metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊6D10, D20.26D6, C60.28C23, D60.11C22, Q8⋊D5⋊1S3, Q8⋊3(S3×D5), (S3×D20)⋊3C2, C5⋊D24⋊5C2, C5⋊2C8⋊11D6, C15⋊D8⋊6C2, C5⋊7(Q8⋊3D6), (C3×Q8)⋊2D10, (C5×Q8)⋊11D6, Q8⋊3S3⋊1D5, (C4×S3).9D10, C3⋊3(D4⋊D10), C15⋊23(C8⋊C22), Q8⋊2D15⋊4C2, (S3×C10).12D4, C10.148(S3×D4), C30.190(C2×D4), D6.8(C5⋊D4), (C5×D12)⋊6C22, D6.Dic5⋊5C2, (Q8×C15)⋊4C22, C15⋊3C8⋊14C22, C20.28(C22×S3), (C5×Dic3).38D4, (C3×D20).9C22, C12.28(C22×D5), (S3×C20).10C22, Dic3.17(C5⋊D4), C4.28(C2×S3×D5), (C3×Q8⋊D5)⋊2C2, C2.29(S3×C5⋊D4), C6.51(C2×C5⋊D4), (C5×Q8⋊3S3)⋊1C2, (C3×C5⋊2C8)⋊12C22, SmallGroup(480,580)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊D10
G = < a,b,c,d | a12=b2=c10=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a7b, dcd=c-1 >
Subgroups: 892 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C5⋊2C8, C5⋊2C8, D20, D20, C2×C20, C5×D4, C5×Q8, C22×D5, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, C5×Dic3, C60, C60, S3×D5, C6×D5, S3×C10, S3×C10, D30, C4.Dic5, D4⋊D5, Q8⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, Q8⋊3D6, C3×C5⋊2C8, C15⋊3C8, C3⋊D20, C3×D20, S3×C20, S3×C20, C5×D12, C5×D12, D60, Q8×C15, C2×S3×D5, D4⋊D10, D6.Dic5, C15⋊D8, C5⋊D24, C3×Q8⋊D5, Q8⋊2D15, S3×D20, C5×Q8⋊3S3, D12⋊D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q8⋊3D6, C2×S3×D5, D4⋊D10, S3×C5⋊D4, D12⋊D10
►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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)(81 120)(82 119)(83 118)(84 117)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 108)(95 107)(96 106)
(1 22 97 56 74)(2 15 98 49 75 6 23 102 57 79)(3 20 99 54 76 11 24 107 58 84)(4 13 100 59 77)(5 18 101 52 78 9 14 105 60 82)(7 16 103 50 80)(8 21 104 55 81 12 17 108 51 73)(10 19 106 53 83)(25 115 38 69 95 35 117 48 71 93)(26 120 39 62 96 28 118 41 72 86)(27 113 40 67 85 33 119 46 61 91)(29 111 42 65 87 31 109 44 63 89)(30 116 43 70 88 36 110 37 64 94)(32 114 45 68 90 34 112 47 66 92)
(1 74)(2 73)(3 84)(4 83)(5 82)(6 81)(7 80)(8 79)(9 78)(10 77)(11 76)(12 75)(13 53)(14 52)(15 51)(16 50)(17 49)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 120)(85 94)(86 93)(87 92)(88 91)(89 90)(95 96)(98 108)(99 107)(100 106)(101 105)(102 104)
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,120)(82,119)(83,118)(84,117)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,108)(95,107)(96,106), (1,22,97,56,74)(2,15,98,49,75,6,23,102,57,79)(3,20,99,54,76,11,24,107,58,84)(4,13,100,59,77)(5,18,101,52,78,9,14,105,60,82)(7,16,103,50,80)(8,21,104,55,81,12,17,108,51,73)(10,19,106,53,83)(25,115,38,69,95,35,117,48,71,93)(26,120,39,62,96,28,118,41,72,86)(27,113,40,67,85,33,119,46,61,91)(29,111,42,65,87,31,109,44,63,89)(30,116,43,70,88,36,110,37,64,94)(32,114,45,68,90,34,112,47,66,92), (1,74)(2,73)(3,84)(4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,53)(14,52)(15,51)(16,50)(17,49)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,120)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)(98,108)(99,107)(100,106)(101,105)(102,104)>;
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,120)(82,119)(83,118)(84,117)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,108)(95,107)(96,106), (1,22,97,56,74)(2,15,98,49,75,6,23,102,57,79)(3,20,99,54,76,11,24,107,58,84)(4,13,100,59,77)(5,18,101,52,78,9,14,105,60,82)(7,16,103,50,80)(8,21,104,55,81,12,17,108,51,73)(10,19,106,53,83)(25,115,38,69,95,35,117,48,71,93)(26,120,39,62,96,28,118,41,72,86)(27,113,40,67,85,33,119,46,61,91)(29,111,42,65,87,31,109,44,63,89)(30,116,43,70,88,36,110,37,64,94)(32,114,45,68,90,34,112,47,66,92), (1,74)(2,73)(3,84)(4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,53)(14,52)(15,51)(16,50)(17,49)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,120)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)(98,108)(99,107)(100,106)(101,105)(102,104) );
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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,54),(26,53),(27,52),(28,51),(29,50),(30,49),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(73,116),(74,115),(75,114),(76,113),(77,112),(78,111),(79,110),(80,109),(81,120),(82,119),(83,118),(84,117),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,108),(95,107),(96,106)], [(1,22,97,56,74),(2,15,98,49,75,6,23,102,57,79),(3,20,99,54,76,11,24,107,58,84),(4,13,100,59,77),(5,18,101,52,78,9,14,105,60,82),(7,16,103,50,80),(8,21,104,55,81,12,17,108,51,73),(10,19,106,53,83),(25,115,38,69,95,35,117,48,71,93),(26,120,39,62,96,28,118,41,72,86),(27,113,40,67,85,33,119,46,61,91),(29,111,42,65,87,31,109,44,63,89),(30,116,43,70,88,36,110,37,64,94),(32,114,45,68,90,34,112,47,66,92)], [(1,74),(2,73),(3,84),(4,83),(5,82),(6,81),(7,80),(8,79),(9,78),(10,77),(11,76),(12,75),(13,53),(14,52),(15,51),(16,50),(17,49),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,119),(38,118),(39,117),(40,116),(41,115),(42,114),(43,113),(44,112),(45,111),(46,110),(47,109),(48,120),(85,94),(86,93),(87,92),(88,91),(89,90),(95,96),(98,108),(99,107),(100,106),(101,105),(102,104)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 15A | 15B | 20A | ··· | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 12 | 20 | 60 | 2 | 2 | 4 | 6 | 2 | 2 | 2 | 40 | 20 | 60 | 2 | 2 | 12 | ··· | 12 | 4 | 8 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 20 | 20 | 4 | 4 | 8 | ··· | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | S3×D4 | S3×D5 | Q8⋊3D6 | C2×S3×D5 | D4⋊D10 | S3×C5⋊D4 | D12⋊D10 |
| kernel | D12⋊D10 | D6.Dic5 | C15⋊D8 | C5⋊D24 | C3×Q8⋊D5 | Q8⋊2D15 | S3×D20 | C5×Q8⋊3S3 | Q8⋊D5 | C5×Dic3 | S3×C10 | Q8⋊3S3 | C5⋊2C8 | D20 | C5×Q8 | C4×S3 | D12 | C3×Q8 | Dic3 | D6 | C15 | C10 | Q8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D12⋊D10 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 2 | 2 |
| 0 | 0 | 1 | 0 | 239 | 0 |
| 0 | 0 | 240 | 240 | 1 | 1 |
| 0 | 0 | 1 | 0 | 240 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 232 | 9 |
| 0 | 0 | 0 | 0 | 18 | 9 |
| 0 | 0 | 116 | 125 | 0 | 0 |
| 0 | 0 | 9 | 125 | 0 | 0 |
| 52 | 52 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 52 | 52 | 0 | 0 | 0 | 0 |
| 240 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 240 | 240 | 1 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,240,1,0,0,240,0,240,0,0,0,2,239,1,240,0,0,2,0,1,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,116,9,0,0,0,0,125,125,0,0,232,18,0,0,0,0,9,9,0,0],[52,189,0,0,0,0,52,240,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,240],[52,240,0,0,0,0,52,189,0,0,0,0,0,0,1,240,1,240,0,0,0,240,0,240,0,0,0,0,240,1,0,0,0,0,0,1] >;
D12⋊D10 in GAP, Magma, Sage, TeX
D_{12}\rtimes D_{10} % in TeX
G:=Group("D12:D10"); // GroupNames label
G:=SmallGroup(480,580);
// by ID
G=gap.SmallGroup(480,580);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations