S3xD4:2D5 - GroupNames

G = S₃xD₄:₂D₅ order 480 = 2⁵·3·5

Direct product of S₃ and D₄:₂D₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S₃xD₄:₂D₅, D₁₂:₁₂D₁₀, Dic₁₀:₁₁D₆, C₃₀.₂₇C₂⁴, C₆₀.₅₁C₂³, Dic₃₀:₈C₂², D₃₀.₁₂C₂³, Dic₁₅.₁₄C₂³, (C₄xD₅):₈D₆, D₄:₈(S₃xD₅), (S₃xD₄):₆D₅, C₅:D₄:₃D₆, (C₅xD₄):₁₁D₆, C₃:D₄:₃D₁₀, C₁₅:Q₈:₂C₂², (C₃xD₄):₁₁D₁₀, D₁₂:₅D₅:₄C₂, D₁₂:D₅:₄C₂, D₄:₂D₁₅:₆C₂, (C₂xDic₅):₁₄D₆, (S₃xDic₁₀):₅C₂, (C₄xS₃).₂₇D₁₀, (C₅xD₁₂):₈C₂², (D₄xC₁₅):₉C₂², (C₄xD₁₅):₃C₂², (D₅xC₁₂):₂C₂², C₁₅:D₄:₅C₂², C₅:D₁₂:₄C₂², C₁₅:₇D₄:₃C₂², (C₂xC₃₀).₃C₂³, C₆.₂₇(C₂³xD₅), Dic₃.D₁₀:₃C₂, C₃₀.C₂³:₄C₂, C₂₀.₅₁(C₂²xS₃), C₁₀.₂₇(S₃xC₂³), D₆.₁₂(C₂²xD₅), (C₆xD₅).₁₁C₂³, C₁₂.₅₁(C₂²xD₅), (S₃xC₂₀).₁₈C₂², (S₃xC₁₀).₁₂C₂³, D₃₀.C₂:₁₁C₂², (D₅xDic₃):₁₁C₂², (S₃xDic₅):₁₁C₂², (C₃xDic₁₀):₇C₂², (C₆xDic₅):₁₂C₂², D₁₀.₁₂(C₂²xS₃), (C₂²xS₃).₆₀D₁₀, (C₂xDic₁₅):₁₇C₂², (C₅xDic₃).₁₄C₂³, (C₃xDic₅).₄₅C₂³, Dic₃.₁₃(C₂²xD₅), Dic₅.₅₇(C₂²xS₃), (C₄xS₃xD₅):₃C₂, (C₅xS₃xD₄):₅C₂, C₅:₃(S₃xC₄oD₄), C₄.₅₁(C₂xS₃xD₅), C₁₅:₁₂(C₂xC₄oD₄), C₃:₃(C₂xD₄:₂D₅), (S₃xC₅:D₄):₃C₂, C₂².₃(C₂xS₃xD₅), (C₂xS₃xDic₅):₂₁C₂, (C₅xS₃):₂(C₄oD₄), (C₃xD₄:₂D₅):₅C₂, (C₂xS₃xD₅).₉C₂², C₂.₃₀(C₂²xS₃xD₅), (C₅xC₃:D₄):₃C₂², (C₃xC₅:D₄):₃C₂², (C₂xC₆).₃(C₂²xD₅), (S₃xC₂xC₁₀).₆₀C₂², (C₂xC₁₀).₃(C₂²xS₃), SmallGroup(480,1099)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — S₃xD₄:₂D₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆xD₅ — C₂xS₃xD₅ — C₄xS₃xD₅ — S₃xD₄:₂D₅

Lower central

C₁₅ — C₃₀ — S₃xD₄:₂D₅

Upper central

C₁ — C₂ — D₄

Generators and relations for S₃xD₄:₂D₅
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, C₂xC₄, D₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, D₆, C₂xC₆, C₂xC₆, C₁₅, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, Dic₅, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, Dic₆, C₄xS₃, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xS₃, C₅xS₃, C₅xS₃, C₃xD₅, D₁₅, C₃₀, C₃₀, C₂xC₄oD₄, Dic₁₀, Dic₁₀, C₄xD₅, C₄xD₅, C₂xDic₅, C₂xDic₅, C₅:D₄, C₅:D₄, C₂xC₂₀, C₅xD₄, C₅xD₄, C₂²xD₅, C₂²xC₁₀, S₃xC₂xC₄, C₄oD₁₂, S₃xD₄, S₃xD₄, D₄:₂S₃, S₃xQ₈, Q₈:₃S₃, C₃xC₄oD₄, C₅xDic₃, C₃xDic₅, C₃xDic₅, Dic₁₅, Dic₁₅, C₆₀, S₃xD₅, C₆xD₅, S₃xC₁₀, S₃xC₁₀, S₃xC₁₀, D₃₀, C₂xC₃₀, C₂xDic₁₀, C₂xC₄xD₅, D₄:₂D₅, D₄:₂D₅, C₂²xDic₅, C₂xC₅:D₄, D₄xC₁₀, S₃xC₄oD₄, D₅xDic₃, S₃xDic₅, S₃xDic₅, D₃₀.C₂, C₁₅:D₄, C₅:D₁₂, C₁₅:Q₈, C₃xDic₁₀, D₅xC₁₂, C₆xDic₅, C₃xC₅:D₄, S₃xC₂₀, C₅xD₁₂, C₅xC₃:D₄, Dic₃₀, C₄xD₁₅, C₂xDic₁₅, C₁₅:₇D₄, D₄xC₁₅, C₂xS₃xD₅, S₃xC₂xC₁₀, C₂xD₄:₂D₅, S₃xDic₁₀, D₁₂:D₅, D₁₂:₅D₅, C₄xS₃xD₅, C₃₀.C₂³, C₂xS₃xDic₅, Dic₃.D₁₀, S₃xC₅:D₄, C₃xD₄:₂D₅, C₅xS₃xD₄, D₄:₂D₁₅, S₃xD₄:₂D₅
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₄oD₄, C₂⁴, D₁₀, C₂²xS₃, C₂xC₄oD₄, C₂²xD₅, S₃xC₂³, S₃xD₅, D₄:₂D₅, C₂³xD₅, S₃xC₄oD₄, C₂xS₃xD₅, C₂xD₄:₂D₅, C₂²xS₃xD₅, S₃xD₄:₂D₅

Smallest permutation representation of S₃xD₄:₂D₅
►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	C₄oD₄	D₁₀	D₁₀	D₁₀	D₁₀	D₁₀	S₃xD₅	D₄:₂D₅	S₃xC₄oD₄	C₂xS₃xD₅	C₂xS₃xD₅	S₃xD₄:₂D₅
kernel	S₃xD₄:₂D₅	S₃xDic₁₀	D₁₂:D₅	D₁₂:₅D₅	C₄xS₃xD₅	C₃₀.C₂³	C₂xS₃xDic₅	Dic₃.D₁₀	S₃xC₅:D₄	C₃xD₄:₂D₅	C₅xS₃xD₄	D₄:₂D₁₅	D₄:₂D₅	S₃xD₄	Dic₁₀	C₄xD₅	C₂xDic₅	C₅:D₄	C₅xD₄	C₅xS₃	C₄xS₃	D₁₂	C₃:D₄	C₃xD₄	C₂²xS₃	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 S₃xD₄:₂D₅ ►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] >;

S₃xD₄:₂D₅ 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

G = S3xD4:2D5 order 480 = 25·3·5

Direct product of S3 and D4:2D5

G = S₃xD₄:₂D₅ order 480 = 2⁵·3·5

Direct product of S₃ and D₄:₂D₅