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## G = S3×C8×D5order 480 = 25·3·5

### Direct product of C8, S3 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×C8×D5
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — C4×S3×D5 — S3×C8×D5
 Lower central C15 — S3×C8×D5
 Upper central C1 — C8

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, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C22×C8, C52C8, C52C8, C40, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C8×D5, 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, C4, C22, S3, C8, C2×C4, C23, D5, D6, C2×C8, C22×C4, D10, C4×S3, C22×S3, C22×C8, C4×D5, C22×D5, S3×C8, S3×C2×C4, S3×D5, C8×D5, 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 64 50)(2 57 51)(3 58 52)(4 59 53)(5 60 54)(6 61 55)(7 62 56)(8 63 49)(9 69 39)(10 70 40)(11 71 33)(12 72 34)(13 65 35)(14 66 36)(15 67 37)(16 68 38)(17 48 120)(18 41 113)(19 42 114)(20 43 115)(21 44 116)(22 45 117)(23 46 118)(24 47 119)(25 95 102)(26 96 103)(27 89 104)(28 90 97)(29 91 98)(30 92 99)(31 93 100)(32 94 101)(73 84 105)(74 85 106)(75 86 107)(76 87 108)(77 88 109)(78 81 110)(79 82 111)(80 83 112)
(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(81 110)(82 111)(83 112)(84 105)(85 106)(86 107)(87 108)(88 109)
(1 73 113 65 99)(2 74 114 66 100)(3 75 115 67 101)(4 76 116 68 102)(5 77 117 69 103)(6 78 118 70 104)(7 79 119 71 97)(8 80 120 72 98)(9 96 54 109 45)(10 89 55 110 46)(11 90 56 111 47)(12 91 49 112 48)(13 92 50 105 41)(14 93 51 106 42)(15 94 52 107 43)(16 95 53 108 44)(17 34 29 63 83)(18 35 30 64 84)(19 36 31 57 85)(20 37 32 58 86)(21 38 25 59 87)(22 39 26 60 88)(23 40 27 61 81)(24 33 28 62 82)
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 21)(18 22)(19 23)(20 24)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 86)(34 87)(35 88)(36 81)(37 82)(38 83)(39 84)(40 85)(41 45)(42 46)(43 47)(44 48)(49 95)(50 96)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(65 77)(66 78)(67 79)(68 80)(69 73)(70 74)(71 75)(72 76)(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), (1,64,50)(2,57,51)(3,58,52)(4,59,53)(5,60,54)(6,61,55)(7,62,56)(8,63,49)(9,69,39)(10,70,40)(11,71,33)(12,72,34)(13,65,35)(14,66,36)(15,67,37)(16,68,38)(17,48,120)(18,41,113)(19,42,114)(20,43,115)(21,44,116)(22,45,117)(23,46,118)(24,47,119)(25,95,102)(26,96,103)(27,89,104)(28,90,97)(29,91,98)(30,92,99)(31,93,100)(32,94,101)(73,84,105)(74,85,106)(75,86,107)(76,87,108)(77,88,109)(78,81,110)(79,82,111)(80,83,112), (9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(81,110)(82,111)(83,112)(84,105)(85,106)(86,107)(87,108)(88,109), (1,73,113,65,99)(2,74,114,66,100)(3,75,115,67,101)(4,76,116,68,102)(5,77,117,69,103)(6,78,118,70,104)(7,79,119,71,97)(8,80,120,72,98)(9,96,54,109,45)(10,89,55,110,46)(11,90,56,111,47)(12,91,49,112,48)(13,92,50,105,41)(14,93,51,106,42)(15,94,52,107,43)(16,95,53,108,44)(17,34,29,63,83)(18,35,30,64,84)(19,36,31,57,85)(20,37,32,58,86)(21,38,25,59,87)(22,39,26,60,88)(23,40,27,61,81)(24,33,28,62,82), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,21)(18,22)(19,23)(20,24)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,86)(34,87)(35,88)(36,81)(37,82)(38,83)(39,84)(40,85)(41,45)(42,46)(43,47)(44,48)(49,95)(50,96)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76)(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), (1,64,50)(2,57,51)(3,58,52)(4,59,53)(5,60,54)(6,61,55)(7,62,56)(8,63,49)(9,69,39)(10,70,40)(11,71,33)(12,72,34)(13,65,35)(14,66,36)(15,67,37)(16,68,38)(17,48,120)(18,41,113)(19,42,114)(20,43,115)(21,44,116)(22,45,117)(23,46,118)(24,47,119)(25,95,102)(26,96,103)(27,89,104)(28,90,97)(29,91,98)(30,92,99)(31,93,100)(32,94,101)(73,84,105)(74,85,106)(75,86,107)(76,87,108)(77,88,109)(78,81,110)(79,82,111)(80,83,112), (9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(81,110)(82,111)(83,112)(84,105)(85,106)(86,107)(87,108)(88,109), (1,73,113,65,99)(2,74,114,66,100)(3,75,115,67,101)(4,76,116,68,102)(5,77,117,69,103)(6,78,118,70,104)(7,79,119,71,97)(8,80,120,72,98)(9,96,54,109,45)(10,89,55,110,46)(11,90,56,111,47)(12,91,49,112,48)(13,92,50,105,41)(14,93,51,106,42)(15,94,52,107,43)(16,95,53,108,44)(17,34,29,63,83)(18,35,30,64,84)(19,36,31,57,85)(20,37,32,58,86)(21,38,25,59,87)(22,39,26,60,88)(23,40,27,61,81)(24,33,28,62,82), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,21)(18,22)(19,23)(20,24)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,86)(34,87)(35,88)(36,81)(37,82)(38,83)(39,84)(40,85)(41,45)(42,46)(43,47)(44,48)(49,95)(50,96)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76)(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)], [(1,64,50),(2,57,51),(3,58,52),(4,59,53),(5,60,54),(6,61,55),(7,62,56),(8,63,49),(9,69,39),(10,70,40),(11,71,33),(12,72,34),(13,65,35),(14,66,36),(15,67,37),(16,68,38),(17,48,120),(18,41,113),(19,42,114),(20,43,115),(21,44,116),(22,45,117),(23,46,118),(24,47,119),(25,95,102),(26,96,103),(27,89,104),(28,90,97),(29,91,98),(30,92,99),(31,93,100),(32,94,101),(73,84,105),(74,85,106),(75,86,107),(76,87,108),(77,88,109),(78,81,110),(79,82,111),(80,83,112)], [(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,95),(26,96),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(49,63),(50,64),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62),(81,110),(82,111),(83,112),(84,105),(85,106),(86,107),(87,108),(88,109)], [(1,73,113,65,99),(2,74,114,66,100),(3,75,115,67,101),(4,76,116,68,102),(5,77,117,69,103),(6,78,118,70,104),(7,79,119,71,97),(8,80,120,72,98),(9,96,54,109,45),(10,89,55,110,46),(11,90,56,111,47),(12,91,49,112,48),(13,92,50,105,41),(14,93,51,106,42),(15,94,52,107,43),(16,95,53,108,44),(17,34,29,63,83),(18,35,30,64,84),(19,36,31,57,85),(20,37,32,58,86),(21,38,25,59,87),(22,39,26,60,88),(23,40,27,61,81),(24,33,28,62,82)], [(1,103),(2,104),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,21),(18,22),(19,23),(20,24),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,86),(34,87),(35,88),(36,81),(37,82),(38,83),(39,84),(40,85),(41,45),(42,46),(43,47),(44,48),(49,95),(50,96),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(65,77),(66,78),(67,79),(68,80),(69,73),(70,74),(71,75),(72,76),(113,117),(114,118),(115,119),(116,120)]])

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 8O 8P 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 40A ··· 40H 40I ··· 40P 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 12 12 12 12 15 15 20 20 20 20 20 20 20 20 24 24 24 24 24 24 24 24 30 30 40 ··· 40 40 ··· 40 60 60 60 60 120 ··· 120 size 1 1 3 3 5 5 15 15 2 1 1 3 3 5 5 15 15 2 2 2 10 10 1 1 1 1 3 3 3 3 5 5 5 5 15 15 15 15 2 2 6 6 6 6 2 2 10 10 4 4 2 2 2 2 6 6 6 6 2 2 2 2 10 10 10 10 4 4 2 ··· 2 6 ··· 6 4 4 4 4 4 ··· 4

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 S3 D5 D6 D6 D6 D10 D10 D10 C4×S3 C4×S3 C4×D5 C4×D5 S3×C8 C8×D5 S3×D5 C2×S3×D5 C4×S3×D5 S3×C8×D5 kernel S3×C8×D5 D5×C3⋊C8 S3×C5⋊2C8 D15⋊2C8 D5×C24 S3×C40 C8×D15 C4×S3×D5 D5×Dic3 S3×Dic5 D30.C2 C2×S3×D5 S3×D5 C8×D5 S3×C8 C5⋊2C8 C40 C4×D5 C3⋊C8 C24 C4×S3 Dic5 D10 Dic3 D6 D5 S3 C8 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 16 1 2 1 1 1 2 2 2 2 2 4 4 8 16 2 2 4 8

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

 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 49 0 0 177 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 64 240
,
 0 1 0 0 240 189 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
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

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