 0 0 60 0 2 0 0 0 0 0 18 1 25 59 0 0 0 0 0 0 1 0 0 0 0 0 0 0 43 60
,
 36 34 15 35 0 0 0 0 23 25 20 26 0 0 0 0 40 40 2 0 0 0 0 0 49 9 34 59 0 0 0 0 0 0 0 0 25 57 0 0 0 0 0 0 34 36 0 0 0 0 0 0 25 57 36 4 0 0 0 0 34 36 27 25

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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