direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C8⋊S3, C40⋊16D6, C24⋊26D10, C120⋊22C22, C60.167C23, C3⋊C8⋊19D10, (C8×D5)⋊7S3, C8⋊12(S3×D5), C5⋊2C8⋊28D6, D6.5(C4×D5), C40⋊S3⋊9C2, C3⋊1(D5×M4(2)), (D5×C24)⋊10C2, (C4×D5).99D6, C15⋊8(C2×M4(2)), D30.16(C2×C4), (C4×S3).30D10, D10.29(C4×S3), (C3×D5)⋊2M4(2), D6.Dic5⋊9C2, D30.C2.1C4, (D5×Dic3).1C4, Dic3.9(C4×D5), (S3×Dic5).1C4, C15⋊3C8⋊23C22, D30.5C4⋊9C2, C30.33(C22×C4), Dic5.34(C4×S3), (S3×C20).30C22, C20.164(C22×S3), Dic15.17(C2×C4), (C4×D15).36C22, (D5×C12).98C22, C12.164(C22×D5), (D5×C3⋊C8)⋊8C2, C6.2(C2×C4×D5), C2.5(C4×S3×D5), C5⋊4(C2×C8⋊S3), (C4×S3×D5).6C2, (C2×S3×D5).1C4, C10.33(S3×C2×C4), C4.137(C2×S3×D5), (C5×C8⋊S3)⋊5C2, (C5×C3⋊C8)⋊19C22, (C6×D5).31(C2×C4), (S3×C10).16(C2×C4), (C3×C5⋊2C8)⋊33C22, (C3×Dic5).36(C2×C4), (C5×Dic3).17(C2×C4), SmallGroup(480,320)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C8⋊S3
G = < a,b,c,d,e | a5=b2=c8=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >
Subgroups: 636 in 136 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, C23, D5, 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, C3⋊C8, C24, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C8⋊S3, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, C2×C8⋊S3, C5×C3⋊C8, C3×C5⋊2C8, C15⋊3C8, C120, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5×M4(2), D5×C3⋊C8, D6.Dic5, D30.5C4, D5×C24, C5×C8⋊S3, C40⋊S3, C4×S3×D5, D5×C8⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, C2×M4(2), C4×D5, C22×D5, C8⋊S3, S3×C2×C4, S3×D5, C2×C4×D5, C2×C8⋊S3, C2×S3×D5, D5×M4(2), C4×S3×D5, D5×C8⋊S3
►On 120 points
Generators in S120
(1 28 113 88 99)(2 29 114 81 100)(3 30 115 82 101)(4 31 116 83 102)(5 32 117 84 103)(6 25 118 85 104)(7 26 119 86 97)(8 27 120 87 98)(9 22 39 72 43)(10 23 40 65 44)(11 24 33 66 45)(12 17 34 67 46)(13 18 35 68 47)(14 19 36 69 48)(15 20 37 70 41)(16 21 38 71 42)(49 112 78 95 61)(50 105 79 96 62)(51 106 80 89 63)(52 107 73 90 64)(53 108 74 91 57)(54 109 75 92 58)(55 110 76 93 59)(56 111 77 94 60)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 65)(24 66)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 105)
(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 50 11)(2 51 12)(3 52 13)(4 53 14)(5 54 15)(6 55 16)(7 56 9)(8 49 10)(17 29 106)(18 30 107)(19 31 108)(20 32 109)(21 25 110)(22 26 111)(23 27 112)(24 28 105)(33 113 79)(34 114 80)(35 115 73)(36 116 74)(37 117 75)(38 118 76)(39 119 77)(40 120 78)(41 103 58)(42 104 59)(43 97 60)(44 98 61)(45 99 62)(46 100 63)(47 101 64)(48 102 57)(65 87 95)(66 88 96)(67 81 89)(68 82 90)(69 83 91)(70 84 92)(71 85 93)(72 86 94)
(2 6)(4 8)(9 56)(10 53)(11 50)(12 55)(13 52)(14 49)(15 54)(16 51)(17 110)(18 107)(19 112)(20 109)(21 106)(22 111)(23 108)(24 105)(25 29)(27 31)(33 79)(34 76)(35 73)(36 78)(37 75)(38 80)(39 77)(40 74)(41 58)(42 63)(43 60)(44 57)(45 62)(46 59)(47 64)(48 61)(65 91)(66 96)(67 93)(68 90)(69 95)(70 92)(71 89)(72 94)(81 85)(83 87)(98 102)(100 104)(114 118)(116 120)
G:=sub<Sym(120)| (1,28,113,88,99)(2,29,114,81,100)(3,30,115,82,101)(4,31,116,83,102)(5,32,117,84,103)(6,25,118,85,104)(7,26,119,86,97)(8,27,120,87,98)(9,22,39,72,43)(10,23,40,65,44)(11,24,33,66,45)(12,17,34,67,46)(13,18,35,68,47)(14,19,36,69,48)(15,20,37,70,41)(16,21,38,71,42)(49,112,78,95,61)(50,105,79,96,62)(51,106,80,89,63)(52,107,73,90,64)(53,108,74,91,57)(54,109,75,92,58)(55,110,76,93,59)(56,111,77,94,60), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,65)(24,66)(25,85)(26,86)(27,87)(28,88)(29,81)(30,82)(31,83)(32,84)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,105), (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), (1,50,11)(2,51,12)(3,52,13)(4,53,14)(5,54,15)(6,55,16)(7,56,9)(8,49,10)(17,29,106)(18,30,107)(19,31,108)(20,32,109)(21,25,110)(22,26,111)(23,27,112)(24,28,105)(33,113,79)(34,114,80)(35,115,73)(36,116,74)(37,117,75)(38,118,76)(39,119,77)(40,120,78)(41,103,58)(42,104,59)(43,97,60)(44,98,61)(45,99,62)(46,100,63)(47,101,64)(48,102,57)(65,87,95)(66,88,96)(67,81,89)(68,82,90)(69,83,91)(70,84,92)(71,85,93)(72,86,94), (2,6)(4,8)(9,56)(10,53)(11,50)(12,55)(13,52)(14,49)(15,54)(16,51)(17,110)(18,107)(19,112)(20,109)(21,106)(22,111)(23,108)(24,105)(25,29)(27,31)(33,79)(34,76)(35,73)(36,78)(37,75)(38,80)(39,77)(40,74)(41,58)(42,63)(43,60)(44,57)(45,62)(46,59)(47,64)(48,61)(65,91)(66,96)(67,93)(68,90)(69,95)(70,92)(71,89)(72,94)(81,85)(83,87)(98,102)(100,104)(114,118)(116,120)>;
G:=Group( (1,28,113,88,99)(2,29,114,81,100)(3,30,115,82,101)(4,31,116,83,102)(5,32,117,84,103)(6,25,118,85,104)(7,26,119,86,97)(8,27,120,87,98)(9,22,39,72,43)(10,23,40,65,44)(11,24,33,66,45)(12,17,34,67,46)(13,18,35,68,47)(14,19,36,69,48)(15,20,37,70,41)(16,21,38,71,42)(49,112,78,95,61)(50,105,79,96,62)(51,106,80,89,63)(52,107,73,90,64)(53,108,74,91,57)(54,109,75,92,58)(55,110,76,93,59)(56,111,77,94,60), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,65)(24,66)(25,85)(26,86)(27,87)(28,88)(29,81)(30,82)(31,83)(32,84)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,105), (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), (1,50,11)(2,51,12)(3,52,13)(4,53,14)(5,54,15)(6,55,16)(7,56,9)(8,49,10)(17,29,106)(18,30,107)(19,31,108)(20,32,109)(21,25,110)(22,26,111)(23,27,112)(24,28,105)(33,113,79)(34,114,80)(35,115,73)(36,116,74)(37,117,75)(38,118,76)(39,119,77)(40,120,78)(41,103,58)(42,104,59)(43,97,60)(44,98,61)(45,99,62)(46,100,63)(47,101,64)(48,102,57)(65,87,95)(66,88,96)(67,81,89)(68,82,90)(69,83,91)(70,84,92)(71,85,93)(72,86,94), (2,6)(4,8)(9,56)(10,53)(11,50)(12,55)(13,52)(14,49)(15,54)(16,51)(17,110)(18,107)(19,112)(20,109)(21,106)(22,111)(23,108)(24,105)(25,29)(27,31)(33,79)(34,76)(35,73)(36,78)(37,75)(38,80)(39,77)(40,74)(41,58)(42,63)(43,60)(44,57)(45,62)(46,59)(47,64)(48,61)(65,91)(66,96)(67,93)(68,90)(69,95)(70,92)(71,89)(72,94)(81,85)(83,87)(98,102)(100,104)(114,118)(116,120) );
G=PermutationGroup([[(1,28,113,88,99),(2,29,114,81,100),(3,30,115,82,101),(4,31,116,83,102),(5,32,117,84,103),(6,25,118,85,104),(7,26,119,86,97),(8,27,120,87,98),(9,22,39,72,43),(10,23,40,65,44),(11,24,33,66,45),(12,17,34,67,46),(13,18,35,68,47),(14,19,36,69,48),(15,20,37,70,41),(16,21,38,71,42),(49,112,78,95,61),(50,105,79,96,62),(51,106,80,89,63),(52,107,73,90,64),(53,108,74,91,57),(54,109,75,92,58),(55,110,76,93,59),(56,111,77,94,60)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,97),(8,98),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,41),(16,42),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,65),(24,66),(25,85),(26,86),(27,87),(28,88),(29,81),(30,82),(31,83),(32,84),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,105)], [(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)], [(1,50,11),(2,51,12),(3,52,13),(4,53,14),(5,54,15),(6,55,16),(7,56,9),(8,49,10),(17,29,106),(18,30,107),(19,31,108),(20,32,109),(21,25,110),(22,26,111),(23,27,112),(24,28,105),(33,113,79),(34,114,80),(35,115,73),(36,116,74),(37,117,75),(38,118,76),(39,119,77),(40,120,78),(41,103,58),(42,104,59),(43,97,60),(44,98,61),(45,99,62),(46,100,63),(47,101,64),(48,102,57),(65,87,95),(66,88,96),(67,81,89),(68,82,90),(69,83,91),(70,84,92),(71,85,93),(72,86,94)], [(2,6),(4,8),(9,56),(10,53),(11,50),(12,55),(13,52),(14,49),(15,54),(16,51),(17,110),(18,107),(19,112),(20,109),(21,106),(22,111),(23,108),(24,105),(25,29),(27,31),(33,79),(34,76),(35,73),(36,78),(37,75),(38,80),(39,77),(40,74),(41,58),(42,63),(43,60),(44,57),(45,62),(46,59),(47,64),(48,61),(65,91),(66,96),(67,93),(68,90),(69,95),(70,92),(71,89),(72,94),(81,85),(83,87),(98,102),(100,104),(114,118),(116,120)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 6 | 30 | 2 | 1 | 1 | 5 | 5 | 6 | 30 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 12 | 12 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
72 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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D6 | M4(2) | D10 | D10 | D10 | C4×S3 | C4×S3 | C4×D5 | C4×D5 | C8⋊S3 | S3×D5 | C2×S3×D5 | D5×M4(2) | C4×S3×D5 | D5×C8⋊S3 |
| kernel | D5×C8⋊S3 | D5×C3⋊C8 | D6.Dic5 | D30.5C4 | D5×C24 | C5×C8⋊S3 | C40⋊S3 | C4×S3×D5 | D5×Dic3 | S3×Dic5 | D30.C2 | C2×S3×D5 | C8×D5 | C8⋊S3 | C5⋊2C8 | C40 | C4×D5 | C3×D5 | C3⋊C8 | C24 | C4×S3 | Dic5 | D10 | Dic3 | D6 | D5 | C8 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D5×C8⋊S3 ►in GL6(𝔽241)
| 51 | 1 | 0 | 0 | 0 | 0 |
| 240 | 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 |
| 1 | 51 | 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 | 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 | 64 | 239 |
| 0 | 0 | 0 | 0 | 88 | 177 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 240 | 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 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 64 | 240 |
G:=sub<GL(6,GF(241))| [51,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,51,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,88,0,0,0,0,239,177],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,0,1,64,0,0,0,0,0,240] >;
D5×C8⋊S3 in GAP, Magma, Sage, TeX
D_5\times C_8\rtimes S_3
% in TeX
G:=Group("D5xC8:S3"); // GroupNames label
G:=SmallGroup(480,320);
// by ID
G=gap.SmallGroup(480,320);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^8=d^3=e^2=1,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=c^5,e*d*e=d^-1>;
// generators/relations