direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×D6⋊C4, D10.26D12, D6⋊9(C4×D5), (C2×C20)⋊15D6, C2.5(D5×D12), D30⋊14(C2×C4), (C2×C12)⋊24D10, (C6×D5).47D4, C30.68(C2×D4), C6.136(D4×D5), D6⋊Dic5⋊22C2, (C2×C60)⋊31C22, (C2×Dic5)⋊18D6, D10.32(C4×S3), C10.27(C2×D12), D30⋊3C4⋊28C2, D30⋊4C4⋊22C2, (C2×Dic3)⋊10D10, C30.67(C22×C4), D10.37(C3⋊D4), (C2×C30).161C23, (C10×Dic3)⋊1C22, (C2×Dic15)⋊5C22, (C6×Dic5)⋊22C22, (C22×S3).48D10, (C22×D5).109D6, (C22×D15).54C22, C5⋊2(C2×D6⋊C4), (C2×S3×D5)⋊1C4, (C2×C4×D5)⋊14S3, C3⋊4(D5×C22⋊C4), C6.35(C2×C4×D5), C2.37(C4×S3×D5), (D5×C2×C12)⋊22C2, (C2×C4)⋊10(S3×D5), C15⋊3(C2×C22⋊C4), C10.68(S3×C2×C4), C2.4(D5×C3⋊D4), (C5×D6⋊C4)⋊29C2, (C2×D5×Dic3)⋊12C2, (S3×C10)⋊14(C2×C4), (C22×S3×D5).1C2, C22.70(C2×S3×D5), (C6×D5).36(C2×C4), C10.39(C2×C3⋊D4), (C3×D5)⋊2(C22⋊C4), (S3×C2×C10).40C22, (D5×C2×C6).109C22, (C2×C6).173(C22×D5), (C2×C10).173(C22×S3), SmallGroup(480,547)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×D6⋊C4
G = < a,b,c,d,e | a5=b2=c6=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c3d >
Subgroups: 1628 in 264 conjugacy classes, 74 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, D6⋊C4, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C4×D5, C23×D5, C2×D6⋊C4, D5×Dic3, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, D5×C22⋊C4, D6⋊Dic5, D30⋊4C4, C5×D6⋊C4, D30⋊3C4, C2×D5×Dic3, D5×C2×C12, C22×S3×D5, D5×D6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22⋊C4, C22×C4, C2×D4, D10, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D5, C22×D5, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, S3×D5, C2×C4×D5, D4×D5, C2×D6⋊C4, C2×S3×D5, D5×C22⋊C4, C4×S3×D5, D5×D12, D5×C3⋊D4, D5×D6⋊C4
►On 120 points
Generators in S120
(1 39 51 48 31)(2 40 52 43 32)(3 41 53 44 33)(4 42 54 45 34)(5 37 49 46 35)(6 38 50 47 36)(7 22 117 30 16)(8 23 118 25 17)(9 24 119 26 18)(10 19 120 27 13)(11 20 115 28 14)(12 21 116 29 15)(55 72 84 75 64)(56 67 79 76 65)(57 68 80 77 66)(58 69 81 78 61)(59 70 82 73 62)(60 71 83 74 63)(85 102 114 105 94)(86 97 109 106 95)(87 98 110 107 96)(88 99 111 108 91)(89 100 112 103 92)(90 101 113 104 93)
(1 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 101)(8 102)(9 97)(10 98)(11 99)(12 100)(13 110)(14 111)(15 112)(16 113)(17 114)(18 109)(19 87)(20 88)(21 89)(22 90)(23 85)(24 86)(25 105)(26 106)(27 107)(28 108)(29 103)(30 104)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)
(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 6)(2 5)(3 4)(7 11)(8 10)(13 17)(14 16)(19 23)(20 22)(25 27)(28 30)(31 36)(32 35)(33 34)(37 40)(38 39)(41 42)(43 46)(44 45)(47 48)(49 52)(50 51)(53 54)(55 60)(56 59)(57 58)(61 66)(62 65)(63 64)(67 70)(68 69)(71 72)(73 76)(74 75)(77 78)(79 82)(80 81)(83 84)(85 87)(88 90)(91 93)(94 96)(98 102)(99 101)(104 108)(105 107)(110 114)(111 113)(115 117)(118 120)
(1 118 58 88)(2 119 59 89)(3 120 60 90)(4 115 55 85)(5 116 56 86)(6 117 57 87)(7 77 107 47)(8 78 108 48)(9 73 103 43)(10 74 104 44)(11 75 105 45)(12 76 106 46)(13 83 113 53)(14 84 114 54)(15 79 109 49)(16 80 110 50)(17 81 111 51)(18 82 112 52)(19 63 93 33)(20 64 94 34)(21 65 95 35)(22 66 96 36)(23 61 91 31)(24 62 92 32)(25 69 99 39)(26 70 100 40)(27 71 101 41)(28 72 102 42)(29 67 97 37)(30 68 98 38)
G:=sub<Sym(120)| (1,39,51,48,31)(2,40,52,43,32)(3,41,53,44,33)(4,42,54,45,34)(5,37,49,46,35)(6,38,50,47,36)(7,22,117,30,16)(8,23,118,25,17)(9,24,119,26,18)(10,19,120,27,13)(11,20,115,28,14)(12,21,116,29,15)(55,72,84,75,64)(56,67,79,76,65)(57,68,80,77,66)(58,69,81,78,61)(59,70,82,73,62)(60,71,83,74,63)(85,102,114,105,94)(86,97,109,106,95)(87,98,110,107,96)(88,99,111,108,91)(89,100,112,103,92)(90,101,113,104,93), (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,101)(8,102)(9,97)(10,98)(11,99)(12,100)(13,110)(14,111)(15,112)(16,113)(17,114)(18,109)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,105)(26,106)(27,107)(28,108)(29,103)(30,104)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120), (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), (1,6)(2,5)(3,4)(7,11)(8,10)(13,17)(14,16)(19,23)(20,22)(25,27)(28,30)(31,36)(32,35)(33,34)(37,40)(38,39)(41,42)(43,46)(44,45)(47,48)(49,52)(50,51)(53,54)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,70)(68,69)(71,72)(73,76)(74,75)(77,78)(79,82)(80,81)(83,84)(85,87)(88,90)(91,93)(94,96)(98,102)(99,101)(104,108)(105,107)(110,114)(111,113)(115,117)(118,120), (1,118,58,88)(2,119,59,89)(3,120,60,90)(4,115,55,85)(5,116,56,86)(6,117,57,87)(7,77,107,47)(8,78,108,48)(9,73,103,43)(10,74,104,44)(11,75,105,45)(12,76,106,46)(13,83,113,53)(14,84,114,54)(15,79,109,49)(16,80,110,50)(17,81,111,51)(18,82,112,52)(19,63,93,33)(20,64,94,34)(21,65,95,35)(22,66,96,36)(23,61,91,31)(24,62,92,32)(25,69,99,39)(26,70,100,40)(27,71,101,41)(28,72,102,42)(29,67,97,37)(30,68,98,38)>;
G:=Group( (1,39,51,48,31)(2,40,52,43,32)(3,41,53,44,33)(4,42,54,45,34)(5,37,49,46,35)(6,38,50,47,36)(7,22,117,30,16)(8,23,118,25,17)(9,24,119,26,18)(10,19,120,27,13)(11,20,115,28,14)(12,21,116,29,15)(55,72,84,75,64)(56,67,79,76,65)(57,68,80,77,66)(58,69,81,78,61)(59,70,82,73,62)(60,71,83,74,63)(85,102,114,105,94)(86,97,109,106,95)(87,98,110,107,96)(88,99,111,108,91)(89,100,112,103,92)(90,101,113,104,93), (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,101)(8,102)(9,97)(10,98)(11,99)(12,100)(13,110)(14,111)(15,112)(16,113)(17,114)(18,109)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,105)(26,106)(27,107)(28,108)(29,103)(30,104)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120), (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), (1,6)(2,5)(3,4)(7,11)(8,10)(13,17)(14,16)(19,23)(20,22)(25,27)(28,30)(31,36)(32,35)(33,34)(37,40)(38,39)(41,42)(43,46)(44,45)(47,48)(49,52)(50,51)(53,54)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,70)(68,69)(71,72)(73,76)(74,75)(77,78)(79,82)(80,81)(83,84)(85,87)(88,90)(91,93)(94,96)(98,102)(99,101)(104,108)(105,107)(110,114)(111,113)(115,117)(118,120), (1,118,58,88)(2,119,59,89)(3,120,60,90)(4,115,55,85)(5,116,56,86)(6,117,57,87)(7,77,107,47)(8,78,108,48)(9,73,103,43)(10,74,104,44)(11,75,105,45)(12,76,106,46)(13,83,113,53)(14,84,114,54)(15,79,109,49)(16,80,110,50)(17,81,111,51)(18,82,112,52)(19,63,93,33)(20,64,94,34)(21,65,95,35)(22,66,96,36)(23,61,91,31)(24,62,92,32)(25,69,99,39)(26,70,100,40)(27,71,101,41)(28,72,102,42)(29,67,97,37)(30,68,98,38) );
G=PermutationGroup([[(1,39,51,48,31),(2,40,52,43,32),(3,41,53,44,33),(4,42,54,45,34),(5,37,49,46,35),(6,38,50,47,36),(7,22,117,30,16),(8,23,118,25,17),(9,24,119,26,18),(10,19,120,27,13),(11,20,115,28,14),(12,21,116,29,15),(55,72,84,75,64),(56,67,79,76,65),(57,68,80,77,66),(58,69,81,78,61),(59,70,82,73,62),(60,71,83,74,63),(85,102,114,105,94),(86,97,109,106,95),(87,98,110,107,96),(88,99,111,108,91),(89,100,112,103,92),(90,101,113,104,93)], [(1,64),(2,65),(3,66),(4,61),(5,62),(6,63),(7,101),(8,102),(9,97),(10,98),(11,99),(12,100),(13,110),(14,111),(15,112),(16,113),(17,114),(18,109),(19,87),(20,88),(21,89),(22,90),(23,85),(24,86),(25,105),(26,106),(27,107),(28,108),(29,103),(30,104),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120)], [(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)], [(1,6),(2,5),(3,4),(7,11),(8,10),(13,17),(14,16),(19,23),(20,22),(25,27),(28,30),(31,36),(32,35),(33,34),(37,40),(38,39),(41,42),(43,46),(44,45),(47,48),(49,52),(50,51),(53,54),(55,60),(56,59),(57,58),(61,66),(62,65),(63,64),(67,70),(68,69),(71,72),(73,76),(74,75),(77,78),(79,82),(80,81),(83,84),(85,87),(88,90),(91,93),(94,96),(98,102),(99,101),(104,108),(105,107),(110,114),(111,113),(115,117),(118,120)], [(1,118,58,88),(2,119,59,89),(3,120,60,90),(4,115,55,85),(5,116,56,86),(6,117,57,87),(7,77,107,47),(8,78,108,48),(9,73,103,43),(10,74,104,44),(11,75,105,45),(12,76,106,46),(13,83,113,53),(14,84,114,54),(15,79,109,49),(16,80,110,50),(17,81,111,51),(18,82,112,52),(19,63,93,33),(20,64,94,34),(21,65,95,35),(22,66,96,36),(23,61,91,31),(24,62,92,32),(25,69,99,39),(26,70,100,40),(27,71,101,41),(28,72,102,42),(29,67,97,37),(30,68,98,38)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4×S3 | D12 | C3⋊D4 | C4×D5 | S3×D5 | D4×D5 | C2×S3×D5 | C4×S3×D5 | D5×D12 | D5×C3⋊D4 |
| kernel | D5×D6⋊C4 | D6⋊Dic5 | D30⋊4C4 | C5×D6⋊C4 | D30⋊3C4 | C2×D5×Dic3 | D5×C2×C12 | C22×S3×D5 | C2×S3×D5 | C2×C4×D5 | C6×D5 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C2×Dic3 | C2×C12 | C22×S3 | D10 | D10 | D10 | D6 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D5×D6⋊C4 ►in GL6(𝔽61)
| 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 | 60 | 1 |
| 0 | 0 | 0 | 0 | 16 | 44 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 60 | 36 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 25 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 27 | 23 | 0 | 0 | 0 | 0 |
| 32 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 15 | 0 | 0 |
| 0 | 0 | 46 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,16,0,0,0,0,1,44],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,16,0,0,0,0,0,1],[60,5,0,0,0,0,36,2,0,0,0,0,0,0,1,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,25,60,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,32,0,0,0,0,23,34,0,0,0,0,0,0,23,46,0,0,0,0,15,38,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
D5×D6⋊C4 in GAP, Magma, Sage, TeX
D_5\times D_6\rtimes C_4
% in TeX
G:=Group("D5xD6:C4"); // GroupNames label
G:=SmallGroup(480,547);
// by ID
G=gap.SmallGroup(480,547);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^6=d^2=e^4=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,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^3*d>;
// generators/relations