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## G = D20⋊Dic3order 480 = 25·3·5

### 1st semidirect product of D20 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20⋊Dic3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C60⋊C4 — D20⋊Dic3
 Lower central C15 — C30 — C60 — D20⋊Dic3
 Upper central C1 — C2 — C4 — D4

Generators and relations for D20⋊Dic3
G = < a,b,c,d | a20=b2=c6=1, d2=c3, bab=a-1, cac-1=a9, dad-1=a7, cbc-1=a8b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 604 in 100 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, D4⋊C4, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C2×C3⋊C8, C4⋊Dic3, C6×D4, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, D5⋊C8, C4⋊F5, D4×D5, D4⋊Dic3, C15⋊C8, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, C2×C3⋊F5, D5×C2×C6, D20⋊C4, C60.C4, C60⋊C4, C3×D4×D5, D20⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, F5, C2×Dic3, C3⋊D4, D4⋊C4, C2×F5, D4⋊S3, D4.S3, C6.D4, C3⋊F5, C22⋊F5, D4⋊Dic3, C2×C3⋊F5, D20⋊C4, D10.D6, D20⋊Dic3

Smallest permutation representation of D20⋊Dic3
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 44)(2 43)(3 42)(4 41)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)
(1 107 21)(2 116 22 10 108 30)(3 105 23 19 109 39)(4 114 24 8 110 28)(5 103 25 17 111 37)(6 112 26)(7 101 27 15 113 35)(9 119 29 13 115 33)(11 117 31)(12 106 32 20 118 40)(14 104 34 18 120 38)(16 102 36)(41 99 65 49 91 73)(42 88 66 58 92 62)(43 97 67 47 93 71)(44 86 68 56 94 80)(45 95 69)(46 84 70 54 96 78)(48 82 72 52 98 76)(50 100 74)(51 89 75 59 81 63)(53 87 77 57 83 61)(55 85 79)(60 90 64)
(2 4 10 8)(3 7 19 15)(5 13 17 9)(6 16)(12 14 20 18)(21 107)(22 110 30 114)(23 113 39 101)(24 116 28 108)(25 119 37 115)(26 102)(27 105 35 109)(29 111 33 103)(31 117)(32 120 40 104)(34 106 38 118)(36 112)(41 58 49 42)(43 44 47 56)(45 50)(46 53 54 57)(48 59 52 51)(55 60)(61 96 77 84)(62 99 66 91)(63 82 75 98)(64 85)(65 88 73 92)(67 94 71 86)(68 97 80 93)(69 100)(70 83 78 87)(72 89 76 81)(74 95)(79 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), (1,44)(2,43)(3,42)(4,41)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101), (1,107,21)(2,116,22,10,108,30)(3,105,23,19,109,39)(4,114,24,8,110,28)(5,103,25,17,111,37)(6,112,26)(7,101,27,15,113,35)(9,119,29,13,115,33)(11,117,31)(12,106,32,20,118,40)(14,104,34,18,120,38)(16,102,36)(41,99,65,49,91,73)(42,88,66,58,92,62)(43,97,67,47,93,71)(44,86,68,56,94,80)(45,95,69)(46,84,70,54,96,78)(48,82,72,52,98,76)(50,100,74)(51,89,75,59,81,63)(53,87,77,57,83,61)(55,85,79)(60,90,64), (2,4,10,8)(3,7,19,15)(5,13,17,9)(6,16)(12,14,20,18)(21,107)(22,110,30,114)(23,113,39,101)(24,116,28,108)(25,119,37,115)(26,102)(27,105,35,109)(29,111,33,103)(31,117)(32,120,40,104)(34,106,38,118)(36,112)(41,58,49,42)(43,44,47,56)(45,50)(46,53,54,57)(48,59,52,51)(55,60)(61,96,77,84)(62,99,66,91)(63,82,75,98)(64,85)(65,88,73,92)(67,94,71,86)(68,97,80,93)(69,100)(70,83,78,87)(72,89,76,81)(74,95)(79,90)>;

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 type + + + + + + + + - - + + + + - + + image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 Dic3 Dic3 D8 SD16 C3⋊D4 C3⋊D4 F5 C2×F5 D4⋊S3 D4.S3 C3⋊F5 C22⋊F5 C2×C3⋊F5 D10.D6 D20⋊C4 D20⋊Dic3 kernel D20⋊Dic3 C60.C4 C60⋊C4 C3×D4×D5 C3×D20 D4×C15 D4×D5 C3×Dic5 C6×D5 C4×D5 D20 C5×D4 C3×D5 C3×D5 Dic5 D10 C3×D4 C12 D5 D5 D4 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 4 1 2

Matrix representation of D20⋊Dic3 in GL6(𝔽241)

 240 96 0 0 0 0 5 1 0 0 0 0 0 0 1 240 0 0 0 0 1 0 240 0 0 0 1 0 0 240 0 0 1 0 0 0
,
 22 149 0 0 0 0 186 219 0 0 0 0 0 0 7 117 124 234 0 0 124 0 117 234 0 0 7 234 117 0 0 0 0 234 124 117
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 114 127 229 0 0 0 0 115 229 12 0 0 229 115 0 126 0 0 229 127 114 12
,
 64 0 0 0 0 0 162 177 0 0 0 0 0 0 1 0 0 240 0 0 0 0 1 240 0 0 0 0 0 240 0 0 0 1 0 240

G:=sub<GL(6,GF(241))| [240,5,0,0,0,0,96,1,0,0,0,0,0,0,1,1,1,1,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0],[22,186,0,0,0,0,149,219,0,0,0,0,0,0,7,124,7,0,0,0,117,0,234,234,0,0,124,117,117,124,0,0,234,234,0,117],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,114,0,229,229,0,0,127,115,115,127,0,0,229,229,0,114,0,0,0,12,126,12],[64,162,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,240,240,240,240] >;

D20⋊Dic3 in GAP, Magma, Sage, TeX

D_{20}\rtimes {\rm Dic}_3
% in TeX

G:=Group("D20:Dic3");
// GroupNames label

G:=SmallGroup(480,312);
// by ID

G=gap.SmallGroup(480,312);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^7,c*b*c^-1=a^8*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

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