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G = S3×C4○D20order 480 = 25·3·5

Direct product of S3 and C4○D20

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×C4○D20, D2028D6, Dic1025D6, D6033C22, C30.19C24, D30.6C23, C60.114C23, Dic3030C22, Dic15.9C23, (C4×D5)⋊12D6, (C2×C20)⋊27D6, (C2×C12)⋊6D10, C5⋊D413D6, (C4×S3)⋊17D10, (S3×D20)⋊13C2, (C2×C60)⋊5C22, C15⋊Q810C22, D205S313C2, D6011C29C2, D60⋊C213C2, (C6×D5).5C23, C6.19(C23×D5), (S3×C20)⋊21C22, (S3×Dic10)⋊13C2, (C2×Dic3)⋊22D10, Dic5.D67C2, (C3×D20)⋊24C22, (C4×D15)⋊15C22, (D5×C12)⋊12C22, C15⋊D412C22, C5⋊D1212C22, C157D414C22, C3⋊D2012C22, C10.19(S3×C23), D30.C28C22, (D5×Dic3)⋊7C22, D10.5(C22×S3), D6.25(C22×D5), D6.D1010C2, (S3×C10).30C23, C20.188(C22×S3), (C2×C30).238C23, (C22×S3).83D10, C12.188(C22×D5), Dic5.8(C22×S3), (C3×Dic5).8C23, (C10×Dic3)⋊27C22, (C3×Dic10)⋊22C22, (S3×Dic5).10C22, (C5×Dic3).29C23, Dic3.34(C22×D5), (S3×C2×C4)⋊6D5, (C4×S3×D5)⋊9C2, C51(S3×C4○D4), (S3×C2×C20)⋊1C2, (C2×C4)⋊9(S3×D5), C34(C2×C4○D20), C159(C2×C4○D4), (S3×C5⋊D4)⋊7C2, C4.161(C2×S3×D5), (C3×C4○D20)⋊5C2, (C5×S3)⋊1(C4○D4), (C2×S3×D5).6C22, C22.14(C2×S3×D5), C2.22(C22×S3×D5), (C3×C5⋊D4)⋊8C22, (C2×C6).10(C22×D5), (S3×C2×C10).103C22, (C2×C10).247(C22×S3), SmallGroup(480,1091)

Series: Derived Chief Lower central Upper central

C1C30 — S3×C4○D20
C1C5C15C30C6×D5C2×S3×D5C4×S3×D5 — S3×C4○D20
C15C30 — S3×C4○D20
C1C4C2×C4

Generators and relations for S3×C4○D20
 G = < a,b,c,d,e | a3=b2=c4=e2=1, d10=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d9 >

Subgroups: 1612 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×2], S3 [×3], C6, C6 [×3], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C2×C6 [×2], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×4], Dic6 [×3], C4×S3 [×4], C4×S3 [×6], D12 [×3], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3, C22×S3 [×2], C5×S3 [×2], C5×S3, C3×D5 [×2], D15 [×2], C30, C30, C2×C4○D4, Dic10, Dic10 [×3], C4×D5 [×2], C4×D5 [×6], D20, D20 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×C10, S3×C2×C4, S3×C2×C4 [×2], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20, C4○D20 [×7], C2×C5⋊D4 [×2], C22×C20, S3×C4○D4, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], S3×C20 [×4], C10×Dic3, Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, C2×S3×D5 [×2], S3×C2×C10, C2×C4○D20, D205S3, S3×Dic10, D60⋊C2, D6.D10 [×2], C4×S3×D5 [×2], S3×D20, Dic5.D6 [×2], S3×C5⋊D4 [×2], C3×C4○D20, S3×C2×C20, D6011C2, S3×C4○D20
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C4○D20 [×2], C23×D5, S3×C4○D4, C2×S3×D5 [×3], C2×C4○D20, C22×S3×D5, S3×C4○D20

Smallest permutation representation of S3×C4○D20
On 120 points
Generators in S120
(1 31 43)(2 32 44)(3 33 45)(4 34 46)(5 35 47)(6 36 48)(7 37 49)(8 38 50)(9 39 51)(10 40 52)(11 21 53)(12 22 54)(13 23 55)(14 24 56)(15 25 57)(16 26 58)(17 27 59)(18 28 60)(19 29 41)(20 30 42)(61 114 100)(62 115 81)(63 116 82)(64 117 83)(65 118 84)(66 119 85)(67 120 86)(68 101 87)(69 102 88)(70 103 89)(71 104 90)(72 105 91)(73 106 92)(74 107 93)(75 108 94)(76 109 95)(77 110 96)(78 111 97)(79 112 98)(80 113 99)
(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)
(1 105 11 115)(2 106 12 116)(3 107 13 117)(4 108 14 118)(5 109 15 119)(6 110 16 120)(7 111 17 101)(8 112 18 102)(9 113 19 103)(10 114 20 104)(21 81 31 91)(22 82 32 92)(23 83 33 93)(24 84 34 94)(25 85 35 95)(26 86 36 96)(27 87 37 97)(28 88 38 98)(29 89 39 99)(30 90 40 100)(41 70 51 80)(42 71 52 61)(43 72 53 62)(44 73 54 63)(45 74 55 64)(46 75 56 65)(47 76 57 66)(48 77 58 67)(49 78 59 68)(50 79 60 69)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D10A···10F10G···10N12A12B12C12D12E15A15B20A···20H20I···20P30A···30F60A···60H
order12222222223444444444455666610···1010···101212121212151520···2020···2030···3060···60
size11233610103030211233610103030222420202···26···62242020442···26···64···44···4

78 irreducible representations

dim111111111111222222222222244444
type++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6C4○D4D10D10D10D10C4○D20S3×D5S3×C4○D4C2×S3×D5C2×S3×D5S3×C4○D20
kernelS3×C4○D20D205S3S3×Dic10D60⋊C2D6.D10C4×S3×D5S3×D20Dic5.D6S3×C5⋊D4C3×C4○D20S3×C2×C20D6011C2C4○D20S3×C2×C4Dic10C4×D5D20C5⋊D4C2×C20C5×S3C4×S3C2×Dic3C2×C12C22×S3S3C2×C4C5C4C22C1
# reps1111221221111212121482221622428

Matrix representation of S3×C4○D20 in GL4(𝔽61) generated by

591500
12100
0010
0001
,
1000
496000
0010
0001
,
1000
0100
00500
00050
,
60000
06000
005429
003259
,
60000
06000
005429
00327
G:=sub<GL(4,GF(61))| [59,12,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[1,49,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,50,0,0,0,0,50],[60,0,0,0,0,60,0,0,0,0,54,32,0,0,29,59],[60,0,0,0,0,60,0,0,0,0,54,32,0,0,29,7] >;

S3×C4○D20 in GAP, Magma, Sage, TeX

S_3\times C_4\circ D_{20}
% in TeX

G:=Group("S3xC4oD20");
// GroupNames label

G:=SmallGroup(480,1091);
// by ID

G=gap.SmallGroup(480,1091);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^2=1,d^10=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^9>;
// generators/relations

׿
×
𝔽