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G = D20⋊S3order 240 = 24·3·5

3rd semidirect product of D20 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D203S3, Dic63D5, D10.2D6, C20.14D6, C12.14D10, C30.3C23, C60.26C22, Dic3.2D10, D30.8C22, Dic15.10C22, (C3×D20)⋊5C2, (C4×D15)⋊5C2, C152(C4○D4), C3⋊D201C2, C4.19(S3×D5), C51(D42S3), (C5×Dic6)⋊5C2, (D5×Dic3)⋊2C2, C32(Q82D5), C6.3(C22×D5), C10.3(C22×S3), (C6×D5).2C22, (C5×Dic3).2C22, C2.7(C2×S3×D5), SmallGroup(240,127)

Series: Derived Chief Lower central Upper central

C1C30 — D20⋊S3
C1C5C15C30C6×D5D5×Dic3 — D20⋊S3
C15C30 — D20⋊S3
C1C2C4

Generators and relations for D20⋊S3
 G = < a,b,c,d | a20=b2=c3=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, dcd=c-1 >

Subgroups: 360 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, Dic3 [×2], Dic3, C12, D6, C2×C6 [×2], C15, C4○D4, Dic5, C20, C20 [×2], D10 [×2], D10, Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C3×D5 [×2], D15, C30, C4×D5 [×3], D20, D20 [×2], C5×Q8, D42S3, C5×Dic3 [×2], Dic15, C60, C6×D5 [×2], D30, Q82D5, D5×Dic3 [×2], C3⋊D20 [×2], C3×D20, C5×Dic6, C4×D15, D20⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, D42S3, S3×D5, Q82D5, C2×S3×D5, D20⋊S3

Smallest permutation representation of D20⋊S3
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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 81)(22 100)(23 99)(24 98)(25 97)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 106)(42 105)(43 104)(44 103)(45 102)(46 101)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 108)(60 107)
(1 23 118)(2 24 119)(3 25 120)(4 26 101)(5 27 102)(6 28 103)(7 29 104)(8 30 105)(9 31 106)(10 32 107)(11 33 108)(12 34 109)(13 35 110)(14 36 111)(15 37 112)(16 38 113)(17 39 114)(18 40 115)(19 21 116)(20 22 117)(41 72 91)(42 73 92)(43 74 93)(44 75 94)(45 76 95)(46 77 96)(47 78 97)(48 79 98)(49 80 99)(50 61 100)(51 62 81)(52 63 82)(53 64 83)(54 65 84)(55 66 85)(56 67 86)(57 68 87)(58 69 88)(59 70 89)(60 71 90)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 120)(22 109)(23 118)(24 107)(25 116)(26 105)(27 114)(28 103)(29 112)(30 101)(31 110)(32 119)(33 108)(34 117)(35 106)(36 115)(37 104)(38 113)(39 102)(40 111)(41 89)(42 98)(43 87)(44 96)(45 85)(46 94)(47 83)(48 92)(49 81)(50 90)(51 99)(52 88)(53 97)(54 86)(55 95)(56 84)(57 93)(58 82)(59 91)(60 100)(61 71)(62 80)(63 69)(64 78)(65 67)(66 76)(68 74)(70 72)(73 79)(75 77)

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

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

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

D20⋊S3 is a maximal subgroup of
C408D6  Dic6.D10  D405S3  D30.3D4  D20.10D6  Dic6⋊D10  D20.16D6  D20.17D6  D20.38D6  D2024D6  D2025D6  D5×D42S3  D2014D6  D20.29D6  S3×Q82D5
D20⋊S3 is a maximal quotient of
Dic157Q8  C4⋊Dic5⋊S3  Dic3.2Dic10  (C4×D15)⋊8C4  D30.34D4  (C2×C12).D10  (C4×Dic15)⋊C2  C60.88D4  D309Q8  C12.Dic10  D10.19(C4×S3)  Dic1513D4  (C6×D5).D4  Dic3⋊D20  D10.16D12  D101Dic6  D208Dic3  (C2×Dic6)⋊D5  C122D20

33 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A10B 12 15A15B20A20B20C20D20E20F30A30B60A60B60C60D
order12222344444556661010121515202020202020303060606060
size11101030226615152222020224444412121212444444

33 irreducible representations

dim111111222222244444
type++++++++++++-+++
imageC1C2C2C2C2C2S3D5D6D6C4○D4D10D10D42S3S3×D5Q82D5C2×S3×D5D20⋊S3
kernelD20⋊S3D5×Dic3C3⋊D20C3×D20C5×Dic6C4×D15D20Dic6C20D10C15Dic3C12C5C4C3C2C1
# reps122111121224212224

Matrix representation of D20⋊S3 in GL6(𝔽61)

010000
60170000
0050000
00501100
0000600
0000060
,
010000
100000
0015900
0006000
000010
000001
,
100000
010000
001000
000100
0000141
00005259
,
100000
17600000
001000
0016000
00006020
000001

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,17,0,0,0,0,0,0,50,50,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,52,0,0,0,0,41,59],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,20,1] >;

D20⋊S3 in GAP, Magma, Sage, TeX

D_{20}\rtimes S_3
% in TeX

G:=Group("D20:S3");
// GroupNames label

G:=SmallGroup(240,127);
// by ID

G=gap.SmallGroup(240,127);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations

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