direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4○D12, D12⋊27D10, Dic6⋊25D10, D60⋊39C22, C30.18C24, D30.5C23, C60.138C23, Dic30⋊36C22, Dic15.8C23, (C2×C20)⋊6D6, (C4×D5)⋊17D6, C15⋊Q8⋊9C22, (C4×S3)⋊12D10, (D5×D12)⋊13C2, (C2×C12)⋊27D10, C3⋊D4⋊12D10, (C2×C60)⋊6C22, (D5×Dic6)⋊13C2, (C2×Dic5)⋊22D6, C12.28D10⋊13C2, D12⋊5D5⋊13C2, D6.5(C22×D5), C6.18(C23×D5), (S3×C20)⋊12C22, Dic3.D10⋊7C2, D6.D10⋊9C2, (C4×D15)⋊14C22, (C5×D12)⋊24C22, (D5×C12)⋊21C22, C15⋊7D4⋊13C22, C15⋊D4⋊11C22, C5⋊D12⋊11C22, C3⋊D20⋊11C22, (S3×C10).5C23, C10.18(S3×C23), D30.C2⋊7C22, D60⋊11C2⋊11C2, (S3×Dic5)⋊7C22, (C6×D5).43C23, (C22×D5).99D6, C20.187(C22×S3), (C2×C30).237C23, (C6×Dic5)⋊27C22, (C5×Dic6)⋊22C22, D10.55(C22×S3), C12.187(C22×D5), Dic3.8(C22×D5), (C5×Dic3).8C23, (C3×Dic5).43C23, (D5×Dic3).10C22, Dic5.43(C22×S3), (C4×S3×D5)⋊8C2, (C2×C4×D5)⋊6S3, C3⋊1(D5×C4○D4), (D5×C2×C12)⋊1C2, C15⋊8(C2×C4○D4), C5⋊4(C2×C4○D12), (C2×C4)⋊12(S3×D5), (D5×C3⋊D4)⋊7C2, C4.160(C2×S3×D5), (C5×C4○D12)⋊5C2, C22.9(C2×S3×D5), (C3×D5)⋊1(C4○D4), (C2×S3×D5).5C22, C2.21(C22×S3×D5), (C5×C3⋊D4)⋊8C22, (D5×C2×C6).120C22, (C2×C10).10(C22×S3), (C2×C6).247(C22×D5), SmallGroup(480,1090)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1628 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×4], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×2], D5 [×3], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6 [×3], C4×S3 [×2], C4×S3 [×6], D12, D12 [×3], C2×Dic3 [×2], C3⋊D4 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×C6, C5×S3 [×2], C3×D5 [×2], C3×D5, D15 [×2], C30, C30, C2×C4○D4, Dic10 [×3], C4×D5 [×4], C4×D5 [×6], D20 [×3], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5, C22×D5 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12, C4○D12 [×7], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×C4×D5, C2×C4×D5 [×2], C4○D20 [×3], D4×D5 [×3], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C5×C4○D4, C2×C4○D12, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], D5×C12 [×4], C6×Dic5, C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15 [×2], D60, C15⋊7D4 [×2], C2×C60, C2×S3×D5 [×2], D5×C2×C6, D5×C4○D4, D5×Dic6, D6.D10 [×2], D12⋊5D5, C12.28D10, C4×S3×D5 [×2], D5×D12, Dic3.D10 [×2], D5×C3⋊D4 [×2], D5×C2×C12, C5×C4○D12, D60⋊11C2, D5×C4○D12
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], C4○D12 [×2], S3×C23, S3×D5, C23×D5, C2×C4○D12, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D5×C4○D12
Generators and relations
G = < a,b,c,d,e | a5=b2=c4=e2=1, d6=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >
►On 120 points
(1 64 88 108 75)(2 65 89 97 76)(3 66 90 98 77)(4 67 91 99 78)(5 68 92 100 79)(6 69 93 101 80)(7 70 94 102 81)(8 71 95 103 82)(9 72 96 104 83)(10 61 85 105 84)(11 62 86 106 73)(12 63 87 107 74)(13 112 40 57 33)(14 113 41 58 34)(15 114 42 59 35)(16 115 43 60 36)(17 116 44 49 25)(18 117 45 50 26)(19 118 46 51 27)(20 119 47 52 28)(21 120 48 53 29)(22 109 37 54 30)(23 110 38 55 31)(24 111 39 56 32)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 73)(12 74)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(61 105)(62 106)(63 107)(64 108)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 94 91 88)(86 95 92 89)(87 96 93 90)(97 106 103 100)(98 107 104 101)(99 108 105 102)(109 112 115 118)(110 113 116 119)(111 114 117 120)
(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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 108)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 96)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 72)(73 113)(74 112)(75 111)(76 110)(77 109)(78 120)(79 119)(80 118)(81 117)(82 116)(83 115)(84 114)
G:=sub<Sym(120)| (1,64,88,108,75)(2,65,89,97,76)(3,66,90,98,77)(4,67,91,99,78)(5,68,92,100,79)(6,69,93,101,80)(7,70,94,102,81)(8,71,95,103,82)(9,72,96,104,83)(10,61,85,105,84)(11,62,86,106,73)(12,63,87,107,74)(13,112,40,57,33)(14,113,41,58,34)(15,114,42,59,35)(16,115,43,60,36)(17,116,44,49,25)(18,117,45,50,26)(19,118,46,51,27)(20,119,47,52,28)(21,120,48,53,29)(22,109,37,54,30)(23,110,38,55,31)(24,111,39,56,32), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,73)(12,74)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(61,105)(62,106)(63,107)(64,108)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,94,91,88)(86,95,92,89)(87,96,93,90)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,112,115,118)(110,113,116,119)(111,114,117,120), (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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,108)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,96)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(73,113)(74,112)(75,111)(76,110)(77,109)(78,120)(79,119)(80,118)(81,117)(82,116)(83,115)(84,114)>;
G:=Group( (1,64,88,108,75)(2,65,89,97,76)(3,66,90,98,77)(4,67,91,99,78)(5,68,92,100,79)(6,69,93,101,80)(7,70,94,102,81)(8,71,95,103,82)(9,72,96,104,83)(10,61,85,105,84)(11,62,86,106,73)(12,63,87,107,74)(13,112,40,57,33)(14,113,41,58,34)(15,114,42,59,35)(16,115,43,60,36)(17,116,44,49,25)(18,117,45,50,26)(19,118,46,51,27)(20,119,47,52,28)(21,120,48,53,29)(22,109,37,54,30)(23,110,38,55,31)(24,111,39,56,32), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,73)(12,74)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(61,105)(62,106)(63,107)(64,108)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,94,91,88)(86,95,92,89)(87,96,93,90)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,112,115,118)(110,113,116,119)(111,114,117,120), (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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,108)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,96)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(73,113)(74,112)(75,111)(76,110)(77,109)(78,120)(79,119)(80,118)(81,117)(82,116)(83,115)(84,114) );
G=PermutationGroup([(1,64,88,108,75),(2,65,89,97,76),(3,66,90,98,77),(4,67,91,99,78),(5,68,92,100,79),(6,69,93,101,80),(7,70,94,102,81),(8,71,95,103,82),(9,72,96,104,83),(10,61,85,105,84),(11,62,86,106,73),(12,63,87,107,74),(13,112,40,57,33),(14,113,41,58,34),(15,114,42,59,35),(16,115,43,60,36),(17,116,44,49,25),(18,117,45,50,26),(19,118,46,51,27),(20,119,47,52,28),(21,120,48,53,29),(22,109,37,54,30),(23,110,38,55,31),(24,111,39,56,32)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,73),(12,74),(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(61,105),(62,106),(63,107),(64,108),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,70,67,64),(62,71,68,65),(63,72,69,66),(73,82,79,76),(74,83,80,77),(75,84,81,78),(85,94,91,88),(86,95,92,89),(87,96,93,90),(97,106,103,100),(98,107,104,101),(99,108,105,102),(109,112,115,118),(110,113,116,119),(111,114,117,120)], [(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)], [(1,39),(2,38),(3,37),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,108),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,96),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,72),(73,113),(74,112),(75,111),(76,110),(77,109),(78,120),(79,119),(80,118),(81,117),(82,116),(83,115),(84,114)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 44 | 1 | 0 | 0 |
| 16 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 21 | 0 |
| 0 | 0 | 60 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 30 |
| 0 | 0 | 28 | 36 |
G:=sub<GL(4,GF(61))| [44,16,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,11,0,0,0,0,11],[60,0,0,0,0,60,0,0,0,0,21,60,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,25,28,0,0,30,36] >;
72 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 | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 5 | 5 | 6 | 6 | 10 | 30 | 30 | 2 | 1 | 1 | 2 | 5 | 5 | 6 | 6 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D12 | S3×D5 | C2×S3×D5 | C2×S3×D5 | D5×C4○D4 | D5×C4○D12 |
| kernel | D5×C4○D12 | D5×Dic6 | D6.D10 | D12⋊5D5 | C12.28D10 | C4×S3×D5 | D5×D12 | Dic3.D10 | D5×C3⋊D4 | D5×C2×C12 | C5×C4○D12 | D60⋊11C2 | C2×C4×D5 | C4○D12 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C3×D5 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | D5 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 8 | 2 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
D_5\times C_4\circ D_{12} % in TeX
G:=Group("D5xC4oD12"); // GroupNames label
G:=SmallGroup(480,1090);
// by ID
G=gap.SmallGroup(480,1090);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^2=1,d^6=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,c*e=e*c,e*d*e=c^2*d^5>;
// generators/relations