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## G = S3×D4⋊D5order 480 = 25·3·5

### Direct product of S3 and D4⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — S3×D4⋊D5
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — S3×D20 — S3×D4⋊D5
 Lower central C15 — C30 — C60 — S3×D4⋊D5
 Upper central C1 — C2 — C4 — D4

Generators and relations for S3×D4⋊D5
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=fcf=c-1, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 972 in 152 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C2×C4, D4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, D8, C2×D4, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×D4, C22×S3, C5×S3, C5×S3, C3×D5, D15, C30, C30, C2×D8, C52C8, C52C8, D20, D20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, S3×D4, C5×Dic3, C60, S3×D5, C6×D5, S3×C10, S3×C10, D30, C2×C30, C2×C52C8, D4⋊D5, D4⋊D5, C2×D20, D4×C10, S3×D8, C3×C52C8, C153C8, C3⋊D20, C3×D20, S3×C20, C5×D12, C5×C3⋊D4, D60, D4×C15, C2×S3×D5, S3×C2×C10, C2×D4⋊D5, S3×C52C8, C15⋊D8, C5⋊D24, C3×D4⋊D5, D4⋊D15, S3×D20, C5×S3×D4, S3×D4⋊D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C22×S3, C2×D8, C5⋊D4, C22×D5, S3×D4, S3×D5, D4⋊D5, C2×C5⋊D4, S3×D8, C2×S3×D5, C2×D4⋊D5, S3×C5⋊D4, S3×D4⋊D5

Smallest permutation representation of S3×D4⋊D5
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 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)
(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 5)(2 4)(6 10)(7 9)(11 15)(12 14)(16 20)(17 19)(21 25)(22 24)(26 30)(27 29)(31 50)(32 49)(33 48)(34 47)(35 46)(36 55)(37 54)(38 53)(39 52)(40 51)(41 60)(42 59)(43 58)(44 57)(45 56)(61 95)(62 94)(63 93)(64 92)(65 91)(66 100)(67 99)(68 98)(69 97)(70 96)(71 105)(72 104)(73 103)(74 102)(75 101)(76 110)(77 109)(78 108)(79 107)(80 106)(81 115)(82 114)(83 113)(84 112)(85 111)(86 120)(87 119)(88 118)(89 117)(90 116)

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 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 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D8 D10 D10 D10 C5⋊D4 C5⋊D4 S3×D4 S3×D5 D4⋊D5 S3×D8 C2×S3×D5 S3×C5⋊D4 S3×D4⋊D5 kernel S3×D4⋊D5 S3×C5⋊2C8 C15⋊D8 C5⋊D24 C3×D4⋊D5 D4⋊D15 S3×D20 C5×S3×D4 D4⋊D5 C5×Dic3 S3×C10 S3×D4 C5⋊2C8 D20 C5×D4 C5×S3 C4×S3 D12 C3×D4 Dic3 D6 C10 D4 S3 C5 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 1 2 4 2 2 4 2

Matrix representation of S3×D4⋊D5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 187 0 0 0 0 67 239 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 0 240 54 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 113 0 0 0 0 177 240
,
 27 138 0 0 0 0 103 214 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 38 0 0 0 0 222 0
,
 0 240 0 0 0 0 1 189 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
,
 52 240 0 0 0 0 52 189 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 0 177 240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,67,0,0,0,0,187,239,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,54,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,177,0,0,0,0,113,240],[27,103,0,0,0,0,138,214,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,222,0,0,0,0,38,0],[0,1,0,0,0,0,240,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,52,0,0,0,0,240,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,177,0,0,0,0,0,240] >;

S3×D4⋊D5 in GAP, Magma, Sage, TeX

S_3\times D_4\rtimes D_5
% in TeX

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

G:=SmallGroup(480,555);
// by ID

G=gap.SmallGroup(480,555);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,675,185,80,1356,18822]);
// 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=f*c*f=c^-1,c*e=e*c,d*e=e*d,f*d*f=c*d,f*e*f=e^-1>;
// generators/relations

׿
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