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## G = S3×D4⋊2D5order 480 = 25·3·5

### Direct product of S3 and D4⋊2D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×D4⋊2D5
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C4×S3×D5 — S3×D4⋊2D5
 Lower central C15 — C30 — S3×D4⋊2D5
 Upper central C1 — C2 — D4

Generators and relations for S3×D42D5
G = < a,b,c,d,e,f | a3=b2=c4=d2=e5=f2=1, bab=a-1, 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, fdf=c2d, fef=e-1 >

Subgroups: 1500 in 328 conjugacy classes, 112 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C5×S3, C5×S3, C3×D5, D15, C30, C30, C2×C4○D4, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, S3×C2×C4, C4○D12, S3×D4, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, Dic15, C60, S3×D5, C6×D5, S3×C10, S3×C10, S3×C10, D30, C2×C30, C2×Dic10, C2×C4×D5, D42D5, D42D5, C22×Dic5, C2×C5⋊D4, D4×C10, S3×C4○D4, D5×Dic3, S3×Dic5, S3×Dic5, D30.C2, C15⋊D4, C5⋊D12, C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C4×D15, C2×Dic15, C157D4, D4×C15, C2×S3×D5, S3×C2×C10, C2×D42D5, S3×Dic10, D12⋊D5, D125D5, C4×S3×D5, C30.C23, C2×S3×Dic5, Dic3.D10, S3×C5⋊D4, C3×D42D5, C5×S3×D4, D42D15, S3×D42D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, S3×C23, S3×D5, D42D5, C23×D5, S3×C4○D4, C2×S3×D5, C2×D42D5, C22×S3×D5, S3×D42D5

Smallest permutation representation of S3×D42D5
On 120 points
Generators in S120
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(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)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)(66 71)(67 72)(68 73)(69 74)(70 75)(81 86)(82 87)(83 88)(84 89)(85 90)(96 101)(97 102)(98 103)(99 104)(100 105)(111 116)(112 117)(113 118)(114 119)(115 120)
(1 46 16 31)(2 47 17 32)(3 48 18 33)(4 49 19 34)(5 50 20 35)(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 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(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 63)(2 62)(3 61)(4 65)(5 64)(6 68)(7 67)(8 66)(9 70)(10 69)(11 73)(12 72)(13 71)(14 75)(15 74)(16 78)(17 77)(18 76)(19 80)(20 79)(21 83)(22 82)(23 81)(24 85)(25 84)(26 88)(27 87)(28 86)(29 90)(30 89)(31 93)(32 92)(33 91)(34 95)(35 94)(36 98)(37 97)(38 96)(39 100)(40 99)(41 103)(42 102)(43 101)(44 105)(45 104)(46 108)(47 107)(48 106)(49 110)(50 109)(51 113)(52 112)(53 111)(54 115)(55 114)(56 118)(57 117)(58 116)(59 120)(60 119)

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

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

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

60 conjugacy classes

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

60 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 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 C4○D4 D10 D10 D10 D10 D10 S3×D5 D4⋊2D5 S3×C4○D4 C2×S3×D5 C2×S3×D5 S3×D4⋊2D5 kernel S3×D4⋊2D5 S3×Dic10 D12⋊D5 D12⋊5D5 C4×S3×D5 C30.C23 C2×S3×Dic5 Dic3.D10 S3×C5⋊D4 C3×D4⋊2D5 C5×S3×D4 D4⋊2D15 D4⋊2D5 S3×D4 Dic10 C4×D5 C2×Dic5 C5⋊D4 C5×D4 C5×S3 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 D4 S3 C5 C4 C22 C1 # reps 1 1 1 1 1 2 2 2 2 1 1 1 1 2 1 1 2 2 1 4 2 2 4 2 4 2 4 2 2 4 2

Matrix representation of S3×D42D5 in GL6(𝔽61)

 0 60 0 0 0 0 1 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 1 17
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 11 0 0 0 0 50 0 0 0 0 0 0 0 60 0 0 0 0 0 17 1

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,50,0,0,0,0,11,0,0,0,0,0,0,0,60,17,0,0,0,0,0,1] >;

S3×D42D5 in GAP, Magma, Sage, TeX

S_3\times D_4\rtimes_2D_5
% in TeX

G:=Group("S3xD4:2D5");
// GroupNames label

G:=SmallGroup(480,1099);
// by ID

G=gap.SmallGroup(480,1099);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,185,1356,18822]);
// Polycyclic

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

׿
×
𝔽