metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊5D10, Dic10⋊5D6, D30.40D4, C60.24C23, D60.7C22, Dic15.14D4, D4⋊S3⋊6D5, C3⋊C8⋊10D10, (D4×D15)⋊3C2, C5⋊2C8⋊10D6, C5⋊D24⋊4C2, C5⋊3(Q8⋊3D6), D4.D5⋊6S3, C6.74(D4×D5), C3⋊4(D8⋊D5), D4.18(S3×D5), (C5×D4).12D6, C10.75(S3×D4), D12⋊D5⋊2C2, C15⋊21(C8⋊C22), (C3×D4).12D10, C30.186(C2×D4), C15⋊SD16⋊4C2, (C5×D12)⋊5C22, D30.5C4⋊4C2, C20.24(C22×S3), (C4×D15).8C22, C12.24(C22×D5), (C3×Dic10)⋊5C22, (D4×C15).18C22, C2.27(D10⋊D6), C4.24(C2×S3×D5), (C5×D4⋊S3)⋊8C2, (C5×C3⋊C8)⋊10C22, (C3×D4.D5)⋊8C2, (C3×C5⋊2C8)⋊10C22, SmallGroup(480,576)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊5D10
G = < a,b,c,d | a12=b2=c10=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a9b, dbd=a7b, dcd=c-1 >
Subgroups: 988 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, C4×S3 [×2], D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, D15 [×2], C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, C3×Dic5, Dic15, C60, S3×C10, D30, D30 [×3], C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, Q8⋊3D6, C5×C3⋊C8, C3×C5⋊2C8, S3×Dic5, C5⋊D12, C3×Dic10, C5×D12, C4×D15, D60, C15⋊7D4, D4×C15, C22×D15, D8⋊D5, D30.5C4, C5⋊D24, C15⋊SD16, C3×D4.D5, C5×D4⋊S3, D12⋊D5, D4×D15, D12⋊5D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, Q8⋊3D6, C2×S3×D5, D8⋊D5, D10⋊D6, D12⋊5D10
►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)
(1 111)(2 110)(3 109)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 97)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 101)(22 100)(23 99)(24 98)(25 40)(26 39)(27 38)(28 37)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 84)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 96)(69 95)(70 94)(71 93)(72 92)
(1 42 52 96 17 4 39 55 93 20)(2 37 53 91 18 11 40 50 94 15)(3 44 54 86 19 6 41 57 95 22)(5 46 56 88 21 8 43 59 85 24)(7 48 58 90 23 10 45 49 87 14)(9 38 60 92 13 12 47 51 89 16)(25 73 70 98 110 31 79 64 104 116)(26 80 71 105 111)(27 75 72 100 112 33 81 66 106 118)(28 82 61 107 113)(29 77 62 102 114 35 83 68 108 120)(30 84 63 97 115)(32 74 65 99 117)(34 76 67 101 119)(36 78 69 103 109)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 24)(10 23)(11 22)(12 21)(25 62)(26 61)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 86)(38 85)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(73 77)(74 76)(78 84)(79 83)(80 82)(97 109)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)
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), (1,111)(2,110)(3,109)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,97)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,101)(22,100)(23,99)(24,98)(25,40)(26,39)(27,38)(28,37)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,96)(69,95)(70,94)(71,93)(72,92), (1,42,52,96,17,4,39,55,93,20)(2,37,53,91,18,11,40,50,94,15)(3,44,54,86,19,6,41,57,95,22)(5,46,56,88,21,8,43,59,85,24)(7,48,58,90,23,10,45,49,87,14)(9,38,60,92,13,12,47,51,89,16)(25,73,70,98,110,31,79,64,104,116)(26,80,71,105,111)(27,75,72,100,112,33,81,66,106,118)(28,82,61,107,113)(29,77,62,102,114,35,83,68,108,120)(30,84,63,97,115)(32,74,65,99,117)(34,76,67,101,119)(36,78,69,103,109), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,24)(10,23)(11,22)(12,21)(25,62)(26,61)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,86)(38,85)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(73,77)(74,76)(78,84)(79,83)(80,82)(97,109)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)>;
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), (1,111)(2,110)(3,109)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,97)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,101)(22,100)(23,99)(24,98)(25,40)(26,39)(27,38)(28,37)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,96)(69,95)(70,94)(71,93)(72,92), (1,42,52,96,17,4,39,55,93,20)(2,37,53,91,18,11,40,50,94,15)(3,44,54,86,19,6,41,57,95,22)(5,46,56,88,21,8,43,59,85,24)(7,48,58,90,23,10,45,49,87,14)(9,38,60,92,13,12,47,51,89,16)(25,73,70,98,110,31,79,64,104,116)(26,80,71,105,111)(27,75,72,100,112,33,81,66,106,118)(28,82,61,107,113)(29,77,62,102,114,35,83,68,108,120)(30,84,63,97,115)(32,74,65,99,117)(34,76,67,101,119)(36,78,69,103,109), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,24)(10,23)(11,22)(12,21)(25,62)(26,61)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,86)(38,85)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(73,77)(74,76)(78,84)(79,83)(80,82)(97,109)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110) );
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)], [(1,111),(2,110),(3,109),(4,120),(5,119),(6,118),(7,117),(8,116),(9,115),(10,114),(11,113),(12,112),(13,97),(14,108),(15,107),(16,106),(17,105),(18,104),(19,103),(20,102),(21,101),(22,100),(23,99),(24,98),(25,40),(26,39),(27,38),(28,37),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,84),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,96),(69,95),(70,94),(71,93),(72,92)], [(1,42,52,96,17,4,39,55,93,20),(2,37,53,91,18,11,40,50,94,15),(3,44,54,86,19,6,41,57,95,22),(5,46,56,88,21,8,43,59,85,24),(7,48,58,90,23,10,45,49,87,14),(9,38,60,92,13,12,47,51,89,16),(25,73,70,98,110,31,79,64,104,116),(26,80,71,105,111),(27,75,72,100,112,33,81,66,106,118),(28,82,61,107,113),(29,77,62,102,114,35,83,68,108,120),(30,84,63,97,115),(32,74,65,99,117),(34,76,67,101,119),(36,78,69,103,109)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,24),(10,23),(11,22),(12,21),(25,62),(26,61),(27,72),(28,71),(29,70),(30,69),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,86),(38,85),(39,96),(40,95),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(73,77),(74,76),(78,84),(79,83),(80,82),(97,109),(98,120),(99,119),(100,118),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110)])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 12 | 30 | 60 | 2 | 2 | 20 | 30 | 2 | 2 | 2 | 8 | 12 | 20 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 40 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | Q8⋊3D6 | C2×S3×D5 | D8⋊D5 | D10⋊D6 | D12⋊5D10 |
| kernel | D12⋊5D10 | D30.5C4 | C5⋊D24 | C15⋊SD16 | C3×D4.D5 | C5×D4⋊S3 | D12⋊D5 | D4×D15 | D4.D5 | Dic15 | D30 | D4⋊S3 | C5⋊2C8 | Dic10 | C5×D4 | C3⋊C8 | D12 | C3×D4 | C15 | C10 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D12⋊5D10 ►in GL8(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 177 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 | 177 | 2 |
| 0 | 0 | 0 | 0 | 177 | 226 | 2 | 192 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 232 | 232 | 194 | 0 |
| 0 | 0 | 0 | 0 | 116 | 9 | 47 | 107 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 200 |
| 0 | 0 | 0 | 0 | 0 | 0 | 47 | 0 |
| 51 | 51 | 0 | 0 | 0 | 0 | 0 | 0 |
| 190 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 190 | 190 | 190 | 190 | 0 | 0 | 0 | 0 |
| 51 | 240 | 51 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 64 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 51 | 51 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 190 | 0 | 0 | 0 | 0 | 0 | 0 |
| 190 | 190 | 190 | 190 | 0 | 0 | 0 | 0 |
| 240 | 51 | 240 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 64 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,240,177,0,0,0,0,0,240,240,226,1,0,0,0,0,177,177,2,0,0,0,0,0,1,2,192,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,240,0,0,0,0,0,0,2,0,240,0,0,0,0,0,0,0,0,232,116,0,0,0,0,0,0,232,9,0,0,0,0,0,0,194,47,0,47,0,0,0,0,0,107,200,0],[51,190,190,51,0,0,0,0,51,1,190,240,0,0,0,0,0,0,190,51,0,0,0,0,0,0,190,240,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,64,240,0,0,0,0,0,1,240,0,0],[51,1,190,240,0,0,0,0,51,190,190,51,0,0,0,0,0,0,190,240,0,0,0,0,0,0,190,51,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,64,0,240,0,0,0,0,0,240,1,0,0] >;
D12⋊5D10 in GAP, Magma, Sage, TeX
D_{12}\rtimes_5D_{10} % in TeX
G:=Group("D12:5D10"); // GroupNames label
G:=SmallGroup(480,576);
// by ID
G=gap.SmallGroup(480,576);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,303,675,346,185,80,1356,18822]);
// 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^7,c*b*c^-1=a^9*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations