direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D8⋊S3, C40⋊20D6, C120⋊31C22, C60.218C23, C8⋊2(S3×C10), (C5×D8)⋊6S3, D8⋊2(C5×S3), C24⋊4(C2×C10), D4⋊S3⋊2C10, (C5×D4)⋊17D6, D4⋊2(S3×C10), (C3×D8)⋊4C10, (S3×D4)⋊2C10, D6.6(C5×D4), C24⋊C2⋊3C10, C8⋊S3⋊3C10, (C15×D8)⋊12C2, D4.S3⋊1C10, C6.28(D4×C10), D4⋊2S3⋊1C10, C15⋊30(C8⋊C22), Dic6⋊1(C2×C10), D12.1(C2×C10), (S3×C10).42D4, C10.182(S3×D4), C30.364(C2×D4), Dic3.8(C5×D4), (D4×C15)⋊19C22, C12.2(C22×C10), (C5×Dic3).45D4, (S3×C20).36C22, C20.191(C22×S3), (C5×Dic6)⋊16C22, (C5×D12).30C22, (C5×S3×D4)⋊9C2, C3⋊C8⋊1(C2×C10), C3⋊2(C5×C8⋊C22), C4.2(S3×C2×C10), C2.16(C5×S3×D4), (C5×D4⋊S3)⋊10C2, (C3×D4)⋊2(C2×C10), (C5×C3⋊C8)⋊23C22, (C5×D4.S3)⋊9C2, (C5×C24⋊C2)⋊11C2, (C5×C8⋊S3)⋊11C2, (C5×D4⋊2S3)⋊8C2, (C4×S3).1(C2×C10), SmallGroup(480,790)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D8⋊S3
G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >
Subgroups: 388 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, M4(2), D8, D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C5×S3, C30, C30, C8⋊C22, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C5×Dic3, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C5×M4(2), C5×D8, C5×D8, C5×SD16, D4×C10, C5×C4○D4, D8⋊S3, C5×C3⋊C8, C120, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, D4×C15, S3×C2×C10, C5×C8⋊C22, C5×C8⋊S3, C5×C24⋊C2, C5×D4⋊S3, C5×D4.S3, C15×D8, C5×S3×D4, C5×D4⋊2S3, C5×D8⋊S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C8⋊C22, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, D8⋊S3, S3×C2×C10, C5×C8⋊C22, C5×S3×D4, C5×D8⋊S3
►On 120 points
Generators in S120
(1 62 31 52 24)(2 63 32 53 17)(3 64 25 54 18)(4 57 26 55 19)(5 58 27 56 20)(6 59 28 49 21)(7 60 29 50 22)(8 61 30 51 23)(9 85 110 35 45)(10 86 111 36 46)(11 87 112 37 47)(12 88 105 38 48)(13 81 106 39 41)(14 82 107 40 42)(15 83 108 33 43)(16 84 109 34 44)(65 93 118 77 102)(66 94 119 78 103)(67 95 120 79 104)(68 96 113 80 97)(69 89 114 73 98)(70 90 115 74 99)(71 91 116 75 100)(72 92 117 76 101)
(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 3)(4 8)(5 7)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 31)(26 30)(27 29)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(57 61)(58 60)(62 64)(65 69)(66 68)(70 72)(73 77)(74 76)(78 80)(81 83)(84 88)(85 87)(89 93)(90 92)(94 96)(97 103)(98 102)(99 101)(105 109)(106 108)(110 112)(113 119)(114 118)(115 117)
(1 99 35)(2 100 36)(3 101 37)(4 102 38)(5 103 39)(6 104 40)(7 97 33)(8 98 34)(9 31 90)(10 32 91)(11 25 92)(12 26 93)(13 27 94)(14 28 95)(15 29 96)(16 30 89)(17 75 111)(18 76 112)(19 77 105)(20 78 106)(21 79 107)(22 80 108)(23 73 109)(24 74 110)(41 58 66)(42 59 67)(43 60 68)(44 61 69)(45 62 70)(46 63 71)(47 64 72)(48 57 65)(49 120 82)(50 113 83)(51 114 84)(52 115 85)(53 116 86)(54 117 87)(55 118 88)(56 119 81)
(2 6)(4 8)(9 90)(10 95)(11 92)(12 89)(13 94)(14 91)(15 96)(16 93)(17 21)(19 23)(26 30)(28 32)(33 97)(34 102)(35 99)(36 104)(37 101)(38 98)(39 103)(40 100)(41 66)(42 71)(43 68)(44 65)(45 70)(46 67)(47 72)(48 69)(49 53)(51 55)(57 61)(59 63)(73 105)(74 110)(75 107)(76 112)(77 109)(78 106)(79 111)(80 108)(81 119)(82 116)(83 113)(84 118)(85 115)(86 120)(87 117)(88 114)
G:=sub<Sym(120)| (1,62,31,52,24)(2,63,32,53,17)(3,64,25,54,18)(4,57,26,55,19)(5,58,27,56,20)(6,59,28,49,21)(7,60,29,50,22)(8,61,30,51,23)(9,85,110,35,45)(10,86,111,36,46)(11,87,112,37,47)(12,88,105,38,48)(13,81,106,39,41)(14,82,107,40,42)(15,83,108,33,43)(16,84,109,34,44)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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), (1,3)(4,8)(5,7)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(73,77)(74,76)(78,80)(81,83)(84,88)(85,87)(89,93)(90,92)(94,96)(97,103)(98,102)(99,101)(105,109)(106,108)(110,112)(113,119)(114,118)(115,117), (1,99,35)(2,100,36)(3,101,37)(4,102,38)(5,103,39)(6,104,40)(7,97,33)(8,98,34)(9,31,90)(10,32,91)(11,25,92)(12,26,93)(13,27,94)(14,28,95)(15,29,96)(16,30,89)(17,75,111)(18,76,112)(19,77,105)(20,78,106)(21,79,107)(22,80,108)(23,73,109)(24,74,110)(41,58,66)(42,59,67)(43,60,68)(44,61,69)(45,62,70)(46,63,71)(47,64,72)(48,57,65)(49,120,82)(50,113,83)(51,114,84)(52,115,85)(53,116,86)(54,117,87)(55,118,88)(56,119,81), (2,6)(4,8)(9,90)(10,95)(11,92)(12,89)(13,94)(14,91)(15,96)(16,93)(17,21)(19,23)(26,30)(28,32)(33,97)(34,102)(35,99)(36,104)(37,101)(38,98)(39,103)(40,100)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(49,53)(51,55)(57,61)(59,63)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114)>;
G:=Group( (1,62,31,52,24)(2,63,32,53,17)(3,64,25,54,18)(4,57,26,55,19)(5,58,27,56,20)(6,59,28,49,21)(7,60,29,50,22)(8,61,30,51,23)(9,85,110,35,45)(10,86,111,36,46)(11,87,112,37,47)(12,88,105,38,48)(13,81,106,39,41)(14,82,107,40,42)(15,83,108,33,43)(16,84,109,34,44)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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), (1,3)(4,8)(5,7)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(73,77)(74,76)(78,80)(81,83)(84,88)(85,87)(89,93)(90,92)(94,96)(97,103)(98,102)(99,101)(105,109)(106,108)(110,112)(113,119)(114,118)(115,117), (1,99,35)(2,100,36)(3,101,37)(4,102,38)(5,103,39)(6,104,40)(7,97,33)(8,98,34)(9,31,90)(10,32,91)(11,25,92)(12,26,93)(13,27,94)(14,28,95)(15,29,96)(16,30,89)(17,75,111)(18,76,112)(19,77,105)(20,78,106)(21,79,107)(22,80,108)(23,73,109)(24,74,110)(41,58,66)(42,59,67)(43,60,68)(44,61,69)(45,62,70)(46,63,71)(47,64,72)(48,57,65)(49,120,82)(50,113,83)(51,114,84)(52,115,85)(53,116,86)(54,117,87)(55,118,88)(56,119,81), (2,6)(4,8)(9,90)(10,95)(11,92)(12,89)(13,94)(14,91)(15,96)(16,93)(17,21)(19,23)(26,30)(28,32)(33,97)(34,102)(35,99)(36,104)(37,101)(38,98)(39,103)(40,100)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(49,53)(51,55)(57,61)(59,63)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114) );
G=PermutationGroup([[(1,62,31,52,24),(2,63,32,53,17),(3,64,25,54,18),(4,57,26,55,19),(5,58,27,56,20),(6,59,28,49,21),(7,60,29,50,22),(8,61,30,51,23),(9,85,110,35,45),(10,86,111,36,46),(11,87,112,37,47),(12,88,105,38,48),(13,81,106,39,41),(14,82,107,40,42),(15,83,108,33,43),(16,84,109,34,44),(65,93,118,77,102),(66,94,119,78,103),(67,95,120,79,104),(68,96,113,80,97),(69,89,114,73,98),(70,90,115,74,99),(71,91,116,75,100),(72,92,117,76,101)], [(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)], [(1,3),(4,8),(5,7),(9,11),(12,16),(13,15),(18,24),(19,23),(20,22),(25,31),(26,30),(27,29),(33,39),(34,38),(35,37),(41,43),(44,48),(45,47),(50,56),(51,55),(52,54),(57,61),(58,60),(62,64),(65,69),(66,68),(70,72),(73,77),(74,76),(78,80),(81,83),(84,88),(85,87),(89,93),(90,92),(94,96),(97,103),(98,102),(99,101),(105,109),(106,108),(110,112),(113,119),(114,118),(115,117)], [(1,99,35),(2,100,36),(3,101,37),(4,102,38),(5,103,39),(6,104,40),(7,97,33),(8,98,34),(9,31,90),(10,32,91),(11,25,92),(12,26,93),(13,27,94),(14,28,95),(15,29,96),(16,30,89),(17,75,111),(18,76,112),(19,77,105),(20,78,106),(21,79,107),(22,80,108),(23,73,109),(24,74,110),(41,58,66),(42,59,67),(43,60,68),(44,61,69),(45,62,70),(46,63,71),(47,64,72),(48,57,65),(49,120,82),(50,113,83),(51,114,84),(52,115,85),(53,116,86),(54,117,87),(55,118,88),(56,119,81)], [(2,6),(4,8),(9,90),(10,95),(11,92),(12,89),(13,94),(14,91),(15,96),(16,93),(17,21),(19,23),(26,30),(28,32),(33,97),(34,102),(35,99),(36,104),(37,101),(38,98),(39,103),(40,100),(41,66),(42,71),(43,68),(44,65),(45,70),(46,67),(47,72),(48,69),(49,53),(51,55),(57,61),(59,63),(73,105),(74,110),(75,107),(76,112),(77,109),(78,106),(79,111),(80,108),(81,119),(82,116),(83,113),(84,118),(85,115),(86,120),(87,117),(88,114)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 10M | 10N | 10O | 10P | 10Q | 10R | 10S | 10T | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 20K | 20L | 24A | 24B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 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 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 4 | 6 | 12 | 2 | 2 | 6 | 12 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 4 | 12 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C8⋊C22 | S3×D4 | D8⋊S3 | C5×C8⋊C22 | C5×S3×D4 | C5×D8⋊S3 |
| kernel | C5×D8⋊S3 | C5×C8⋊S3 | C5×C24⋊C2 | C5×D4⋊S3 | C5×D4.S3 | C15×D8 | C5×S3×D4 | C5×D4⋊2S3 | D8⋊S3 | C8⋊S3 | C24⋊C2 | D4⋊S3 | D4.S3 | C3×D8 | S3×D4 | D4⋊2S3 | C5×D8 | C5×Dic3 | S3×C10 | C40 | C5×D4 | D8 | Dic3 | D6 | C8 | D4 | C15 | C10 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×D8⋊S3 ►in GL6(𝔽241)
| 205 | 0 | 0 | 0 | 0 | 0 |
| 0 | 205 | 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 |
| 240 | 239 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 147 | 194 | 94 |
| 0 | 0 | 94 | 194 | 147 | 47 |
| 0 | 0 | 47 | 147 | 47 | 147 |
| 0 | 0 | 94 | 194 | 94 | 194 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [205,0,0,0,0,0,0,205,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,1,0,0,0,0,239,1,0,0,0,0,0,0,47,94,47,94,0,0,147,194,147,194,0,0,194,147,47,94,0,0,94,47,147,194],[240,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;
C5×D8⋊S3 in GAP, Magma, Sage, TeX
C_5\times D_8\rtimes S_3
% in TeX
G:=Group("C5xD8:S3"); // GroupNames label
G:=SmallGroup(480,790);
// by ID
G=gap.SmallGroup(480,790);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,471,2111,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations