direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C6.D4, (C6×D5).68D4, C6.155(D4×D5), D10⋊7(C2×Dic3), C30.227(C2×D4), (C23×D5).4S3, C22⋊4(D5×Dic3), C23.40(S3×D5), (C2×Dic3)⋊12D10, (C22×D5)⋊6Dic3, (C22×C6).90D10, (C22×C10).44D6, C30.38D4⋊20C2, D10⋊Dic3⋊29C2, D10.42(C3⋊D4), C30.143(C22×C4), (C2×C30).189C23, (C2×Dic15)⋊9C22, (C10×Dic3)⋊5C22, (C22×D5).110D6, (C22×C30).51C22, C10.31(C22×Dic3), (D5×C2×C6)⋊6C4, C3⋊5(D5×C22⋊C4), C6.94(C2×C4×D5), (C2×C6)⋊14(C4×D5), C2.5(D5×C3⋊D4), (C2×C30)⋊14(C2×C4), (C6×D5)⋊27(C2×C4), C15⋊11(C2×C22⋊C4), (C2×D5×Dic3)⋊13C2, C5⋊2(C2×C6.D4), C2.18(C2×D5×Dic3), (D5×C22×C6).2C2, C22.83(C2×S3×D5), (C2×C10)⋊8(C2×Dic3), C10.59(C2×C3⋊D4), (C3×D5)⋊3(C22⋊C4), (C5×C6.D4)⋊8C2, (D5×C2×C6).111C22, (C2×C6).201(C22×D5), (C2×C10).201(C22×S3), SmallGroup(480,623)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C6.D4
G = < a,b,c,d,e | a5=b2=c6=d4=1, e2=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c3d-1 >
Subgroups: 1212 in 264 conjugacy classes, 84 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, D5, D5, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C22×C6, C22×C6, C3×D5, C3×D5, C30, C30, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, C6.D4, C6.D4, C22×Dic3, C23×C6, C5×Dic3, Dic15, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, C2×C6.D4, D5×Dic3, C10×Dic3, C2×Dic15, D5×C2×C6, D5×C2×C6, D5×C2×C6, C22×C30, D5×C22⋊C4, D10⋊Dic3, C5×C6.D4, C30.38D4, C2×D5×Dic3, D5×C22×C6, D5×C6.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, Dic3, D6, C22⋊C4, C22×C4, C2×D4, D10, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D5, C22×D5, C6.D4, C22×Dic3, C2×C3⋊D4, S3×D5, C2×C4×D5, D4×D5, C2×C6.D4, D5×Dic3, C2×S3×D5, D5×C22⋊C4, C2×D5×Dic3, D5×C3⋊D4, D5×C6.D4
►On 120 points
Generators in S120
(1 101 88 95 34)(2 102 89 96 35)(3 97 90 91 36)(4 98 85 92 31)(5 99 86 93 32)(6 100 87 94 33)(7 67 103 56 44)(8 68 104 57 45)(9 69 105 58 46)(10 70 106 59 47)(11 71 107 60 48)(12 72 108 55 43)(13 41 23 30 84)(14 42 24 25 79)(15 37 19 26 80)(16 38 20 27 81)(17 39 21 28 82)(18 40 22 29 83)(49 116 66 74 113)(50 117 61 75 114)(51 118 62 76 109)(52 119 63 77 110)(53 120 64 78 111)(54 115 65 73 112)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 59)(8 60)(9 55)(10 56)(11 57)(12 58)(13 81)(14 82)(15 83)(16 84)(17 79)(18 80)(19 22)(20 23)(21 24)(25 39)(26 40)(27 41)(28 42)(29 37)(30 38)(43 46)(44 47)(45 48)(49 119)(50 120)(51 115)(52 116)(53 117)(54 118)(61 111)(62 112)(63 113)(64 114)(65 109)(66 110)(67 106)(68 107)(69 108)(70 103)(71 104)(72 105)(73 76)(74 77)(75 78)(85 88)(86 89)(87 90)(91 100)(92 101)(93 102)(94 97)(95 98)(96 99)
(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)
(1 116 13 104)(2 115 14 103)(3 120 15 108)(4 119 16 107)(5 118 17 106)(6 117 18 105)(7 96 112 25)(8 95 113 30)(9 94 114 29)(10 93 109 28)(11 92 110 27)(12 91 111 26)(19 43 90 78)(20 48 85 77)(21 47 86 76)(22 46 87 75)(23 45 88 74)(24 44 89 73)(31 52 81 71)(32 51 82 70)(33 50 83 69)(34 49 84 68)(35 54 79 67)(36 53 80 72)(37 55 97 64)(38 60 98 63)(39 59 99 62)(40 58 100 61)(41 57 101 66)(42 56 102 65)
(1 119 4 116)(2 118 5 115)(3 117 6 120)(7 25 10 28)(8 30 11 27)(9 29 12 26)(13 107 16 104)(14 106 17 103)(15 105 18 108)(19 46 22 43)(20 45 23 48)(21 44 24 47)(31 49 34 52)(32 54 35 51)(33 53 36 50)(37 58 40 55)(38 57 41 60)(39 56 42 59)(61 100 64 97)(62 99 65 102)(63 98 66 101)(67 79 70 82)(68 84 71 81)(69 83 72 80)(73 89 76 86)(74 88 77 85)(75 87 78 90)(91 114 94 111)(92 113 95 110)(93 112 96 109)
G:=sub<Sym(120)| (1,101,88,95,34)(2,102,89,96,35)(3,97,90,91,36)(4,98,85,92,31)(5,99,86,93,32)(6,100,87,94,33)(7,67,103,56,44)(8,68,104,57,45)(9,69,105,58,46)(10,70,106,59,47)(11,71,107,60,48)(12,72,108,55,43)(13,41,23,30,84)(14,42,24,25,79)(15,37,19,26,80)(16,38,20,27,81)(17,39,21,28,82)(18,40,22,29,83)(49,116,66,74,113)(50,117,61,75,114)(51,118,62,76,109)(52,119,63,77,110)(53,120,64,78,111)(54,115,65,73,112), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,81)(14,82)(15,83)(16,84)(17,79)(18,80)(19,22)(20,23)(21,24)(25,39)(26,40)(27,41)(28,42)(29,37)(30,38)(43,46)(44,47)(45,48)(49,119)(50,120)(51,115)(52,116)(53,117)(54,118)(61,111)(62,112)(63,113)(64,114)(65,109)(66,110)(67,106)(68,107)(69,108)(70,103)(71,104)(72,105)(73,76)(74,77)(75,78)(85,88)(86,89)(87,90)(91,100)(92,101)(93,102)(94,97)(95,98)(96,99), (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), (1,116,13,104)(2,115,14,103)(3,120,15,108)(4,119,16,107)(5,118,17,106)(6,117,18,105)(7,96,112,25)(8,95,113,30)(9,94,114,29)(10,93,109,28)(11,92,110,27)(12,91,111,26)(19,43,90,78)(20,48,85,77)(21,47,86,76)(22,46,87,75)(23,45,88,74)(24,44,89,73)(31,52,81,71)(32,51,82,70)(33,50,83,69)(34,49,84,68)(35,54,79,67)(36,53,80,72)(37,55,97,64)(38,60,98,63)(39,59,99,62)(40,58,100,61)(41,57,101,66)(42,56,102,65), (1,119,4,116)(2,118,5,115)(3,117,6,120)(7,25,10,28)(8,30,11,27)(9,29,12,26)(13,107,16,104)(14,106,17,103)(15,105,18,108)(19,46,22,43)(20,45,23,48)(21,44,24,47)(31,49,34,52)(32,54,35,51)(33,53,36,50)(37,58,40,55)(38,57,41,60)(39,56,42,59)(61,100,64,97)(62,99,65,102)(63,98,66,101)(67,79,70,82)(68,84,71,81)(69,83,72,80)(73,89,76,86)(74,88,77,85)(75,87,78,90)(91,114,94,111)(92,113,95,110)(93,112,96,109)>;
G:=Group( (1,101,88,95,34)(2,102,89,96,35)(3,97,90,91,36)(4,98,85,92,31)(5,99,86,93,32)(6,100,87,94,33)(7,67,103,56,44)(8,68,104,57,45)(9,69,105,58,46)(10,70,106,59,47)(11,71,107,60,48)(12,72,108,55,43)(13,41,23,30,84)(14,42,24,25,79)(15,37,19,26,80)(16,38,20,27,81)(17,39,21,28,82)(18,40,22,29,83)(49,116,66,74,113)(50,117,61,75,114)(51,118,62,76,109)(52,119,63,77,110)(53,120,64,78,111)(54,115,65,73,112), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,81)(14,82)(15,83)(16,84)(17,79)(18,80)(19,22)(20,23)(21,24)(25,39)(26,40)(27,41)(28,42)(29,37)(30,38)(43,46)(44,47)(45,48)(49,119)(50,120)(51,115)(52,116)(53,117)(54,118)(61,111)(62,112)(63,113)(64,114)(65,109)(66,110)(67,106)(68,107)(69,108)(70,103)(71,104)(72,105)(73,76)(74,77)(75,78)(85,88)(86,89)(87,90)(91,100)(92,101)(93,102)(94,97)(95,98)(96,99), (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), (1,116,13,104)(2,115,14,103)(3,120,15,108)(4,119,16,107)(5,118,17,106)(6,117,18,105)(7,96,112,25)(8,95,113,30)(9,94,114,29)(10,93,109,28)(11,92,110,27)(12,91,111,26)(19,43,90,78)(20,48,85,77)(21,47,86,76)(22,46,87,75)(23,45,88,74)(24,44,89,73)(31,52,81,71)(32,51,82,70)(33,50,83,69)(34,49,84,68)(35,54,79,67)(36,53,80,72)(37,55,97,64)(38,60,98,63)(39,59,99,62)(40,58,100,61)(41,57,101,66)(42,56,102,65), (1,119,4,116)(2,118,5,115)(3,117,6,120)(7,25,10,28)(8,30,11,27)(9,29,12,26)(13,107,16,104)(14,106,17,103)(15,105,18,108)(19,46,22,43)(20,45,23,48)(21,44,24,47)(31,49,34,52)(32,54,35,51)(33,53,36,50)(37,58,40,55)(38,57,41,60)(39,56,42,59)(61,100,64,97)(62,99,65,102)(63,98,66,101)(67,79,70,82)(68,84,71,81)(69,83,72,80)(73,89,76,86)(74,88,77,85)(75,87,78,90)(91,114,94,111)(92,113,95,110)(93,112,96,109) );
G=PermutationGroup([[(1,101,88,95,34),(2,102,89,96,35),(3,97,90,91,36),(4,98,85,92,31),(5,99,86,93,32),(6,100,87,94,33),(7,67,103,56,44),(8,68,104,57,45),(9,69,105,58,46),(10,70,106,59,47),(11,71,107,60,48),(12,72,108,55,43),(13,41,23,30,84),(14,42,24,25,79),(15,37,19,26,80),(16,38,20,27,81),(17,39,21,28,82),(18,40,22,29,83),(49,116,66,74,113),(50,117,61,75,114),(51,118,62,76,109),(52,119,63,77,110),(53,120,64,78,111),(54,115,65,73,112)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,59),(8,60),(9,55),(10,56),(11,57),(12,58),(13,81),(14,82),(15,83),(16,84),(17,79),(18,80),(19,22),(20,23),(21,24),(25,39),(26,40),(27,41),(28,42),(29,37),(30,38),(43,46),(44,47),(45,48),(49,119),(50,120),(51,115),(52,116),(53,117),(54,118),(61,111),(62,112),(63,113),(64,114),(65,109),(66,110),(67,106),(68,107),(69,108),(70,103),(71,104),(72,105),(73,76),(74,77),(75,78),(85,88),(86,89),(87,90),(91,100),(92,101),(93,102),(94,97),(95,98),(96,99)], [(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)], [(1,116,13,104),(2,115,14,103),(3,120,15,108),(4,119,16,107),(5,118,17,106),(6,117,18,105),(7,96,112,25),(8,95,113,30),(9,94,114,29),(10,93,109,28),(11,92,110,27),(12,91,111,26),(19,43,90,78),(20,48,85,77),(21,47,86,76),(22,46,87,75),(23,45,88,74),(24,44,89,73),(31,52,81,71),(32,51,82,70),(33,50,83,69),(34,49,84,68),(35,54,79,67),(36,53,80,72),(37,55,97,64),(38,60,98,63),(39,59,99,62),(40,58,100,61),(41,57,101,66),(42,56,102,65)], [(1,119,4,116),(2,118,5,115),(3,117,6,120),(7,25,10,28),(8,30,11,27),(9,29,12,26),(13,107,16,104),(14,106,17,103),(15,105,18,108),(19,46,22,43),(20,45,23,48),(21,44,24,47),(31,49,34,52),(32,54,35,51),(33,53,36,50),(37,58,40,55),(38,57,41,60),(39,56,42,59),(61,100,64,97),(62,99,65,102),(63,98,66,101),(67,79,70,82),(68,84,71,81),(69,83,72,80),(73,89,76,86),(74,88,77,85),(75,87,78,90),(91,114,94,111),(92,113,95,110),(93,112,96,109)]])
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 | ··· | 6G | 6H | ··· | 6O | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | Dic3 | D6 | D6 | D10 | D10 | C3⋊D4 | C4×D5 | S3×D5 | D4×D5 | D5×Dic3 | C2×S3×D5 | D5×C3⋊D4 |
| kernel | D5×C6.D4 | D10⋊Dic3 | C5×C6.D4 | C30.38D4 | C2×D5×Dic3 | D5×C22×C6 | D5×C2×C6 | C23×D5 | C6×D5 | C6.D4 | C22×D5 | C22×D5 | C22×C10 | C2×Dic3 | C22×C6 | D10 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 4 | 2 | 4 | 2 | 1 | 4 | 2 | 8 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of D5×C6.D4 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 17 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 47 | 60 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 13 | 9 | 0 | 0 |
| 15 | 48 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 13 | 1 | 0 | 0 |
| 15 | 48 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,17],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[47,0,0,0,60,13,0,0,0,0,60,0,0,0,0,60],[13,15,0,0,9,48,0,0,0,0,11,0,0,0,0,11],[13,15,0,0,1,48,0,0,0,0,50,0,0,0,0,50] >;
D5×C6.D4 in GAP, Magma, Sage, TeX
D_5\times C_6.D_4
% in TeX
G:=Group("D5xC6.D4"); // GroupNames label
G:=SmallGroup(480,623);
// by ID
G=gap.SmallGroup(480,623);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^6=d^4=1,e^2=c^3,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^-1=e*c*e^-1=c^-1,e*d*e^-1=c^3*d^-1>;
// generators/relations