metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊5D5, D6.2D10, C20.17D6, D10.10D6, C12.27D10, C30.9C23, Dic30⋊10C2, C60.20C22, Dic5.19D6, Dic15.4C22, (C4×D5)⋊1S3, (C5×D12)⋊2C2, (D5×C12)⋊1C2, C5⋊3(C4○D12), C15⋊6(C4○D4), C15⋊D4⋊2C2, C4.13(S3×D5), C3⋊2(D4⋊2D5), (S3×Dic5)⋊2C2, C6.9(C22×D5), C10.9(C22×S3), (S3×C10).2C22, (C6×D5).11C22, (C3×Dic5).13C22, C2.13(C2×S3×D5), SmallGroup(240,133)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊5D5
G = < a,b,c,d | a12=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 320 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, D10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, C30, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C4○D12, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D4⋊2D5, S3×Dic5, C15⋊D4, D5×C12, C5×D12, Dic30, D12⋊5D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, C4○D12, S3×D5, D4⋊2D5, C2×S3×D5, D12⋊5D5
►On 120 points
Generators in S120
(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 87)(2 86)(3 85)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 101)(26 100)(27 99)(28 98)(29 97)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(49 110)(50 109)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 84)(70 83)(71 82)(72 81)
(1 22 55 27 79)(2 23 56 28 80)(3 24 57 29 81)(4 13 58 30 82)(5 14 59 31 83)(6 15 60 32 84)(7 16 49 33 73)(8 17 50 34 74)(9 18 51 35 75)(10 19 52 36 76)(11 20 53 25 77)(12 21 54 26 78)(37 112 107 70 95)(38 113 108 71 96)(39 114 97 72 85)(40 115 98 61 86)(41 116 99 62 87)(42 117 100 63 88)(43 118 101 64 89)(44 119 102 65 90)(45 120 103 66 91)(46 109 104 67 92)(47 110 105 68 93)(48 111 106 69 94)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 97)(46 98)(47 99)(48 100)(61 92)(62 93)(63 94)(64 95)(65 96)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
G:=sub<Sym(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), (1,87)(2,86)(3,85)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,101)(26,100)(27,99)(28,98)(29,97)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(49,110)(50,109)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,84)(70,83)(71,82)(72,81), (1,22,55,27,79)(2,23,56,28,80)(3,24,57,29,81)(4,13,58,30,82)(5,14,59,31,83)(6,15,60,32,84)(7,16,49,33,73)(8,17,50,34,74)(9,18,51,35,75)(10,19,52,36,76)(11,20,53,25,77)(12,21,54,26,78)(37,112,107,70,95)(38,113,108,71,96)(39,114,97,72,85)(40,115,98,61,86)(41,116,99,62,87)(42,117,100,63,88)(43,118,101,64,89)(44,119,102,65,90)(45,120,103,66,91)(46,109,104,67,92)(47,110,105,68,93)(48,111,106,69,94), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,97)(46,98)(47,99)(48,100)(61,92)(62,93)(63,94)(64,95)(65,96)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)>;
G:=Group( (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), (1,87)(2,86)(3,85)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,101)(26,100)(27,99)(28,98)(29,97)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(49,110)(50,109)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,84)(70,83)(71,82)(72,81), (1,22,55,27,79)(2,23,56,28,80)(3,24,57,29,81)(4,13,58,30,82)(5,14,59,31,83)(6,15,60,32,84)(7,16,49,33,73)(8,17,50,34,74)(9,18,51,35,75)(10,19,52,36,76)(11,20,53,25,77)(12,21,54,26,78)(37,112,107,70,95)(38,113,108,71,96)(39,114,97,72,85)(40,115,98,61,86)(41,116,99,62,87)(42,117,100,63,88)(43,118,101,64,89)(44,119,102,65,90)(45,120,103,66,91)(46,109,104,67,92)(47,110,105,68,93)(48,111,106,69,94), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,97)(46,98)(47,99)(48,100)(61,92)(62,93)(63,94)(64,95)(65,96)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120) );
G=PermutationGroup([[(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)], [(1,87),(2,86),(3,85),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,90),(11,89),(12,88),(13,38),(14,37),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,101),(26,100),(27,99),(28,98),(29,97),(30,108),(31,107),(32,106),(33,105),(34,104),(35,103),(36,102),(49,110),(50,109),(51,120),(52,119),(53,118),(54,117),(55,116),(56,115),(57,114),(58,113),(59,112),(60,111),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,84),(70,83),(71,82),(72,81)], [(1,22,55,27,79),(2,23,56,28,80),(3,24,57,29,81),(4,13,58,30,82),(5,14,59,31,83),(6,15,60,32,84),(7,16,49,33,73),(8,17,50,34,74),(9,18,51,35,75),(10,19,52,36,76),(11,20,53,25,77),(12,21,54,26,78),(37,112,107,70,95),(38,113,108,71,96),(39,114,97,72,85),(40,115,98,61,86),(41,116,99,62,87),(42,117,100,63,88),(43,118,101,64,89),(44,119,102,65,90),(45,120,103,66,91),(46,109,104,67,92),(47,110,105,68,93),(48,111,106,69,94)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,25),(21,26),(22,27),(23,28),(24,29),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,97),(46,98),(47,99),(48,100),(61,92),(62,93),(63,94),(64,95),(65,96),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120)]])
D12⋊5D5 is a maximal subgroup of
D12⋊4F5 D12⋊2F5 D24⋊D5 Dic60⋊C2 C40.31D6 D24⋊7D5 D12⋊10D10 D12.24D10 D12.27D10 D20.14D6 D12.2F5 D12.F5 D20.39D6 D5×C4○D12 D20⋊26D6 S3×D4⋊2D5 D20⋊13D6 D12.29D10 D20⋊16D6
D12⋊5D5 is a maximal quotient of
Dic30⋊17C4 Dic15.Q8 C60⋊5C4⋊C2 (C4×D5)⋊Dic3 C60.68D4 (C2×C12).D10 C5⋊(C42⋊3S3) C60.69D4 C20.Dic6 D6.(C4×D5) Dic5×D12 (C2×D12).D5 D6⋊3Dic10 D6⋊(C4×D5) D6⋊C4⋊D5 C60⋊D4 Dic15⋊2D4 D6.9D20 Dic15.10D4
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 6 | 6 | 10 | 2 | 2 | 5 | 5 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C4○D12 | S3×D5 | D4⋊2D5 | C2×S3×D5 | D12⋊5D5 |
| kernel | D12⋊5D5 | S3×Dic5 | C15⋊D4 | D5×C12 | C5×D12 | Dic30 | C4×D5 | D12 | Dic5 | C20 | D10 | C15 | C12 | D6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of D12⋊5D5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 21 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 21 |
| 0 | 0 | 32 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,21],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,21,0],[0,60,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,60] >;
D12⋊5D5 in GAP, Magma, Sage, TeX
D_{12}\rtimes_5D_5 % in TeX
G:=Group("D12:5D5"); // GroupNames label
G:=SmallGroup(240,133);
// by ID
G=gap.SmallGroup(240,133);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations