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## G = D20⋊19D6order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20⋊19D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C3⋊D40 — D20⋊19D6
 Lower central C15 — C30 — C60 — D20⋊19D6
 Upper central C1 — C2 — C2×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 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 20E 20F 30A ··· 30F 40A ··· 40H 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 6 6 8 8 10 10 10 10 12 12 12 12 12 15 15 20 20 20 20 20 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 20 60 60 2 2 2 20 2 2 2 4 20 20 12 12 2 2 4 4 2 2 4 20 20 4 4 2 2 2 2 4 4 4 ··· 4 12 ··· 12 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 C3⋊D4 C3⋊D4 D20 D20 C8⋊C22 S3×D5 D4⋊D6 C3⋊D20 C2×S3×D5 C3⋊D20 C8⋊D10 D20⋊19D6 kernel D20⋊19D6 C3⋊D40 C15⋊SD16 C5×C4.Dic3 C3×C4○D20 C2×D60 C4○D20 C60 C2×C30 C4.Dic3 Dic10 D20 C2×C20 C3⋊C8 C2×C12 C20 C2×C10 C12 C2×C6 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 1 2 2 2 2 2 4 8

Matrix representation of D2019D6 in GL4(𝔽241) generated by

 3 44 0 0 197 78 0 0 0 0 3 44 0 0 197 78
,
 0 0 3 44 0 0 197 238 3 44 0 0 197 238 0 0
,
 147 84 0 0 157 93 0 0 0 0 148 84 0 0 157 94
,
 147 84 0 0 110 94 0 0 0 0 6 119 0 0 184 235
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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