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G = S3×D42D5order 480 = 25·3·5

Direct product of S3 and D42D5

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

Aliases: S3×D42D5, D1212D10, Dic1011D6, C30.27C24, C60.51C23, Dic308C22, D30.12C23, Dic15.14C23, (C4×D5)⋊8D6, D48(S3×D5), (S3×D4)⋊6D5, C5⋊D43D6, (C5×D4)⋊11D6, C3⋊D43D10, C15⋊Q82C22, (C3×D4)⋊11D10, D125D54C2, D12⋊D54C2, D42D156C2, (C2×Dic5)⋊14D6, (S3×Dic10)⋊5C2, (C4×S3).27D10, (C5×D12)⋊8C22, (D4×C15)⋊9C22, (C4×D15)⋊3C22, (D5×C12)⋊2C22, C15⋊D45C22, C5⋊D124C22, C157D43C22, (C2×C30).3C23, C6.27(C23×D5), Dic3.D103C2, C30.C234C2, C20.51(C22×S3), C10.27(S3×C23), D6.12(C22×D5), (C6×D5).11C23, C12.51(C22×D5), (S3×C20).18C22, (S3×C10).12C23, D30.C211C22, (D5×Dic3)⋊11C22, (S3×Dic5)⋊11C22, (C3×Dic10)⋊7C22, (C6×Dic5)⋊12C22, D10.12(C22×S3), (C22×S3).60D10, (C2×Dic15)⋊17C22, (C5×Dic3).14C23, (C3×Dic5).45C23, Dic3.13(C22×D5), Dic5.57(C22×S3), (C4×S3×D5)⋊3C2, (C5×S3×D4)⋊5C2, C53(S3×C4○D4), C4.51(C2×S3×D5), C1512(C2×C4○D4), C33(C2×D42D5), (S3×C5⋊D4)⋊3C2, C22.3(C2×S3×D5), (C2×S3×Dic5)⋊21C2, (C5×S3)⋊2(C4○D4), (C3×D42D5)⋊5C2, (C2×S3×D5).9C22, C2.30(C22×S3×D5), (C5×C3⋊D4)⋊3C22, (C3×C5⋊D4)⋊3C22, (C2×C6).3(C22×D5), (S3×C2×C10).60C22, (C2×C10).3(C22×S3), SmallGroup(480,1099)

Series: Derived Chief Lower central Upper central

C1C30 — S3×D42D5
C1C5C15C30C6×D5C2×S3×D5C4×S3×D5 — S3×D42D5
C15C30 — S3×D42D5
C1C2D4

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 [×8], C3, C4, C4 [×7], C22 [×2], C22 [×11], C5, S3 [×2], S3 [×3], C6, C6 [×3], C2×C4 [×16], D4, D4 [×11], Q8 [×4], C23 [×3], D5 [×2], C10, C10 [×6], Dic3, Dic3 [×3], C12, C12 [×3], D6, D6 [×2], D6 [×7], C2×C6 [×2], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5, Dic5 [×2], Dic5 [×3], C20, C20, D10, D10 [×3], C2×C10 [×2], C2×C10 [×7], Dic6 [×3], C4×S3, C4×S3 [×9], D12, D12 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C22×S3 [×2], C22×S3, C5×S3 [×2], C5×S3 [×2], C3×D5, D15, C30, C30 [×2], C2×C4○D4, Dic10, Dic10 [×3], C4×D5, C4×D5 [×3], C2×Dic5 [×2], C2×Dic5 [×9], C5⋊D4 [×2], C5⋊D4 [×6], C2×C20, C5×D4, C5×D4 [×3], C22×D5, C22×C10 [×2], S3×C2×C4 [×3], C4○D12 [×3], S3×D4, S3×D4 [×2], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5 [×2], Dic15, Dic15 [×2], C60, S3×D5 [×2], C6×D5, S3×C10, S3×C10 [×2], S3×C10 [×4], D30, C2×C30 [×2], C2×Dic10, C2×C4×D5, D42D5, D42D5 [×7], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, S3×C4○D4, D5×Dic3, S3×Dic5, S3×Dic5 [×6], D30.C2, C15⋊D4 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], S3×C20, C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15, C2×Dic15 [×2], C157D4 [×2], D4×C15, C2×S3×D5, S3×C2×C10 [×2], C2×D42D5, S3×Dic10, D12⋊D5, D125D5, C4×S3×D5, C30.C23 [×2], C2×S3×Dic5 [×2], Dic3.D10 [×2], S3×C5⋊D4 [×2], C3×D42D5, C5×S3×D4, D42D15, S3×D42D5
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, D42D5 [×2], C23×D5, S3×C4○D4, C2×S3×D5 [×3], 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 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D10A10B10C10D10E10F10G10H10I10J10K10L10M10N12A12B12C12D12E15A15B20A20B20C20D30A30B30C30D30E30F60A60B
order122222222234444444444556666101010101010101010101010101012121212121515202020203030303030306060
size1122336610302255610101515303022244202244446666121212124101020204444121244888888

60 irreducible representations

dim1111111111112222222222222444448
type+++++++++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6C4○D4D10D10D10D10D10S3×D5D42D5S3×C4○D4C2×S3×D5C2×S3×D5S3×D42D5
kernelS3×D42D5S3×Dic10D12⋊D5D125D5C4×S3×D5C30.C23C2×S3×Dic5Dic3.D10S3×C5⋊D4C3×D42D5C5×S3×D4D42D15D42D5S3×D4Dic10C4×D5C2×Dic5C5⋊D4C5×D4C5×S3C4×S3D12C3⋊D4C3×D4C22×S3D4S3C5C4C22C1
# reps1111122221111211221422424242242

Matrix representation of S3×D42D5 in GL6(𝔽61)

0600000
1600000
001000
000100
000010
000001
,
010000
100000
001000
000100
000010
000001
,
6000000
0600000
000100
0060000
000010
000001
,
6000000
0600000
0006000
0060000
000010
000001
,
100000
010000
001000
000100
0000060
0000117
,
100000
010000
0001100
0050000
0000600
0000171

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

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