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## G = S3×C2×C4×D5order 480 = 25·3·5

### Direct product of C2×C4, S3 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×C2×C4×D5
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C22×S3×D5 — S3×C2×C4×D5
 Lower central C15 — S3×C2×C4×D5
 Upper central C1 — C2×C4

Generators and relations for S3×C2×C4×D5
G = < a,b,c,d,e,f | a2=b4=c3=d2=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2140 in 472 conjugacy classes, 180 normal (36 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, C23×C4, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C2×C30, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, S3×C22×C4, D5×Dic3, S3×Dic5, D30.C2, D5×C12, C6×Dic5, S3×C20, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, D5×C22×C4, C4×S3×D5, C2×D5×Dic3, C2×S3×Dic5, C2×D30.C2, D5×C2×C12, S3×C2×C20, C2×C4×D15, C22×S3×D5, S3×C2×C4×D5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C24, D10, C4×S3, C22×S3, C23×C4, C4×D5, C22×D5, S3×C2×C4, S3×C23, S3×D5, C2×C4×D5, C23×D5, S3×C22×C4, C2×S3×D5, D5×C22×C4, C4×S3×D5, C22×S3×D5, S3×C2×C4×D5

Smallest permutation representation of S3×C2×C4×D5
On 120 points
Generators in S120
(1 64)(2 65)(3 61)(4 62)(5 63)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 106 76 91)(62 107 77 92)(63 108 78 93)(64 109 79 94)(65 110 80 95)(66 111 81 96)(67 112 82 97)(68 113 83 98)(69 114 84 99)(70 115 85 100)(71 116 86 101)(72 117 87 102)(73 118 88 103)(74 119 89 104)(75 120 90 105)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(1 64)(2 65)(3 61)(4 62)(5 63)(6 71)(7 72)(8 73)(9 74)(10 75)(11 66)(12 67)(13 68)(14 69)(15 70)(16 76)(17 77)(18 78)(19 79)(20 80)(21 86)(22 87)(23 88)(24 89)(25 90)(26 81)(27 82)(28 83)(29 84)(30 85)(31 91)(32 92)(33 93)(34 94)(35 95)(36 101)(37 102)(38 103)(39 104)(40 105)(41 96)(42 97)(43 98)(44 99)(45 100)(46 106)(47 107)(48 108)(49 109)(50 110)(51 116)(52 117)(53 118)(54 119)(55 120)(56 111)(57 112)(58 113)(59 114)(60 115)
(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 63)(2 62)(3 61)(4 65)(5 64)(6 66)(7 70)(8 69)(9 68)(10 67)(11 71)(12 75)(13 74)(14 73)(15 72)(16 76)(17 80)(18 79)(19 78)(20 77)(21 81)(22 85)(23 84)(24 83)(25 82)(26 86)(27 90)(28 89)(29 88)(30 87)(31 91)(32 95)(33 94)(34 93)(35 92)(36 96)(37 100)(38 99)(39 98)(40 97)(41 101)(42 105)(43 104)(44 103)(45 102)(46 106)(47 110)(48 109)(49 108)(50 107)(51 111)(52 115)(53 114)(54 113)(55 112)(56 116)(57 120)(58 119)(59 118)(60 117)

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

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

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

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 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 C2 C4 S3 D5 D6 D6 D6 D6 D10 D10 D10 D10 C4×S3 C4×D5 S3×D5 C2×S3×D5 C2×S3×D5 C4×S3×D5 kernel S3×C2×C4×D5 C4×S3×D5 C2×D5×Dic3 C2×S3×Dic5 C2×D30.C2 D5×C2×C12 S3×C2×C20 C2×C4×D15 C22×S3×D5 C2×S3×D5 C2×C4×D5 S3×C2×C4 C4×D5 C2×Dic5 C2×C20 C22×D5 C4×S3 C2×Dic3 C2×C12 C22×S3 D10 D6 C2×C4 C4 C22 C2 # reps 1 8 1 1 1 1 1 1 1 16 1 2 4 1 1 1 8 2 2 2 8 16 2 4 2 8

Matrix representation of S3×C2×C4×D5 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 11 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 1 0 0 0 0 1 15 0 0 12 59
,
 60 0 0 0 0 60 0 0 0 0 60 46 0 0 0 1
,
 17 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 60 44 0 0 0 1 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,1,12,0,0,15,59],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,46,1],[17,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,44,1,0,0,0,0,1,0,0,0,0,1] >;

S3×C2×C4×D5 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_4\times D_5
% in TeX

G:=Group("S3xC2xC4xD5");
// GroupNames label

G:=SmallGroup(480,1086);
// by ID

G=gap.SmallGroup(480,1086);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^4=c^3=d^2=e^5=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

׿
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𝔽