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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, 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, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, 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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 94)(14 93)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 96)(24 95)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 108)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)
(1 75 94 71 59 4 84 85 68 50)(2 82 95 66 60 11 73 92 69 57)(3 77 96 61 49 6 74 87 70 52)(5 79 86 63 51 8 76 89 72 54)(7 81 88 65 53 10 78 91 62 56)(9 83 90 67 55 12 80 93 64 58)(13 30 97 39 109)(14 25 98 46 110 20 31 104 40 116)(15 32 99 41 111)(16 27 100 48 112 22 33 106 42 118)(17 34 101 43 113)(18 29 102 38 114 24 35 108 44 120)(19 36 103 45 115)(21 26 105 47 117)(23 28 107 37 119)
(1 50)(2 49)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 15)(16 24)(17 23)(18 22)(19 21)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 120)(34 119)(35 118)(36 117)(37 101)(38 100)(39 99)(40 98)(41 97)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 84)(72 83)(85 94)(86 93)(87 92)(88 91)(89 90)(95 96)
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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,96)(24,95)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,108)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109), (1,75,94,71,59,4,84,85,68,50)(2,82,95,66,60,11,73,92,69,57)(3,77,96,61,49,6,74,87,70,52)(5,79,86,63,51,8,76,89,72,54)(7,81,88,65,53,10,78,91,62,56)(9,83,90,67,55,12,80,93,64,58)(13,30,97,39,109)(14,25,98,46,110,20,31,104,40,116)(15,32,99,41,111)(16,27,100,48,112,22,33,106,42,118)(17,34,101,43,113)(18,29,102,38,114,24,35,108,44,120)(19,36,103,45,115)(21,26,105,47,117)(23,28,107,37,119), (1,50)(2,49)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,15)(16,24)(17,23)(18,22)(19,21)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,120)(34,119)(35,118)(36,117)(37,101)(38,100)(39,99)(40,98)(41,97)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)>;
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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,96)(24,95)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,108)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109), (1,75,94,71,59,4,84,85,68,50)(2,82,95,66,60,11,73,92,69,57)(3,77,96,61,49,6,74,87,70,52)(5,79,86,63,51,8,76,89,72,54)(7,81,88,65,53,10,78,91,62,56)(9,83,90,67,55,12,80,93,64,58)(13,30,97,39,109)(14,25,98,46,110,20,31,104,40,116)(15,32,99,41,111)(16,27,100,48,112,22,33,106,42,118)(17,34,101,43,113)(18,29,102,38,114,24,35,108,44,120)(19,36,103,45,115)(21,26,105,47,117)(23,28,107,37,119), (1,50)(2,49)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,15)(16,24)(17,23)(18,22)(19,21)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,120)(34,119)(35,118)(36,117)(37,101)(38,100)(39,99)(40,98)(41,97)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96) );
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)], [(1,39),(2,38),(3,37),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,94),(14,93),(15,92),(16,91),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,96),(24,95),(25,61),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100),(57,99),(58,98),(59,97),(60,108),(73,120),(74,119),(75,118),(76,117),(77,116),(78,115),(79,114),(80,113),(81,112),(82,111),(83,110),(84,109)], [(1,75,94,71,59,4,84,85,68,50),(2,82,95,66,60,11,73,92,69,57),(3,77,96,61,49,6,74,87,70,52),(5,79,86,63,51,8,76,89,72,54),(7,81,88,65,53,10,78,91,62,56),(9,83,90,67,55,12,80,93,64,58),(13,30,97,39,109),(14,25,98,46,110,20,31,104,40,116),(15,32,99,41,111),(16,27,100,48,112,22,33,106,42,118),(17,34,101,43,113),(18,29,102,38,114,24,35,108,44,120),(19,36,103,45,115),(21,26,105,47,117),(23,28,107,37,119)], [(1,50),(2,49),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,15),(16,24),(17,23),(18,22),(19,21),(25,116),(26,115),(27,114),(28,113),(29,112),(30,111),(31,110),(32,109),(33,120),(34,119),(35,118),(36,117),(37,101),(38,100),(39,99),(40,98),(41,97),(42,108),(43,107),(44,106),(45,105),(46,104),(47,103),(48,102),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,84),(72,83),(85,94),(86,93),(87,92),(88,91),(89,90),(95,96)]])
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