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, (D5×D12)⋊13C2, (C4×S3)⋊12D10, (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, C3⋊D20⋊11C22, C5⋊D12⋊11C22, C15⋊D4⋊11C22, C15⋊7D4⋊13C22, (S3×C10).5C23, C10.18(S3×C23), D30.C2⋊7C22, D60⋊11C2⋊11C2, (S3×Dic5)⋊7C22, (C22×D5).99D6, (C6×D5).43C23, C20.187(C22×S3), (C2×C30).237C23, (C5×Dic6)⋊22C22, (C6×Dic5)⋊27C22, D10.55(C22×S3), C12.187(C22×D5), Dic3.8(C22×D5), (C5×Dic3).8C23, (D5×Dic3).10C22, (C3×Dic5).43C23, Dic5.43(C22×S3), (C2×C4×D5)⋊6S3, (C4×S3×D5)⋊8C2, 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
Generators and relations for D5×C4○D12
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 >
Subgroups: 1628 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C3×D5, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, C6×D5, S3×C10, D30, C2×C30, C2×C4×D5, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, C2×C4○D12, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C6×Dic5, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C2×S3×D5, D5×C2×C6, D5×C4○D4, 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, D5×C4○D12
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, C4○D12, S3×C23, S3×D5, C23×D5, C2×C4○D12, C2×S3×D5, D5×C4○D4, C22×S3×D5, D5×C4○D12
►On 120 points
(1 62 112 15 54)(2 63 113 16 55)(3 64 114 17 56)(4 65 115 18 57)(5 66 116 19 58)(6 67 117 20 59)(7 68 118 21 60)(8 69 119 22 49)(9 70 120 23 50)(10 71 109 24 51)(11 72 110 13 52)(12 61 111 14 53)(25 47 78 100 90)(26 48 79 101 91)(27 37 80 102 92)(28 38 81 103 93)(29 39 82 104 94)(30 40 83 105 95)(31 41 84 106 96)(32 42 73 107 85)(33 43 74 108 86)(34 44 75 97 87)(35 45 76 98 88)(36 46 77 99 89)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 49)(9 50)(10 51)(11 52)(12 53)(13 72)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 85)(33 86)(34 87)(35 88)(36 89)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 97)(45 98)(46 99)(47 100)(48 101)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(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 58 55 52)(50 59 56 53)(51 60 57 54)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 118 115 112)(110 119 116 113)(111 120 117 114)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 98)(14 97)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 72)(46 71)(47 70)(48 69)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 96)(57 95)(58 94)(59 93)(60 92)(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)| (1,62,112,15,54)(2,63,113,16,55)(3,64,114,17,56)(4,65,115,18,57)(5,66,116,19,58)(6,67,117,20,59)(7,68,118,21,60)(8,69,119,22,49)(9,70,120,23,50)(10,71,109,24,51)(11,72,110,13,52)(12,61,111,14,53)(25,47,78,100,90)(26,48,79,101,91)(27,37,80,102,92)(28,38,81,103,93)(29,39,82,104,94)(30,40,83,105,95)(31,41,84,106,96)(32,42,73,107,85)(33,43,74,108,86)(34,44,75,97,87)(35,45,76,98,88)(36,46,77,99,89), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,49)(9,50)(10,51)(11,52)(12,53)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,85)(33,86)(34,87)(35,88)(36,89)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,97)(45,98)(46,99)(47,100)(48,101), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,118,115,112)(110,119,116,113)(111,120,117,114), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,98)(14,97)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,72)(46,71)(47,70)(48,69)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,96)(57,95)(58,94)(59,93)(60,92)(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( (1,62,112,15,54)(2,63,113,16,55)(3,64,114,17,56)(4,65,115,18,57)(5,66,116,19,58)(6,67,117,20,59)(7,68,118,21,60)(8,69,119,22,49)(9,70,120,23,50)(10,71,109,24,51)(11,72,110,13,52)(12,61,111,14,53)(25,47,78,100,90)(26,48,79,101,91)(27,37,80,102,92)(28,38,81,103,93)(29,39,82,104,94)(30,40,83,105,95)(31,41,84,106,96)(32,42,73,107,85)(33,43,74,108,86)(34,44,75,97,87)(35,45,76,98,88)(36,46,77,99,89), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,49)(9,50)(10,51)(11,52)(12,53)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,85)(33,86)(34,87)(35,88)(36,89)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,97)(45,98)(46,99)(47,100)(48,101), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,118,115,112)(110,119,116,113)(111,120,117,114), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,98)(14,97)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,72)(46,71)(47,70)(48,69)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,96)(57,95)(58,94)(59,93)(60,92)(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([[(1,62,112,15,54),(2,63,113,16,55),(3,64,114,17,56),(4,65,115,18,57),(5,66,116,19,58),(6,67,117,20,59),(7,68,118,21,60),(8,69,119,22,49),(9,70,120,23,50),(10,71,109,24,51),(11,72,110,13,52),(12,61,111,14,53),(25,47,78,100,90),(26,48,79,101,91),(27,37,80,102,92),(28,38,81,103,93),(29,39,82,104,94),(30,40,83,105,95),(31,41,84,106,96),(32,42,73,107,85),(33,43,74,108,86),(34,44,75,97,87),(35,45,76,98,88),(36,46,77,99,89)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,49),(9,50),(10,51),(11,52),(12,53),(13,72),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,85),(33,86),(34,87),(35,88),(36,89),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,97),(45,98),(46,99),(47,100),(48,101)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,58,55,52),(50,59,56,53),(51,60,57,54),(61,70,67,64),(62,71,68,65),(63,72,69,66),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,100,103,106),(98,101,104,107),(99,102,105,108),(109,118,115,112),(110,119,116,113),(111,120,117,114)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,36),(11,35),(12,34),(13,98),(14,97),(15,108),(16,107),(17,106),(18,105),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,72),(46,71),(47,70),(48,69),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,96),(57,95),(58,94),(59,93),(60,92),(73,113),(74,112),(75,111),(76,110),(77,109),(78,120),(79,119),(80,118),(81,117),(82,116),(83,115),(84,114)]])
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 |
Matrix representation of D5×C4○D12 ►in 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] >;
D5×C4○D12 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