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G = D205S3order 240 = 24·3·5

The semidirect product of D20 and S3 acting through Inn(D20)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D205S3, D10.1D6, D6.5D10, C20.26D6, Dic306C2, C12.13D10, C30.2C23, C60.12C22, Dic3.10D10, Dic15.2C22, (C4×S3)⋊1D5, C4.5(S3×D5), (S3×C20)⋊1C2, (C3×D20)⋊2C2, C151(C4○D4), C33(C4○D20), C15⋊D41C2, C52(D42S3), (D5×Dic3)⋊1C2, C6.2(C22×D5), C10.2(C22×S3), (C6×D5).1C22, (S3×C10).5C22, (C5×Dic3).7C22, C2.6(C2×S3×D5), SmallGroup(240,126)

Series: Derived Chief Lower central Upper central

C1C30 — D205S3
C1C5C15C30C6×D5D5×Dic3 — D205S3
C15C30 — D205S3
C1C2C4

Generators and relations for D205S3
 G = < a,b,c,d | a20=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 328 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3, Dic3 [×2], C12, D6, C2×C6 [×2], C15, C4○D4, Dic5 [×2], C20, C20, D10 [×2], C2×C10, Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C5×S3, C3×D5 [×2], C30, Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, D42S3, C5×Dic3, Dic15 [×2], C60, C6×D5 [×2], S3×C10, C4○D20, D5×Dic3 [×2], C15⋊D4 [×2], C3×D20, S3×C20, Dic30, D205S3
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, D42S3, S3×D5, C4○D20, C2×S3×D5, D205S3

Smallest permutation representation of D205S3
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)
(1 105 96)(2 106 97)(3 107 98)(4 108 99)(5 109 100)(6 110 81)(7 111 82)(8 112 83)(9 113 84)(10 114 85)(11 115 86)(12 116 87)(13 117 88)(14 118 89)(15 119 90)(16 120 91)(17 101 92)(18 102 93)(19 103 94)(20 104 95)(21 69 49)(22 70 50)(23 71 51)(24 72 52)(25 73 53)(26 74 54)(27 75 55)(28 76 56)(29 77 57)(30 78 58)(31 79 59)(32 80 60)(33 61 41)(34 62 42)(35 63 43)(36 64 44)(37 65 45)(38 66 46)(39 67 47)(40 68 48)
(1 77)(2 78)(3 79)(4 80)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 117)(22 118)(23 119)(24 120)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 100)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113), (1,105,96)(2,106,97)(3,107,98)(4,108,99)(5,109,100)(6,110,81)(7,111,82)(8,112,83)(9,113,84)(10,114,85)(11,115,86)(12,116,87)(13,117,88)(14,118,89)(15,119,90)(16,120,91)(17,101,92)(18,102,93)(19,103,94)(20,104,95)(21,69,49)(22,70,50)(23,71,51)(24,72,52)(25,73,53)(26,74,54)(27,75,55)(28,76,56)(29,77,57)(30,78,58)(31,79,59)(32,80,60)(33,61,41)(34,62,42)(35,63,43)(36,64,44)(37,65,45)(38,66,46)(39,67,47)(40,68,48), (1,77)(2,78)(3,79)(4,80)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,117)(22,118)(23,119)(24,120)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,100)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)>;

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

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

D205S3 is a maximal subgroup of
D40⋊S3  D6.1D20  D407S3  C40.2D6  D20.24D6  D2010D6  D20.27D6  D20.28D6  D20.39D6  S3×C4○D20  D2025D6  D5×D42S3  D2013D6  D20.29D6  D2016D6
D205S3 is a maximal quotient of
(S3×C20)⋊5C4  Dic15.2Q8  Dic3014C4  D6⋊C4.D5  (C2×C60).C22  (C4×Dic3)⋊D5  C60.44D4  C60.45D4  C60.6Q8  (D5×Dic3)⋊C4  (C6×D5).D4  Dic15⋊D4  D10.17D12  Dic3×D20  D104Dic6  C1517(C4×D4)  D10⋊C4⋊S3  C604D4  Dic15.10D4

39 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A10B10C10D10E10F 12 15A15B20A20B20C20D20E20F20G20H30A30B60A60B60C60D
order12222344444556661010101010101215152020202020202020303060606060
size116101022333030222202022666644422226666444444

39 irreducible representations

dim1111112222222224444
type+++++++++++++-++-
imageC1C2C2C2C2C2S3D5D6D6C4○D4D10D10D10C4○D20D42S3S3×D5C2×S3×D5D205S3
kernelD205S3D5×Dic3C15⋊D4C3×D20S3×C20Dic30D20C4×S3C20D10C15Dic3C12D6C3C5C4C2C1
# reps1221111212222281224

Matrix representation of D205S3 in GL4(𝔽61) generated by

60000
06000
00436
002527
,
1000
0100
00060
00600
,
606000
1000
0010
0001
,
606000
0100
003044
001731
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,4,25,0,0,36,27],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,30,17,0,0,44,31] >;

D205S3 in GAP, Magma, Sage, TeX

D_{20}\rtimes_5S_3
% in TeX

G:=Group("D20:5S3");
// GroupNames label

G:=SmallGroup(240,126);
// by ID

G=gap.SmallGroup(240,126);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,50,490,6917]);
// 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,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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