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G = D20⋊Dic3order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D201Dic3, D41(C3⋊F5), (C3×D4)⋊1F5, (D4×C15)⋊1C4, (C3×D20)⋊1C4, C5⋊(D4⋊Dic3), C60⋊C45C2, (C3×D5).6D8, (D4×D5).2S3, C12.9(C2×F5), C60.33(C2×C4), C33(D20⋊C4), (C5×D4)⋊1Dic3, (C6×D5).76D4, (C4×D5).27D6, C60.C46C2, C159(D4⋊C4), D5.3(D4⋊S3), (C3×D5).8SD16, C20.1(C2×Dic3), D5.3(D4.S3), (C3×Dic5).37D4, C6.20(C22⋊F5), D10.33(C3⋊D4), C30.20(C22⋊C4), Dic5.6(C3⋊D4), (D5×C12).68C22, C10.5(C6.D4), C2.6(D10.D6), C4.1(C2×C3⋊F5), (C3×D4×D5).2C2, SmallGroup(480,312)

Series: Derived Chief Lower central Upper central

C1C60 — D20⋊Dic3
C1C5C15C30C6×D5D5×C12C60⋊C4 — D20⋊Dic3
C15C30C60 — D20⋊Dic3
C1C2C4D4

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 [×4], C3, C4, C4 [×2], C22 [×5], C5, C6, C6 [×4], C8, C2×C4 [×2], D4, D4 [×2], C23, D5 [×2], D5, C10, C10, Dic3, C12, C12, C2×C6 [×5], C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10 [×3], C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4 [×2], C22×C6, C3×D5 [×2], 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 [×3], 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 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D8, SD16, F5, C2×Dic3, C3⋊D4 [×2], 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 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 100)(9 99)(10 98)(11 97)(12 96)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 109)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 101)(30 120)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 61)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)
(1 68 106)(2 77 107 10 69 115)(3 66 108 19 70 104)(4 75 109 8 71 113)(5 64 110 17 72 102)(6 73 111)(7 62 112 15 74 120)(9 80 114 13 76 118)(11 78 116)(12 67 117 20 79 105)(14 65 119 18 61 103)(16 63 101)(21 92 51 29 84 59)(22 81 52 38 85 48)(23 90 53 27 86 57)(24 99 54 36 87 46)(25 88 55)(26 97 56 34 89 44)(28 95 58 32 91 42)(30 93 60)(31 82 41 39 94 49)(33 100 43 37 96 47)(35 98 45)(40 83 50)
(2 4 10 8)(3 7 19 15)(5 13 17 9)(6 16)(12 14 20 18)(21 48 29 52)(22 51 38 59)(23 54 27 46)(24 57 36 53)(25 60)(26 43 34 47)(28 49 32 41)(30 55)(31 58 39 42)(33 44 37 56)(35 50)(40 45)(61 105 65 117)(62 108 74 104)(63 111)(64 114 72 118)(66 120 70 112)(67 103 79 119)(68 106)(69 109 77 113)(71 115 75 107)(73 101)(76 110 80 102)(78 116)(81 92 85 84)(82 95 94 91)(83 98)(86 87 90 99)(88 93)(89 96 97 100)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D 5 6A6B6C6D6E6F6G8A8B8C8D10A10B10C12A12B15A15B 20 30A30B30C30D30E30F60A60B
order1222223444456666666888810101012121515203030303030306060
size1145520221060604244101020203030303048842044844888888

39 irreducible representations

dim11111122222222224444444488
type++++++++--++++-++
imageC1C2C2C2C4C4S3D4D4D6Dic3Dic3D8SD16C3⋊D4C3⋊D4F5C2×F5D4⋊S3D4.S3C3⋊F5C22⋊F5C2×C3⋊F5D10.D6D20⋊C4D20⋊Dic3
kernelD20⋊Dic3C60.C4C60⋊C4C3×D4×D5C3×D20D4×C15D4×D5C3×Dic5C6×D5C4×D5D20C5×D4C3×D5C3×D5Dic5D10C3×D4C12D5D5D4C6C4C2C3C1
# reps11112211111122221111222412

Matrix representation of D20⋊Dic3 in GL6(𝔽241)

240960000
510000
00124000
00102400
00100240
001000
,
221490000
1862190000
007117124234
001240117234
0072341170
000234124117
,
24000000
02400000
001141272290
00011522912
002291150126
0022912711412
,
6400000
1621770000
00100240
00001240
00000240
00010240

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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