direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×D4⋊2S3, D20⋊12D6, Dic6⋊11D10, C60.50C23, C30.26C24, Dic30⋊7C22, D30.11C23, Dic15.13C23, D4⋊7(S3×D5), (D4×D5)⋊6S3, C5⋊D4⋊2D6, (C4×S3)⋊8D10, (C5×D4)⋊10D6, C3⋊D4⋊2D10, C15⋊Q8⋊1C22, (C3×D4)⋊10D10, (D5×Dic6)⋊5C2, (C4×D5).51D6, D20⋊S3⋊4C2, D20⋊5S3⋊4C2, D4⋊2D15⋊5C2, (S3×C20)⋊2C22, (C4×D15)⋊2C22, (D4×C15)⋊8C22, (C3×D20)⋊8C22, C15⋊7D4⋊2C22, C15⋊D4⋊4C22, C3⋊D20⋊4C22, (C2×C30).2C23, C6.26(C23×D5), (C2×Dic3)⋊14D10, Dic5.D6⋊3C2, C30.C23⋊3C2, C10.26(S3×C23), C20.50(C22×S3), (C5×Dic6)⋊7C22, (C22×D5).69D6, D6.11(C22×D5), (C6×D5).44C23, C12.50(C22×D5), (S3×C10).11C23, D30.C2⋊10C22, (D5×Dic3)⋊10C22, (S3×Dic5)⋊10C22, (D5×C12).18C22, D10.11(C22×S3), (C10×Dic3)⋊12C22, (C2×Dic15)⋊16C22, (C3×Dic5).12C23, Dic3.12(C22×D5), Dic5.13(C22×S3), (C5×Dic3).13C23, (C4×S3×D5)⋊2C2, (C3×D4×D5)⋊5C2, C3⋊3(D5×C4○D4), C4.50(C2×S3×D5), C15⋊11(C2×C4○D4), C5⋊3(C2×D4⋊2S3), (D5×C3⋊D4)⋊3C2, C22.2(C2×S3×D5), (C2×D5×Dic3)⋊21C2, (C3×D5)⋊2(C4○D4), (C5×D4⋊2S3)⋊5C2, (C2×S3×D5).8C22, C2.29(C22×S3×D5), (C3×C5⋊D4)⋊2C22, (C5×C3⋊D4)⋊2C22, (D5×C2×C6).60C22, (C2×C6).2(C22×D5), (C2×C10).2(C22×S3), SmallGroup(480,1098)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×D4⋊2S3
G = < a,b,c,d,e,f | a5=b2=c4=d2=e3=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: 1548 in 328 conjugacy classes, 112 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, C23, D5, D5, C10, C10, Dic3, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C5×S3, C3×D5, C3×D5, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C2×Dic6, S3×C2×C4, D4⋊2S3, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, S3×D5, C6×D5, C6×D5, C6×D5, S3×C10, D30, C2×C30, C2×C4×D5, C4○D20, D4×D5, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, C2×D4⋊2S3, D5×Dic3, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, C2×S3×D5, D5×C2×C6, D5×C4○D4, D5×Dic6, D20⋊5S3, D20⋊S3, C4×S3×D5, C2×D5×Dic3, Dic5.D6, C30.C23, D5×C3⋊D4, C3×D4×D5, C5×D4⋊2S3, D4⋊2D15, D5×D4⋊2S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, D4⋊2S3, S3×C23, S3×D5, C23×D5, C2×D4⋊2S3, C2×S3×D5, D5×C4○D4, C22×S3×D5, D5×D4⋊2S3
►On 120 points
Generators in S120
(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)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)
(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 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(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 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)
G:=sub<Sym(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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(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)(6,66)(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)>;
G:=Group( (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)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(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)(6,66)(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) );
G=PermutationGroup([[(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),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(62,65),(63,64),(67,70),(68,69),(72,75),(73,74),(77,80),(78,79),(82,85),(83,84),(87,90),(88,89),(92,95),(93,94),(97,100),(98,99),(102,105),(103,104),(107,110),(108,109),(112,115),(113,114),(117,120),(118,119)], [(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)], [(1,34),(2,35),(3,31),(4,32),(5,33),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55),(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),(6,66),(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)]])
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 | 6E | 6F | 6G | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 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 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 5 | 5 | 6 | 10 | 10 | 30 | 2 | 2 | 3 | 3 | 6 | 6 | 10 | 15 | 15 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 20 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 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 | D4⋊2S3 | S3×D5 | C2×S3×D5 | C2×S3×D5 | D5×C4○D4 | D5×D4⋊2S3 |
| kernel | D5×D4⋊2S3 | D5×Dic6 | D20⋊5S3 | D20⋊S3 | C4×S3×D5 | C2×D5×Dic3 | Dic5.D6 | C30.C23 | D5×C3⋊D4 | C3×D4×D5 | C5×D4⋊2S3 | D4⋊2D15 | D4×D5 | D4⋊2S3 | C4×D5 | D20 | C5⋊D4 | C5×D4 | C22×D5 | C3×D5 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | D5 | D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D5×D4⋊2S3 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 43 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 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 | 1 | 3 |
| 0 | 0 | 0 | 0 | 40 | 60 |
| 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 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 46 | 0 | 0 | 0 | 0 |
| 49 | 59 | 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 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 27 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 33 |
| 0 | 0 | 0 | 0 | 13 | 50 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,3,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,60],[1,49,0,0,0,0,46,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,27,0,0,0,0,13,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,13,0,0,0,0,33,50] >;
D5×D4⋊2S3 in GAP, Magma, Sage, TeX
D_5\times D_4\rtimes_2S_3
% in TeX
G:=Group("D5xD4:2S3"); // GroupNames label
G:=SmallGroup(480,1098);
// by ID
G=gap.SmallGroup(480,1098);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,346,185,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^4=d^2=e^3=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