direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4.F5, C5⋊C8⋊1D6, C5⋊1(S3×M4(2)), (C4×S3).1F5, C4.20(S3×F5), (S3×C20).1C4, C20.20(C4×S3), C60.20(C2×C4), D30.8(C2×C4), (C4×D5).65D6, (C4×D15).1C4, C15⋊1(C2×M4(2)), D6.12(C2×F5), C12.15(C2×F5), C12.F5⋊5C2, C15⋊C8⋊1C22, C6.7(C22×F5), D10.21(C4×S3), (C5×S3)⋊1M4(2), Dic3.F5⋊1C2, C30.7(C22×C4), Dic3.8(C2×F5), (D5×Dic3).5C4, Dic15.10(C2×C4), (D5×C12).51C22, D30.C2.13C22, (C3×Dic5).25C23, Dic5.27(C22×S3), (S3×Dic5).13C22, (S3×C5⋊C8)⋊2C2, C3⋊1(C2×C4.F5), C10.7(S3×C2×C4), (C2×S3×D5).5C4, (C4×S3×D5).3C2, C2.11(C2×S3×F5), (C3×C5⋊C8)⋊1C22, (C3×C4.F5)⋊5C2, (S3×C10).8(C2×C4), (C6×D5).16(C2×C4), (C5×Dic3).10(C2×C4), SmallGroup(480,988)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4.F5
G = < a,b,c,d,e | a3=b2=c4=d5=1, e4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >
Subgroups: 676 in 136 conjugacy classes, 50 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C2×C4, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C4.F5, C4.F5, C2×C5⋊C8, C2×C4×D5, S3×M4(2), C3×C5⋊C8, C15⋊C8, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, C2×C4.F5, S3×C5⋊C8, Dic3.F5, C3×C4.F5, C12.F5, C4×S3×D5, S3×C4.F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, F5, C4×S3, C22×S3, C2×M4(2), C2×F5, S3×C2×C4, C4.F5, C22×F5, S3×M4(2), S3×F5, C2×C4.F5, C2×S3×F5, S3×C4.F5
►On 120 points
Generators in S120
(1 64 50)(2 57 51)(3 58 52)(4 59 53)(5 60 54)(6 61 55)(7 62 56)(8 63 49)(9 69 39)(10 70 40)(11 71 33)(12 72 34)(13 65 35)(14 66 36)(15 67 37)(16 68 38)(17 48 120)(18 41 113)(19 42 114)(20 43 115)(21 44 116)(22 45 117)(23 46 118)(24 47 119)(25 95 102)(26 96 103)(27 89 104)(28 90 97)(29 91 98)(30 92 99)(31 93 100)(32 94 101)(73 84 105)(74 85 106)(75 86 107)(76 87 108)(77 88 109)(78 81 110)(79 82 111)(80 83 112)
(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(81 110)(82 111)(83 112)(84 105)(85 106)(86 107)(87 108)(88 109)
(1 7 5 3)(2 4 6 8)(9 43 13 47)(10 48 14 44)(11 45 15 41)(12 42 16 46)(17 36 21 40)(18 33 22 37)(19 38 23 34)(20 35 24 39)(25 81 29 85)(26 86 30 82)(27 83 31 87)(28 88 32 84)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 119 69 115)(66 116 70 120)(67 113 71 117)(68 118 72 114)(73 97 77 101)(74 102 78 98)(75 99 79 103)(76 104 80 100)(89 112 93 108)(90 109 94 105)(91 106 95 110)(92 111 96 107)
(1 73 113 65 99)(2 66 74 100 114)(3 101 67 115 75)(4 116 102 76 68)(5 77 117 69 103)(6 70 78 104 118)(7 97 71 119 79)(8 120 98 80 72)(9 96 54 109 45)(10 110 89 46 55)(11 47 111 56 90)(12 49 48 91 112)(13 92 50 105 41)(14 106 93 42 51)(15 43 107 52 94)(16 53 44 95 108)(17 29 83 34 63)(18 35 30 64 84)(19 57 36 85 31)(20 86 58 32 37)(21 25 87 38 59)(22 39 26 60 88)(23 61 40 81 27)(24 82 62 28 33)
(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)
G:=sub<Sym(120)| (1,64,50)(2,57,51)(3,58,52)(4,59,53)(5,60,54)(6,61,55)(7,62,56)(8,63,49)(9,69,39)(10,70,40)(11,71,33)(12,72,34)(13,65,35)(14,66,36)(15,67,37)(16,68,38)(17,48,120)(18,41,113)(19,42,114)(20,43,115)(21,44,116)(22,45,117)(23,46,118)(24,47,119)(25,95,102)(26,96,103)(27,89,104)(28,90,97)(29,91,98)(30,92,99)(31,93,100)(32,94,101)(73,84,105)(74,85,106)(75,86,107)(76,87,108)(77,88,109)(78,81,110)(79,82,111)(80,83,112), (9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(81,110)(82,111)(83,112)(84,105)(85,106)(86,107)(87,108)(88,109), (1,7,5,3)(2,4,6,8)(9,43,13,47)(10,48,14,44)(11,45,15,41)(12,42,16,46)(17,36,21,40)(18,33,22,37)(19,38,23,34)(20,35,24,39)(25,81,29,85)(26,86,30,82)(27,83,31,87)(28,88,32,84)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,119,69,115)(66,116,70,120)(67,113,71,117)(68,118,72,114)(73,97,77,101)(74,102,78,98)(75,99,79,103)(76,104,80,100)(89,112,93,108)(90,109,94,105)(91,106,95,110)(92,111,96,107), (1,73,113,65,99)(2,66,74,100,114)(3,101,67,115,75)(4,116,102,76,68)(5,77,117,69,103)(6,70,78,104,118)(7,97,71,119,79)(8,120,98,80,72)(9,96,54,109,45)(10,110,89,46,55)(11,47,111,56,90)(12,49,48,91,112)(13,92,50,105,41)(14,106,93,42,51)(15,43,107,52,94)(16,53,44,95,108)(17,29,83,34,63)(18,35,30,64,84)(19,57,36,85,31)(20,86,58,32,37)(21,25,87,38,59)(22,39,26,60,88)(23,61,40,81,27)(24,82,62,28,33), (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)>;
G:=Group( (1,64,50)(2,57,51)(3,58,52)(4,59,53)(5,60,54)(6,61,55)(7,62,56)(8,63,49)(9,69,39)(10,70,40)(11,71,33)(12,72,34)(13,65,35)(14,66,36)(15,67,37)(16,68,38)(17,48,120)(18,41,113)(19,42,114)(20,43,115)(21,44,116)(22,45,117)(23,46,118)(24,47,119)(25,95,102)(26,96,103)(27,89,104)(28,90,97)(29,91,98)(30,92,99)(31,93,100)(32,94,101)(73,84,105)(74,85,106)(75,86,107)(76,87,108)(77,88,109)(78,81,110)(79,82,111)(80,83,112), (9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(81,110)(82,111)(83,112)(84,105)(85,106)(86,107)(87,108)(88,109), (1,7,5,3)(2,4,6,8)(9,43,13,47)(10,48,14,44)(11,45,15,41)(12,42,16,46)(17,36,21,40)(18,33,22,37)(19,38,23,34)(20,35,24,39)(25,81,29,85)(26,86,30,82)(27,83,31,87)(28,88,32,84)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,119,69,115)(66,116,70,120)(67,113,71,117)(68,118,72,114)(73,97,77,101)(74,102,78,98)(75,99,79,103)(76,104,80,100)(89,112,93,108)(90,109,94,105)(91,106,95,110)(92,111,96,107), (1,73,113,65,99)(2,66,74,100,114)(3,101,67,115,75)(4,116,102,76,68)(5,77,117,69,103)(6,70,78,104,118)(7,97,71,119,79)(8,120,98,80,72)(9,96,54,109,45)(10,110,89,46,55)(11,47,111,56,90)(12,49,48,91,112)(13,92,50,105,41)(14,106,93,42,51)(15,43,107,52,94)(16,53,44,95,108)(17,29,83,34,63)(18,35,30,64,84)(19,57,36,85,31)(20,86,58,32,37)(21,25,87,38,59)(22,39,26,60,88)(23,61,40,81,27)(24,82,62,28,33), (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) );
G=PermutationGroup([[(1,64,50),(2,57,51),(3,58,52),(4,59,53),(5,60,54),(6,61,55),(7,62,56),(8,63,49),(9,69,39),(10,70,40),(11,71,33),(12,72,34),(13,65,35),(14,66,36),(15,67,37),(16,68,38),(17,48,120),(18,41,113),(19,42,114),(20,43,115),(21,44,116),(22,45,117),(23,46,118),(24,47,119),(25,95,102),(26,96,103),(27,89,104),(28,90,97),(29,91,98),(30,92,99),(31,93,100),(32,94,101),(73,84,105),(74,85,106),(75,86,107),(76,87,108),(77,88,109),(78,81,110),(79,82,111),(80,83,112)], [(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,95),(26,96),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(49,63),(50,64),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62),(81,110),(82,111),(83,112),(84,105),(85,106),(86,107),(87,108),(88,109)], [(1,7,5,3),(2,4,6,8),(9,43,13,47),(10,48,14,44),(11,45,15,41),(12,42,16,46),(17,36,21,40),(18,33,22,37),(19,38,23,34),(20,35,24,39),(25,81,29,85),(26,86,30,82),(27,83,31,87),(28,88,32,84),(49,51,53,55),(50,56,54,52),(57,59,61,63),(58,64,62,60),(65,119,69,115),(66,116,70,120),(67,113,71,117),(68,118,72,114),(73,97,77,101),(74,102,78,98),(75,99,79,103),(76,104,80,100),(89,112,93,108),(90,109,94,105),(91,106,95,110),(92,111,96,107)], [(1,73,113,65,99),(2,66,74,100,114),(3,101,67,115,75),(4,116,102,76,68),(5,77,117,69,103),(6,70,78,104,118),(7,97,71,119,79),(8,120,98,80,72),(9,96,54,109,45),(10,110,89,46,55),(11,47,111,56,90),(12,49,48,91,112),(13,92,50,105,41),(14,106,93,42,51),(15,43,107,52,94),(16,53,44,95,108),(17,29,83,34,63),(18,35,30,64,84),(19,57,36,85,31),(20,86,58,32,37),(21,25,87,38,59),(22,39,26,60,88),(23,61,40,81,27),(24,82,62,28,33)], [(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 15 | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 3 | 3 | 10 | 30 | 2 | 2 | 5 | 5 | 6 | 15 | 15 | 4 | 2 | 20 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 12 | 12 | 4 | 10 | 10 | 8 | 4 | 4 | 12 | 12 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | F5 | C2×F5 | C2×F5 | C2×F5 | C4.F5 | S3×M4(2) | S3×F5 | C2×S3×F5 | S3×C4.F5 |
| kernel | S3×C4.F5 | S3×C5⋊C8 | Dic3.F5 | C3×C4.F5 | C12.F5 | C4×S3×D5 | D5×Dic3 | S3×C20 | C4×D15 | C2×S3×D5 | C4.F5 | C5⋊C8 | C4×D5 | C5×S3 | C20 | D10 | C4×S3 | Dic3 | C12 | D6 | S3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 2 |
Matrix representation of S3×C4.F5 ►in GL8(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 64 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 240 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 88 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 130 | 111 | 196 | 0 |
| 0 | 0 | 0 | 0 | 85 | 111 | 0 | 130 |
| 0 | 0 | 0 | 0 | 130 | 0 | 111 | 85 |
| 0 | 0 | 0 | 0 | 0 | 196 | 111 | 130 |
G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[64,64,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[240,88,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,130,85,130,0,0,0,0,0,111,111,0,196,0,0,0,0,196,0,111,111,0,0,0,0,0,130,85,130] >;
S3×C4.F5 in GAP, Magma, Sage, TeX
S_3\times C_4.F_5
% in TeX
G:=Group("S3xC4.F5"); // GroupNames label
G:=SmallGroup(480,988);
// by ID
G=gap.SmallGroup(480,988);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,100,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^5=1,e^4=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations