metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.5D6, C30.8D4, C15⋊5SD16, D12.1D5, C10.7D12, Dic30⋊8C2, C12.23D10, C60.16C22, C5⋊2C8⋊2S3, C4.9(S3×D5), C5⋊3(C24⋊C2), C3⋊1(D4.D5), (C5×D12).1C2, C6.2(C5⋊D4), C2.5(C5⋊D12), (C3×C5⋊2C8)⋊2C2, SmallGroup(240,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.D5
G = < a,b,c,d | a12=b2=c5=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Smallest permutation representation of D12.D5
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 23)(14 22)(15 21)(16 20)(17 19)(25 31)(26 30)(27 29)(32 36)(33 35)(37 45)(38 44)(39 43)(40 42)(46 48)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 63)(64 72)(65 71)(66 70)(67 69)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)(86 96)(87 95)(88 94)(89 93)(90 92)(97 104)(98 103)(99 102)(100 101)(105 108)(106 107)(109 118)(110 117)(111 116)(112 115)(113 114)(119 120)
(1 77 114 101 50)(2 78 115 102 51)(3 79 116 103 52)(4 80 117 104 53)(5 81 118 105 54)(6 82 119 106 55)(7 83 120 107 56)(8 84 109 108 57)(9 73 110 97 58)(10 74 111 98 59)(11 75 112 99 60)(12 76 113 100 49)(13 92 63 29 42)(14 93 64 30 43)(15 94 65 31 44)(16 95 66 32 45)(17 96 67 33 46)(18 85 68 34 47)(19 86 69 35 48)(20 87 70 36 37)(21 88 71 25 38)(22 89 72 26 39)(23 90 61 27 40)(24 91 62 28 41)
(1 43 10 40 7 37 4 46)(2 44 11 41 8 38 5 47)(3 45 12 42 9 39 6 48)(13 58 22 55 19 52 16 49)(14 59 23 56 20 53 17 50)(15 60 24 57 21 54 18 51)(25 81 34 78 31 75 28 84)(26 82 35 79 32 76 29 73)(27 83 36 80 33 77 30 74)(61 120 70 117 67 114 64 111)(62 109 71 118 68 115 65 112)(63 110 72 119 69 116 66 113)(85 102 94 99 91 108 88 105)(86 103 95 100 92 97 89 106)(87 104 96 101 93 98 90 107)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,45)(38,44)(39,43)(40,42)(46,48)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,63)(64,72)(65,71)(66,70)(67,69)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)(86,96)(87,95)(88,94)(89,93)(90,92)(97,104)(98,103)(99,102)(100,101)(105,108)(106,107)(109,118)(110,117)(111,116)(112,115)(113,114)(119,120), (1,77,114,101,50)(2,78,115,102,51)(3,79,116,103,52)(4,80,117,104,53)(5,81,118,105,54)(6,82,119,106,55)(7,83,120,107,56)(8,84,109,108,57)(9,73,110,97,58)(10,74,111,98,59)(11,75,112,99,60)(12,76,113,100,49)(13,92,63,29,42)(14,93,64,30,43)(15,94,65,31,44)(16,95,66,32,45)(17,96,67,33,46)(18,85,68,34,47)(19,86,69,35,48)(20,87,70,36,37)(21,88,71,25,38)(22,89,72,26,39)(23,90,61,27,40)(24,91,62,28,41), (1,43,10,40,7,37,4,46)(2,44,11,41,8,38,5,47)(3,45,12,42,9,39,6,48)(13,58,22,55,19,52,16,49)(14,59,23,56,20,53,17,50)(15,60,24,57,21,54,18,51)(25,81,34,78,31,75,28,84)(26,82,35,79,32,76,29,73)(27,83,36,80,33,77,30,74)(61,120,70,117,67,114,64,111)(62,109,71,118,68,115,65,112)(63,110,72,119,69,116,66,113)(85,102,94,99,91,108,88,105)(86,103,95,100,92,97,89,106)(87,104,96,101,93,98,90,107)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,45)(38,44)(39,43)(40,42)(46,48)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,63)(64,72)(65,71)(66,70)(67,69)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)(86,96)(87,95)(88,94)(89,93)(90,92)(97,104)(98,103)(99,102)(100,101)(105,108)(106,107)(109,118)(110,117)(111,116)(112,115)(113,114)(119,120), (1,77,114,101,50)(2,78,115,102,51)(3,79,116,103,52)(4,80,117,104,53)(5,81,118,105,54)(6,82,119,106,55)(7,83,120,107,56)(8,84,109,108,57)(9,73,110,97,58)(10,74,111,98,59)(11,75,112,99,60)(12,76,113,100,49)(13,92,63,29,42)(14,93,64,30,43)(15,94,65,31,44)(16,95,66,32,45)(17,96,67,33,46)(18,85,68,34,47)(19,86,69,35,48)(20,87,70,36,37)(21,88,71,25,38)(22,89,72,26,39)(23,90,61,27,40)(24,91,62,28,41), (1,43,10,40,7,37,4,46)(2,44,11,41,8,38,5,47)(3,45,12,42,9,39,6,48)(13,58,22,55,19,52,16,49)(14,59,23,56,20,53,17,50)(15,60,24,57,21,54,18,51)(25,81,34,78,31,75,28,84)(26,82,35,79,32,76,29,73)(27,83,36,80,33,77,30,74)(61,120,70,117,67,114,64,111)(62,109,71,118,68,115,65,112)(63,110,72,119,69,116,66,113)(85,102,94,99,91,108,88,105)(86,103,95,100,92,97,89,106)(87,104,96,101,93,98,90,107) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,23),(14,22),(15,21),(16,20),(17,19),(25,31),(26,30),(27,29),(32,36),(33,35),(37,45),(38,44),(39,43),(40,42),(46,48),(49,50),(51,60),(52,59),(53,58),(54,57),(55,56),(61,63),(64,72),(65,71),(66,70),(67,69),(73,80),(74,79),(75,78),(76,77),(81,84),(82,83),(86,96),(87,95),(88,94),(89,93),(90,92),(97,104),(98,103),(99,102),(100,101),(105,108),(106,107),(109,118),(110,117),(111,116),(112,115),(113,114),(119,120)], [(1,77,114,101,50),(2,78,115,102,51),(3,79,116,103,52),(4,80,117,104,53),(5,81,118,105,54),(6,82,119,106,55),(7,83,120,107,56),(8,84,109,108,57),(9,73,110,97,58),(10,74,111,98,59),(11,75,112,99,60),(12,76,113,100,49),(13,92,63,29,42),(14,93,64,30,43),(15,94,65,31,44),(16,95,66,32,45),(17,96,67,33,46),(18,85,68,34,47),(19,86,69,35,48),(20,87,70,36,37),(21,88,71,25,38),(22,89,72,26,39),(23,90,61,27,40),(24,91,62,28,41)], [(1,43,10,40,7,37,4,46),(2,44,11,41,8,38,5,47),(3,45,12,42,9,39,6,48),(13,58,22,55,19,52,16,49),(14,59,23,56,20,53,17,50),(15,60,24,57,21,54,18,51),(25,81,34,78,31,75,28,84),(26,82,35,79,32,76,29,73),(27,83,36,80,33,77,30,74),(61,120,70,117,67,114,64,111),(62,109,71,118,68,115,65,112),(63,110,72,119,69,116,66,113),(85,102,94,99,91,108,88,105),(86,103,95,100,92,97,89,106),(87,104,96,101,93,98,90,107)]])
D12.D5 is a maximal subgroup of
D5×C24⋊C2 D24⋊D5 Dic60⋊C2 D24⋊7D5 C20.60D12 D60⋊36C22 D12.33D10 D30.8D4 S3×D4.D5 D20⋊10D6 D30.11D4 D15⋊SD16 Dic10.26D6 D20.27D6 D30.44D4
D12.D5 is a maximal quotient of
C10.D24 Dic30⋊15C4 C60.7Q8
33 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 12 | 2 | 2 | 60 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | SD16 | D10 | D12 | C5⋊D4 | C24⋊C2 | S3×D5 | D4.D5 | C5⋊D12 | D12.D5 |
| kernel | D12.D5 | C3×C5⋊2C8 | C5×D12 | Dic30 | C5⋊2C8 | C30 | D12 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of D12.D5 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 43 | 99 |
| 0 | 0 | 142 | 142 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 99 | 99 |
| 0 | 0 | 198 | 142 |
| 0 | 1 | 0 | 0 |
| 240 | 51 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 173 | 160 | 0 | 0 |
| 66 | 68 | 0 | 0 |
| 0 | 0 | 66 | 147 |
| 0 | 0 | 94 | 213 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,43,142,0,0,99,142],[1,0,0,0,0,1,0,0,0,0,99,198,0,0,99,142],[0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[173,66,0,0,160,68,0,0,0,0,66,94,0,0,147,213] >;
D12.D5 in GAP, Magma, Sage, TeX
D_{12}.D_5 % in TeX
G:=Group("D12.D5"); // GroupNames label
G:=SmallGroup(240,20);
// by ID
G=gap.SmallGroup(240,20);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,169,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^5=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations
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