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## G = S3×C8⋊D5order 480 = 25·3·5

### Direct product of S3 and C8⋊D5

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C8⋊D5
G = < a,b,c,d,e | a3=b2=c8=d5=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >

Subgroups: 636 in 136 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), 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, C2×M4(2), C52C8, C52C8, C40, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), 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×M4(2), C5×C3⋊C8, C3×C52C8, C153C8, C120, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, C2×C8⋊D5, S3×C52C8, C20.32D6, D30.5C4, C3×C8⋊D5, S3×C40, C40⋊S3, C4×S3×D5, S3×C8⋊D5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, C2×M4(2), C4×D5, C22×D5, S3×C2×C4, S3×D5, C8⋊D5, C2×C4×D5, S3×M4(2), C2×S3×D5, C2×C8⋊D5, C4×S3×D5, S3×C8⋊D5

Smallest permutation representation of S3×C8⋊D5
On 120 points
Generators in S120
(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 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 99 65 113 73)(2 100 66 114 74)(3 101 67 115 75)(4 102 68 116 76)(5 103 69 117 77)(6 104 70 118 78)(7 97 71 119 79)(8 98 72 120 80)(9 45 109 54 96)(10 46 110 55 89)(11 47 111 56 90)(12 48 112 49 91)(13 41 105 50 92)(14 42 106 51 93)(15 43 107 52 94)(16 44 108 53 95)(17 83 63 29 34)(18 84 64 30 35)(19 85 57 31 36)(20 86 58 32 37)(21 87 59 25 38)(22 88 60 26 39)(23 81 61 27 40)(24 82 62 28 33)
(1 73)(2 78)(3 75)(4 80)(5 77)(6 74)(7 79)(8 76)(10 14)(12 16)(17 25)(18 30)(19 27)(20 32)(21 29)(22 26)(23 31)(24 28)(34 38)(36 40)(41 92)(42 89)(43 94)(44 91)(45 96)(46 93)(47 90)(48 95)(49 108)(50 105)(51 110)(52 107)(53 112)(54 109)(55 106)(56 111)(57 81)(58 86)(59 83)(60 88)(61 85)(62 82)(63 87)(64 84)(66 70)(68 72)(97 119)(98 116)(99 113)(100 118)(101 115)(102 120)(103 117)(104 114)

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

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

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

78 conjugacy classes

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

78 irreducible representations

 dim 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 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D5 D6 D6 D6 M4(2) D10 D10 D10 C4×S3 C4×S3 C4×D5 C4×D5 C8⋊D5 S3×D5 S3×M4(2) C2×S3×D5 C4×S3×D5 S3×C8⋊D5 kernel S3×C8⋊D5 S3×C5⋊2C8 C20.32D6 D30.5C4 C3×C8⋊D5 S3×C40 C40⋊S3 C4×S3×D5 D5×Dic3 S3×Dic5 D30.C2 C2×S3×D5 C8⋊D5 S3×C8 C5⋊2C8 C40 C4×D5 C5×S3 C3⋊C8 C24 C4×S3 Dic5 D10 Dic3 D6 S3 C8 C5 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 4 2 2 2 2 2 4 4 16 2 2 2 4 8

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

 240 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 216 101 0 0 140 25
,
 1 0 0 0 0 1 0 0 0 0 189 240 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 189 240 0 0 52 52
G:=sub<GL(4,GF(241))| [240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,216,140,0,0,101,25],[1,0,0,0,0,1,0,0,0,0,189,1,0,0,240,0],[1,0,0,0,0,1,0,0,0,0,189,52,0,0,240,52] >;

S3×C8⋊D5 in GAP, Magma, Sage, TeX

S_3\times C_8\rtimes D_5
% in TeX

G:=Group("S3xC8:D5");
// GroupNames label

G:=SmallGroup(480,321);
// by ID

G=gap.SmallGroup(480,321);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=d^5=e^2=1,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,e*c*e=c^5,e*d*e=d^-1>;
// generators/relations

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