metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.9D10, D60⋊5C22, Dic10⋊10D6, C60.20C23, (S3×D4)⋊3D5, C5⋊2C8⋊8D6, D4⋊D15⋊8C2, C5⋊D24⋊3C2, C5⋊6(Q8⋊3D6), D4.D5⋊4S3, D4.6(S3×D5), (C3×D4).8D10, (C4×S3).8D10, (C5×D4).23D6, D60⋊C2⋊2C2, C15⋊19(C8⋊C22), (S3×C10).34D4, C10.146(S3×D4), C20.D6⋊4C2, C30.182(C2×D4), D6.Dic5⋊4C2, C15⋊3C8⋊10C22, D6.14(C5⋊D4), C3⋊3(D4.D10), (S3×C20).8C22, C20.20(C22×S3), (C5×Dic3).14D4, (C5×D12).6C22, C12.20(C22×D5), (C3×Dic10)⋊4C22, (D4×C15).14C22, Dic3.11(C5⋊D4), (C5×S3×D4)⋊3C2, C4.20(C2×S3×D5), (C3×D4.D5)⋊6C2, C2.27(S3×C5⋊D4), C6.49(C2×C5⋊D4), (C3×C5⋊2C8)⋊8C22, SmallGroup(480,572)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.9D10
G = < a,b,c,d | a12=b2=d2=1, c10=a6, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a7b, dcd=c9 >
Subgroups: 732 in 136 conjugacy classes, 40 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, C10, C10 [×3], Dic3, C12, C12, D6, D6 [×4], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10 [×5], C3⋊C8, C24, C4×S3, C4×S3, D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], D15, C30, C30, C8⋊C22, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4 [×2], C22×C10, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, C5×Dic3, C3×Dic5, C60, S3×C10, S3×C10 [×3], D30, C2×C30, C4.Dic5, D4⋊D5 [×2], D4.D5, D4.D5, C4○D20, D4×C10, Q8⋊3D6, C3×C5⋊2C8, C15⋊3C8, D30.C2, C5⋊D12, C3×Dic10, S3×C20, C5×D12, C5×C3⋊D4, D60, D4×C15, S3×C2×C10, D4.D10, D6.Dic5, C5⋊D24, C20.D6, C3×D4.D5, D4⋊D15, D60⋊C2, C5×S3×D4, D12.9D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q8⋊3D6, C2×S3×D5, D4.D10, S3×C5⋊D4, D12.9D10
►On 120 points
Generators in S120
(1 53 48 6 92 87 11 43 58 16 82 97)(2 88 83 7 59 54 12 98 93 17 49 44)(3 55 50 8 94 89 13 45 60 18 84 99)(4 90 85 9 41 56 14 100 95 19 51 46)(5 57 52 10 96 91 15 47 42 20 86 81)(21 66 71 36 106 111 31 76 61 26 116 101)(22 112 117 37 62 67 32 102 107 27 72 77)(23 68 73 38 108 113 33 78 63 28 118 103)(24 114 119 39 64 69 34 104 109 29 74 79)(25 70 75 40 110 115 35 80 65 30 120 105)
(1 97)(2 54)(3 99)(4 56)(5 81)(6 58)(7 83)(8 60)(9 85)(10 42)(11 87)(12 44)(13 89)(14 46)(15 91)(16 48)(17 93)(18 50)(19 95)(20 52)(21 71)(22 107)(23 73)(24 109)(25 75)(26 111)(27 77)(28 113)(29 79)(30 115)(31 61)(32 117)(33 63)(34 119)(35 65)(36 101)(37 67)(38 103)(39 69)(40 105)(41 90)(43 92)(45 94)(47 96)(49 98)(51 100)(53 82)(55 84)(57 86)(59 88)(102 112)(104 114)(106 116)(108 118)(110 120)
(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 38)(2 27)(3 36)(4 25)(5 34)(6 23)(7 32)(8 21)(9 30)(10 39)(11 28)(12 37)(13 26)(14 35)(15 24)(16 33)(17 22)(18 31)(19 40)(20 29)(41 65)(42 74)(43 63)(44 72)(45 61)(46 70)(47 79)(48 68)(49 77)(50 66)(51 75)(52 64)(53 73)(54 62)(55 71)(56 80)(57 69)(58 78)(59 67)(60 76)(81 104)(82 113)(83 102)(84 111)(85 120)(86 109)(87 118)(88 107)(89 116)(90 105)(91 114)(92 103)(93 112)(94 101)(95 110)(96 119)(97 108)(98 117)(99 106)(100 115)
G:=sub<Sym(120)| (1,53,48,6,92,87,11,43,58,16,82,97)(2,88,83,7,59,54,12,98,93,17,49,44)(3,55,50,8,94,89,13,45,60,18,84,99)(4,90,85,9,41,56,14,100,95,19,51,46)(5,57,52,10,96,91,15,47,42,20,86,81)(21,66,71,36,106,111,31,76,61,26,116,101)(22,112,117,37,62,67,32,102,107,27,72,77)(23,68,73,38,108,113,33,78,63,28,118,103)(24,114,119,39,64,69,34,104,109,29,74,79)(25,70,75,40,110,115,35,80,65,30,120,105), (1,97)(2,54)(3,99)(4,56)(5,81)(6,58)(7,83)(8,60)(9,85)(10,42)(11,87)(12,44)(13,89)(14,46)(15,91)(16,48)(17,93)(18,50)(19,95)(20,52)(21,71)(22,107)(23,73)(24,109)(25,75)(26,111)(27,77)(28,113)(29,79)(30,115)(31,61)(32,117)(33,63)(34,119)(35,65)(36,101)(37,67)(38,103)(39,69)(40,105)(41,90)(43,92)(45,94)(47,96)(49,98)(51,100)(53,82)(55,84)(57,86)(59,88)(102,112)(104,114)(106,116)(108,118)(110,120), (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), (1,38)(2,27)(3,36)(4,25)(5,34)(6,23)(7,32)(8,21)(9,30)(10,39)(11,28)(12,37)(13,26)(14,35)(15,24)(16,33)(17,22)(18,31)(19,40)(20,29)(41,65)(42,74)(43,63)(44,72)(45,61)(46,70)(47,79)(48,68)(49,77)(50,66)(51,75)(52,64)(53,73)(54,62)(55,71)(56,80)(57,69)(58,78)(59,67)(60,76)(81,104)(82,113)(83,102)(84,111)(85,120)(86,109)(87,118)(88,107)(89,116)(90,105)(91,114)(92,103)(93,112)(94,101)(95,110)(96,119)(97,108)(98,117)(99,106)(100,115)>;
G:=Group( (1,53,48,6,92,87,11,43,58,16,82,97)(2,88,83,7,59,54,12,98,93,17,49,44)(3,55,50,8,94,89,13,45,60,18,84,99)(4,90,85,9,41,56,14,100,95,19,51,46)(5,57,52,10,96,91,15,47,42,20,86,81)(21,66,71,36,106,111,31,76,61,26,116,101)(22,112,117,37,62,67,32,102,107,27,72,77)(23,68,73,38,108,113,33,78,63,28,118,103)(24,114,119,39,64,69,34,104,109,29,74,79)(25,70,75,40,110,115,35,80,65,30,120,105), (1,97)(2,54)(3,99)(4,56)(5,81)(6,58)(7,83)(8,60)(9,85)(10,42)(11,87)(12,44)(13,89)(14,46)(15,91)(16,48)(17,93)(18,50)(19,95)(20,52)(21,71)(22,107)(23,73)(24,109)(25,75)(26,111)(27,77)(28,113)(29,79)(30,115)(31,61)(32,117)(33,63)(34,119)(35,65)(36,101)(37,67)(38,103)(39,69)(40,105)(41,90)(43,92)(45,94)(47,96)(49,98)(51,100)(53,82)(55,84)(57,86)(59,88)(102,112)(104,114)(106,116)(108,118)(110,120), (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), (1,38)(2,27)(3,36)(4,25)(5,34)(6,23)(7,32)(8,21)(9,30)(10,39)(11,28)(12,37)(13,26)(14,35)(15,24)(16,33)(17,22)(18,31)(19,40)(20,29)(41,65)(42,74)(43,63)(44,72)(45,61)(46,70)(47,79)(48,68)(49,77)(50,66)(51,75)(52,64)(53,73)(54,62)(55,71)(56,80)(57,69)(58,78)(59,67)(60,76)(81,104)(82,113)(83,102)(84,111)(85,120)(86,109)(87,118)(88,107)(89,116)(90,105)(91,114)(92,103)(93,112)(94,101)(95,110)(96,119)(97,108)(98,117)(99,106)(100,115) );
G=PermutationGroup([(1,53,48,6,92,87,11,43,58,16,82,97),(2,88,83,7,59,54,12,98,93,17,49,44),(3,55,50,8,94,89,13,45,60,18,84,99),(4,90,85,9,41,56,14,100,95,19,51,46),(5,57,52,10,96,91,15,47,42,20,86,81),(21,66,71,36,106,111,31,76,61,26,116,101),(22,112,117,37,62,67,32,102,107,27,72,77),(23,68,73,38,108,113,33,78,63,28,118,103),(24,114,119,39,64,69,34,104,109,29,74,79),(25,70,75,40,110,115,35,80,65,30,120,105)], [(1,97),(2,54),(3,99),(4,56),(5,81),(6,58),(7,83),(8,60),(9,85),(10,42),(11,87),(12,44),(13,89),(14,46),(15,91),(16,48),(17,93),(18,50),(19,95),(20,52),(21,71),(22,107),(23,73),(24,109),(25,75),(26,111),(27,77),(28,113),(29,79),(30,115),(31,61),(32,117),(33,63),(34,119),(35,65),(36,101),(37,67),(38,103),(39,69),(40,105),(41,90),(43,92),(45,94),(47,96),(49,98),(51,100),(53,82),(55,84),(57,86),(59,88),(102,112),(104,114),(106,116),(108,118),(110,120)], [(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)], [(1,38),(2,27),(3,36),(4,25),(5,34),(6,23),(7,32),(8,21),(9,30),(10,39),(11,28),(12,37),(13,26),(14,35),(15,24),(16,33),(17,22),(18,31),(19,40),(20,29),(41,65),(42,74),(43,63),(44,72),(45,61),(46,70),(47,79),(48,68),(49,77),(50,66),(51,75),(52,64),(53,73),(54,62),(55,71),(56,80),(57,69),(58,78),(59,67),(60,76),(81,104),(82,113),(83,102),(84,111),(85,120),(86,109),(87,118),(88,107),(89,116),(90,105),(91,114),(92,103),(93,112),(94,101),(95,110),(96,119),(97,108),(98,117),(99,106),(100,115)])
48 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 | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 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 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 6 | 12 | 60 | 2 | 2 | 6 | 20 | 2 | 2 | 2 | 8 | 20 | 60 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 40 | 4 | 4 | 4 | 4 | 12 | 12 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | D12.9D10 |
| kernel | D12.9D10 | D6.Dic5 | C5⋊D24 | C20.D6 | C3×D4.D5 | D4⋊D15 | D60⋊C2 | C5×S3×D4 | D4.D5 | C5×Dic3 | S3×C10 | S3×D4 | C5⋊2C8 | Dic10 | C5×D4 | C4×S3 | D12 | C3×D4 | Dic3 | D6 | C15 | C10 | D4 | 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 D12.9D10 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 2 | 2 |
| 0 | 0 | 240 | 0 | 239 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 2 | 2 |
| 0 | 0 | 0 | 240 | 0 | 239 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 51 | 51 | 0 | 0 | 0 | 0 |
| 190 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 240 | 240 | 239 | 239 |
| 0 | 0 | 240 | 0 | 240 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 231 | 59 | 0 | 0 | 0 | 0 |
| 31 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 | 9 | 18 |
| 0 | 0 | 9 | 232 | 9 | 232 |
| 0 | 0 | 116 | 232 | 232 | 223 |
| 0 | 0 | 116 | 125 | 232 | 9 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,240,240,1,0,0,1,0,240,0,0,0,2,239,240,1,0,0,2,0,240,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,240,0,0,0,0,2,0,240,0,0,0,2,239,240,1],[51,190,0,0,0,0,51,1,0,0,0,0,0,0,1,240,240,1,0,0,0,240,0,1,0,0,2,239,240,1,0,0,0,239,0,1],[231,31,0,0,0,0,59,10,0,0,0,0,0,0,9,9,116,116,0,0,18,232,232,125,0,0,9,9,232,232,0,0,18,232,223,9] >;
D12.9D10 in GAP, Magma, Sage, TeX
D_{12}._9D_{10} % in TeX
G:=Group("D12.9D10"); // GroupNames label
G:=SmallGroup(480,572);
// by ID
G=gap.SmallGroup(480,572);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^10=a^6,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^9>;
// generators/relations