metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.3D12, (C22×S3)⋊F5, C22⋊F5⋊1S3, C3⋊1(C23⋊F5), C15⋊1(C23⋊C4), (C6×D5).28D4, C22.3(S3×F5), (C2×Dic15)⋊4C4, C2.12(D6⋊F5), C10.12(D6⋊C4), D10.3(C3⋊D4), (C22×D5).34D6, C5⋊2(C23.6D6), C6.12(C22⋊F5), D10.D6⋊1C2, C30.12(C22⋊C4), (S3×C2×C10)⋊2C4, (C2×C6).1(C2×F5), (C2×C10).8(C4×S3), (C2×C30).6(C2×C4), (C3×C22⋊F5)⋊1C2, (D5×C2×C6).64C22, (C2×C15⋊D4).10C2, SmallGroup(480,248)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.D12
G = < a,b,c,d | a10=b2=c12=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a-1bc-1 >
Subgroups: 708 in 104 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, F5, D10, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C30, C23⋊C4, C2×Dic5, C5⋊D4, C2×F5, C22×D5, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, Dic15, C3×F5, C3⋊F5, C6×D5, C6×D5, S3×C10, C2×C30, C22⋊F5, C22⋊F5, C2×C5⋊D4, C23.6D6, C15⋊D4, C2×Dic15, C6×F5, C2×C3⋊F5, D5×C2×C6, S3×C2×C10, C23⋊F5, C3×C22⋊F5, D10.D6, C2×C15⋊D4, D10.D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C23⋊C4, C2×F5, D6⋊C4, C22⋊F5, C23.6D6, S3×F5, C23⋊F5, D6⋊F5, D10.D12
►On 120 points
Generators in S120
(1 58 71 93 104 21 43 32 79 116)(2 94 44 117 72 22 80 59 105 33)(3 118 81 34 45 23 106 95 61 60)(4 35 107 49 82 24 62 119 46 96)(5 50 63 85 108 13 47 36 83 120)(6 86 48 109 64 14 84 51 97 25)(7 110 73 26 37 15 98 87 65 52)(8 27 99 53 74 16 66 111 38 88)(9 54 67 89 100 17 39 28 75 112)(10 90 40 113 68 18 76 55 101 29)(11 114 77 30 41 19 102 91 69 56)(12 31 103 57 78 20 70 115 42 92)
(1 116)(2 72)(3 60)(4 82)(5 120)(6 64)(7 52)(8 74)(9 112)(10 68)(11 56)(12 78)(13 108)(14 25)(15 37)(16 88)(17 100)(18 29)(19 41)(20 92)(21 104)(22 33)(23 45)(24 96)(26 98)(27 53)(28 67)(30 102)(31 57)(32 71)(34 106)(35 49)(36 63)(38 66)(39 89)(42 70)(43 93)(46 62)(47 85)(50 83)(54 75)(58 79)(61 118)(65 110)(69 114)(73 87)(76 101)(77 91)(80 105)(81 95)(84 97)(86 109)(90 113)(94 117)
(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 9)(2 16 22 8)(3 15)(4 6 24 14)(7 23)(10 20 18 12)(11 19)(17 21)(25 46 84 119)(26 106 87 45)(27 59 74 105)(28 116 89 58)(29 42 76 115)(30 102 91 41)(31 55 78 101)(32 112 93 54)(33 38 80 111)(34 98 95 37)(35 51 82 97)(36 120 85 50)(39 71 100 79)(40 92 113 70)(43 67 104 75)(44 88 117 66)(47 63 108 83)(48 96 109 62)(49 86 107 64)(52 61 110 81)(53 94 99 72)(56 69 114 77)(57 90 103 68)(60 65 118 73)
G:=sub<Sym(120)| (1,58,71,93,104,21,43,32,79,116)(2,94,44,117,72,22,80,59,105,33)(3,118,81,34,45,23,106,95,61,60)(4,35,107,49,82,24,62,119,46,96)(5,50,63,85,108,13,47,36,83,120)(6,86,48,109,64,14,84,51,97,25)(7,110,73,26,37,15,98,87,65,52)(8,27,99,53,74,16,66,111,38,88)(9,54,67,89,100,17,39,28,75,112)(10,90,40,113,68,18,76,55,101,29)(11,114,77,30,41,19,102,91,69,56)(12,31,103,57,78,20,70,115,42,92), (1,116)(2,72)(3,60)(4,82)(5,120)(6,64)(7,52)(8,74)(9,112)(10,68)(11,56)(12,78)(13,108)(14,25)(15,37)(16,88)(17,100)(18,29)(19,41)(20,92)(21,104)(22,33)(23,45)(24,96)(26,98)(27,53)(28,67)(30,102)(31,57)(32,71)(34,106)(35,49)(36,63)(38,66)(39,89)(42,70)(43,93)(46,62)(47,85)(50,83)(54,75)(58,79)(61,118)(65,110)(69,114)(73,87)(76,101)(77,91)(80,105)(81,95)(84,97)(86,109)(90,113)(94,117), (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), (1,9)(2,16,22,8)(3,15)(4,6,24,14)(7,23)(10,20,18,12)(11,19)(17,21)(25,46,84,119)(26,106,87,45)(27,59,74,105)(28,116,89,58)(29,42,76,115)(30,102,91,41)(31,55,78,101)(32,112,93,54)(33,38,80,111)(34,98,95,37)(35,51,82,97)(36,120,85,50)(39,71,100,79)(40,92,113,70)(43,67,104,75)(44,88,117,66)(47,63,108,83)(48,96,109,62)(49,86,107,64)(52,61,110,81)(53,94,99,72)(56,69,114,77)(57,90,103,68)(60,65,118,73)>;
G:=Group( (1,58,71,93,104,21,43,32,79,116)(2,94,44,117,72,22,80,59,105,33)(3,118,81,34,45,23,106,95,61,60)(4,35,107,49,82,24,62,119,46,96)(5,50,63,85,108,13,47,36,83,120)(6,86,48,109,64,14,84,51,97,25)(7,110,73,26,37,15,98,87,65,52)(8,27,99,53,74,16,66,111,38,88)(9,54,67,89,100,17,39,28,75,112)(10,90,40,113,68,18,76,55,101,29)(11,114,77,30,41,19,102,91,69,56)(12,31,103,57,78,20,70,115,42,92), (1,116)(2,72)(3,60)(4,82)(5,120)(6,64)(7,52)(8,74)(9,112)(10,68)(11,56)(12,78)(13,108)(14,25)(15,37)(16,88)(17,100)(18,29)(19,41)(20,92)(21,104)(22,33)(23,45)(24,96)(26,98)(27,53)(28,67)(30,102)(31,57)(32,71)(34,106)(35,49)(36,63)(38,66)(39,89)(42,70)(43,93)(46,62)(47,85)(50,83)(54,75)(58,79)(61,118)(65,110)(69,114)(73,87)(76,101)(77,91)(80,105)(81,95)(84,97)(86,109)(90,113)(94,117), (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), (1,9)(2,16,22,8)(3,15)(4,6,24,14)(7,23)(10,20,18,12)(11,19)(17,21)(25,46,84,119)(26,106,87,45)(27,59,74,105)(28,116,89,58)(29,42,76,115)(30,102,91,41)(31,55,78,101)(32,112,93,54)(33,38,80,111)(34,98,95,37)(35,51,82,97)(36,120,85,50)(39,71,100,79)(40,92,113,70)(43,67,104,75)(44,88,117,66)(47,63,108,83)(48,96,109,62)(49,86,107,64)(52,61,110,81)(53,94,99,72)(56,69,114,77)(57,90,103,68)(60,65,118,73) );
G=PermutationGroup([[(1,58,71,93,104,21,43,32,79,116),(2,94,44,117,72,22,80,59,105,33),(3,118,81,34,45,23,106,95,61,60),(4,35,107,49,82,24,62,119,46,96),(5,50,63,85,108,13,47,36,83,120),(6,86,48,109,64,14,84,51,97,25),(7,110,73,26,37,15,98,87,65,52),(8,27,99,53,74,16,66,111,38,88),(9,54,67,89,100,17,39,28,75,112),(10,90,40,113,68,18,76,55,101,29),(11,114,77,30,41,19,102,91,69,56),(12,31,103,57,78,20,70,115,42,92)], [(1,116),(2,72),(3,60),(4,82),(5,120),(6,64),(7,52),(8,74),(9,112),(10,68),(11,56),(12,78),(13,108),(14,25),(15,37),(16,88),(17,100),(18,29),(19,41),(20,92),(21,104),(22,33),(23,45),(24,96),(26,98),(27,53),(28,67),(30,102),(31,57),(32,71),(34,106),(35,49),(36,63),(38,66),(39,89),(42,70),(43,93),(46,62),(47,85),(50,83),(54,75),(58,79),(61,118),(65,110),(69,114),(73,87),(76,101),(77,91),(80,105),(81,95),(84,97),(86,109),(90,113),(94,117)], [(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)], [(1,9),(2,16,22,8),(3,15),(4,6,24,14),(7,23),(10,20,18,12),(11,19),(17,21),(25,46,84,119),(26,106,87,45),(27,59,74,105),(28,116,89,58),(29,42,76,115),(30,102,91,41),(31,55,78,101),(32,112,93,54),(33,38,80,111),(34,98,95,37),(35,51,82,97),(36,120,85,50),(39,71,100,79),(40,92,113,70),(43,67,104,75),(44,88,117,66),(47,63,108,83),(48,96,109,62),(49,86,107,64),(52,61,110,81),(53,94,99,72),(56,69,114,77),(57,90,103,68),(60,65,118,73)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 12A | 12B | 12C | 12D | 15 | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 10 | 10 | 12 | 2 | 20 | 20 | 60 | 60 | 60 | 4 | 2 | 4 | 10 | 10 | 20 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | F5 | C23⋊C4 | C2×F5 | C22⋊F5 | C23.6D6 | C23⋊F5 | S3×F5 | D6⋊F5 | D10.D12 |
| kernel | D10.D12 | C3×C22⋊F5 | D10.D6 | C2×C15⋊D4 | C2×Dic15 | S3×C2×C10 | C22⋊F5 | C6×D5 | C22×D5 | D10 | D10 | C2×C10 | C22×S3 | C15 | C2×C6 | C6 | C5 | C3 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of D10.D12 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 50 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 7 | 37 | 51 |
| 0 | 0 | 13 | 58 | 27 | 27 |
| 0 | 0 | 3 | 34 | 34 | 3 |
| 0 | 0 | 10 | 10 | 24 | 54 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 50 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 1 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0],[0,50,0,0,0,0,11,11,0,0,0,0,0,0,37,13,3,10,0,0,7,58,34,10,0,0,37,27,34,24,0,0,51,27,3,54],[50,50,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60,0,0,0,1,0,0] >;
D10.D12 in GAP, Magma, Sage, TeX
D_{10}.D_{12} % in TeX
G:=Group("D10.D12"); // GroupNames label
G:=SmallGroup(480,248);
// by ID
G=gap.SmallGroup(480,248);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,675,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations