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G = S3×C8×D5order 480 = 25·3·5

Direct product of C8, S3 and D5

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

Aliases: S3×C8×D5, C4024D6, C2425D10, C12021C22, C60.166C23, C3⋊C831D10, (S3×C40)⋊6C2, D153(C2×C8), (D5×C24)⋊9C2, C52C831D6, C154(C22×C8), D6.9(C4×D5), (C8×D15)⋊12C2, (C4×D5).98D6, D10.28(C4×S3), (C4×S3).46D10, D30.26(C2×C4), D30.C2.7C4, (D5×Dic3).8C4, (S3×Dic5).7C4, D152C814C2, C153C840C22, C30.32(C22×C4), Dic5.33(C4×S3), Dic3.13(C4×D5), (S3×C20).49C22, C20.163(C22×S3), Dic15.33(C2×C4), (D5×C12).97C22, (C4×D15).60C22, C12.163(C22×D5), C54(S3×C2×C8), C31(D5×C2×C8), C6.1(C2×C4×D5), C2.1(C4×S3×D5), (D5×C3⋊C8)⋊14C2, (C2×S3×D5).8C4, (C5×S3)⋊3(C2×C8), (C3×D5)⋊2(C2×C8), C10.32(S3×C2×C4), (C4×S3×D5).13C2, C4.136(C2×S3×D5), (S3×C52C8)⋊14C2, (C5×C3⋊C8)⋊32C22, (C6×D5).30(C2×C4), (S3×C10).24(C2×C4), (C3×C52C8)⋊32C22, (C3×Dic5).35(C2×C4), (C5×Dic3).29(C2×C4), SmallGroup(480,319)

Series: Derived Chief Lower central Upper central

C1C15 — S3×C8×D5
C1C5C15C30C60D5×C12C4×S3×D5 — S3×C8×D5
C15 — S3×C8×D5
C1C8

Generators and relations for S3×C8×D5
 G = < a,b,c,d,e | a8=b3=c2=d5=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: 636 in 152 conjugacy classes, 64 normal (50 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C22 [×7], C5, S3 [×2], S3 [×2], C6, C6 [×2], C8, C8 [×3], C2×C4 [×6], C23, D5 [×2], D5 [×2], C10, C10 [×2], Dic3, Dic3, C12, C12, D6, D6 [×5], C2×C6, C15, C2×C8 [×6], C22×C4, Dic5, Dic5, C20, C20, D10, D10 [×5], C2×C10, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3 [×3], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C22×C8, C52C8, C52C8, C40, C40, C4×D5, C4×D5 [×3], C2×Dic5, C2×C20, C22×D5, S3×C8, S3×C8 [×3], C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×4], C6×D5, S3×C10, D30, C8×D5, C8×D5 [×3], C2×C52C8, C2×C40, C2×C4×D5, S3×C2×C8, C5×C3⋊C8, C3×C52C8, C153C8, C120, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5×C2×C8, D5×C3⋊C8, S3×C52C8, D152C8, D5×C24, S3×C40, C8×D15, C4×S3×D5, S3×C8×D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, D5, D6 [×3], C2×C8 [×6], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C22×C8, C4×D5 [×2], C22×D5, S3×C8 [×2], S3×C2×C4, S3×D5, C8×D5 [×2], C2×C4×D5, S3×C2×C8, C2×S3×D5, D5×C2×C8, C4×S3×D5, S3×C8×D5

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

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

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

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

96 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A8B8C8D8E8F8G8H8I8J8K8L8M8N8O8P10A10B10C10D10E10F12A12B12C12D15A15B20A20B20C20D20E20F20G20H24A24B24C24D24E24F24G24H30A30B40A···40H40I···40P60A60B60C60D120A···120H
order1222222234444444455666888888888888888810101010101012121212151520202020202020202424242424242424303040···4040···4060606060120···120
size1133551515211335515152221010111133335555151515152266662210104422226666222210101010442···26···644444···4

96 irreducible representations

dim1111111111111222222222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4C8S3D5D6D6D6D10D10D10C4×S3C4×S3C4×D5C4×D5S3×C8C8×D5S3×D5C2×S3×D5C4×S3×D5S3×C8×D5
kernelS3×C8×D5D5×C3⋊C8S3×C52C8D152C8D5×C24S3×C40C8×D15C4×S3×D5D5×Dic3S3×Dic5D30.C2C2×S3×D5S3×D5C8×D5S3×C8C52C8C40C4×D5C3⋊C8C24C4×S3Dic5D10Dic3D6D5S3C8C4C2C1
# reps111111112222161211122222448162248

Matrix representation of S3×C8×D5 in GL4(𝔽241) generated by

8000
0800
0010
0001
,
1000
0100
0023949
001771
,
1000
0100
0010
0064240
,
0100
24018900
0010
0001
,
024000
240000
0010
0001
G:=sub<GL(4,GF(241))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,239,177,0,0,49,1],[1,0,0,0,0,1,0,0,0,0,1,64,0,0,0,240],[0,240,0,0,1,189,0,0,0,0,1,0,0,0,0,1],[0,240,0,0,240,0,0,0,0,0,1,0,0,0,0,1] >;

S3×C8×D5 in GAP, Magma, Sage, TeX

S_3\times C_8\times D_5
% in TeX

G:=Group("S3xC8xD5");
// GroupNames label

G:=SmallGroup(480,319);
// by ID

G=gap.SmallGroup(480,319);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^3=c^2=d^5=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