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G = S3×D4×C10order 480 = 25·3·5

Direct product of C10, S3 and D4

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

Aliases: S3×D4×C10, C608C23, C30.88C24, C62(D4×C10), (C6×D4)⋊5C10, C3015(C2×D4), (C2×C20)⋊29D6, C12⋊(C22×C10), (D4×C30)⋊19C2, D127(C2×C10), C207(C22×S3), C234(S3×C10), (C2×C30)⋊8C23, C1516(C22×D4), (C10×D12)⋊27C2, (C2×D12)⋊11C10, (S3×C23)⋊4C10, (C2×C60)⋊27C22, D62(C22×C10), (C22×C10)⋊13D6, C6.5(C23×C10), (S3×C10)⋊11C23, (S3×C20)⋊22C22, (D4×C15)⋊36C22, (C5×D12)⋊37C22, C10.73(S3×C23), (C5×Dic3)⋊9C23, (C22×C30)⋊16C22, Dic31(C22×C10), (C10×Dic3)⋊35C22, C32(D4×C2×C10), C41(S3×C2×C10), (S3×C2×C4)⋊3C10, (S3×C2×C20)⋊13C2, (C2×C4)⋊6(S3×C10), (C2×C6)⋊(C22×C10), C222(S3×C2×C10), (C4×S3)⋊3(C2×C10), (C2×C12)⋊2(C2×C10), (C3×D4)⋊5(C2×C10), (C2×C3⋊D4)⋊9C10, C3⋊D41(C2×C10), C2.6(S3×C22×C10), (C10×C3⋊D4)⋊24C2, (S3×C22×C10)⋊10C2, (S3×C2×C10)⋊22C22, (C22×C6)⋊4(C2×C10), (C2×C10)⋊8(C22×S3), (C22×S3)⋊6(C2×C10), (C2×Dic3)⋊8(C2×C10), (C5×C3⋊D4)⋊17C22, SmallGroup(480,1154)

Series: Derived Chief Lower central Upper central

C1C6 — S3×D4×C10
C1C3C6C30S3×C10S3×C2×C10S3×C22×C10 — S3×D4×C10
C3C6 — S3×D4×C10
C1C2×C10D4×C10

Generators and relations for S3×D4×C10
 G = < a,b,c,d,e | a10=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1124 in 472 conjugacy classes, 194 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, S3 [×4], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], C10, C10 [×2], C10 [×12], Dic3 [×2], C12 [×2], D6 [×10], D6 [×20], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×34], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×10], C22×S3 [×8], C22×C6 [×2], C5×S3 [×4], C5×S3 [×4], C30, C30 [×2], C30 [×4], C22×D4, C2×C20, C2×C20 [×5], C5×D4 [×4], C5×D4 [×12], C22×C10 [×2], C22×C10 [×19], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C5×Dic3 [×2], C60 [×2], S3×C10 [×10], S3×C10 [×20], C2×C30, C2×C30 [×4], C2×C30 [×4], C22×C20, D4×C10, D4×C10 [×11], C23×C10 [×2], C2×S3×D4, S3×C20 [×4], C5×D12 [×4], C10×Dic3, C5×C3⋊D4 [×8], C2×C60, D4×C15 [×4], S3×C2×C10, S3×C2×C10 [×10], S3×C2×C10 [×8], C22×C30 [×2], D4×C2×C10, S3×C2×C20, C10×D12, C5×S3×D4 [×8], C10×C3⋊D4 [×2], D4×C30, S3×C22×C10 [×2], S3×D4×C10
Quotients: C1, C2 [×15], C22 [×35], C5, S3, D4 [×4], C23 [×15], C10 [×15], D6 [×7], C2×D4 [×6], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C22×D4, C5×D4 [×4], C22×C10 [×15], S3×D4 [×2], S3×C23, S3×C10 [×7], D4×C10 [×6], C23×C10, C2×S3×D4, S3×C2×C10 [×7], D4×C2×C10, C5×S3×D4 [×2], S3×C22×C10, S3×D4×C10

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

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

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

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

150 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A4B4C4D5A5B5C5D6A6B6C6D6E6F6G10A···10L10M···10AB10AC···10AR10AS···10BH12A12B15A15B15C15D20A···20H20I···20P30A···30L30M···30AB60A···60H
order1222222222222222344445555666666610···1010···1010···1010···1012121515151520···2020···2030···3030···3060···60
size111122223333666622266111122244441···12···23···36···64422222···26···62···24···44···4

150 irreducible representations

dim11111111111111222222222244
type+++++++++++++
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10S3D4D6D6D6C5×S3C5×D4S3×C10S3×C10S3×C10S3×D4C5×S3×D4
kernelS3×D4×C10S3×C2×C20C10×D12C5×S3×D4C10×C3⋊D4D4×C30S3×C22×C10C2×S3×D4S3×C2×C4C2×D12S3×D4C2×C3⋊D4C6×D4S3×C23D4×C10S3×C10C2×C20C5×D4C22×C10C2×D4D6C2×C4D4C23C10C2
# reps11182124443284814142416416828

Matrix representation of S3×D4×C10 in GL4(𝔽61) generated by

41000
04100
0010
0001
,
606000
1000
0010
0001
,
60000
1100
00600
00060
,
60000
06000
006059
0011
,
60000
06000
006059
0001
G:=sub<GL(4,GF(61))| [41,0,0,0,0,41,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,1,0,0,59,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,59,1] >;

S3×D4×C10 in GAP, Magma, Sage, TeX

S_3\times D_4\times C_{10}
% in TeX

G:=Group("S3xD4xC10");
// GroupNames label

G:=SmallGroup(480,1154);
// by ID

G=gap.SmallGroup(480,1154);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,633,15686]);
// Polycyclic

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

׿
×
𝔽