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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, D15, C30, C30, C8⋊C22, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×C10, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, C5×Dic3, C3×Dic5, C60, S3×C10, S3×C10, D30, C2×C30, C4.Dic5, D4⋊D5, 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, 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, D12.9D10
►On 120 points
Generators in S120
(1 108 103 6 99 94 11 118 113 16 89 84)(2 95 90 7 114 109 12 85 100 17 104 119)(3 110 105 8 81 96 13 120 115 18 91 86)(4 97 92 9 116 111 14 87 82 19 106 101)(5 112 107 10 83 98 15 102 117 20 93 88)(21 43 48 36 74 79 31 53 58 26 64 69)(22 80 65 37 59 44 32 70 75 27 49 54)(23 45 50 38 76 61 33 55 60 28 66 71)(24 62 67 39 41 46 34 72 77 29 51 56)(25 47 52 40 78 63 35 57 42 30 68 73)
(1 84)(2 109)(3 86)(4 111)(5 88)(6 113)(7 90)(8 115)(9 92)(10 117)(11 94)(12 119)(13 96)(14 101)(15 98)(16 103)(17 100)(18 105)(19 82)(20 107)(21 58)(22 65)(23 60)(24 67)(25 42)(26 69)(27 44)(28 71)(29 46)(30 73)(31 48)(32 75)(33 50)(34 77)(35 52)(36 79)(37 54)(38 61)(39 56)(40 63)(41 51)(43 53)(45 55)(47 57)(49 59)(81 120)(83 102)(85 104)(87 106)(89 108)(91 110)(93 112)(95 114)(97 116)(99 118)
(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 33)(2 22)(3 31)(4 40)(5 29)(6 38)(7 27)(8 36)(9 25)(10 34)(11 23)(12 32)(13 21)(14 30)(15 39)(16 28)(17 37)(18 26)(19 35)(20 24)(41 98)(42 87)(43 96)(44 85)(45 94)(46 83)(47 92)(48 81)(49 90)(50 99)(51 88)(52 97)(53 86)(54 95)(55 84)(56 93)(57 82)(58 91)(59 100)(60 89)(61 108)(62 117)(63 106)(64 115)(65 104)(66 113)(67 102)(68 111)(69 120)(70 109)(71 118)(72 107)(73 116)(74 105)(75 114)(76 103)(77 112)(78 101)(79 110)(80 119)
G:=sub<Sym(120)| (1,108,103,6,99,94,11,118,113,16,89,84)(2,95,90,7,114,109,12,85,100,17,104,119)(3,110,105,8,81,96,13,120,115,18,91,86)(4,97,92,9,116,111,14,87,82,19,106,101)(5,112,107,10,83,98,15,102,117,20,93,88)(21,43,48,36,74,79,31,53,58,26,64,69)(22,80,65,37,59,44,32,70,75,27,49,54)(23,45,50,38,76,61,33,55,60,28,66,71)(24,62,67,39,41,46,34,72,77,29,51,56)(25,47,52,40,78,63,35,57,42,30,68,73), (1,84)(2,109)(3,86)(4,111)(5,88)(6,113)(7,90)(8,115)(9,92)(10,117)(11,94)(12,119)(13,96)(14,101)(15,98)(16,103)(17,100)(18,105)(19,82)(20,107)(21,58)(22,65)(23,60)(24,67)(25,42)(26,69)(27,44)(28,71)(29,46)(30,73)(31,48)(32,75)(33,50)(34,77)(35,52)(36,79)(37,54)(38,61)(39,56)(40,63)(41,51)(43,53)(45,55)(47,57)(49,59)(81,120)(83,102)(85,104)(87,106)(89,108)(91,110)(93,112)(95,114)(97,116)(99,118), (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), (1,33)(2,22)(3,31)(4,40)(5,29)(6,38)(7,27)(8,36)(9,25)(10,34)(11,23)(12,32)(13,21)(14,30)(15,39)(16,28)(17,37)(18,26)(19,35)(20,24)(41,98)(42,87)(43,96)(44,85)(45,94)(46,83)(47,92)(48,81)(49,90)(50,99)(51,88)(52,97)(53,86)(54,95)(55,84)(56,93)(57,82)(58,91)(59,100)(60,89)(61,108)(62,117)(63,106)(64,115)(65,104)(66,113)(67,102)(68,111)(69,120)(70,109)(71,118)(72,107)(73,116)(74,105)(75,114)(76,103)(77,112)(78,101)(79,110)(80,119)>;
G:=Group( (1,108,103,6,99,94,11,118,113,16,89,84)(2,95,90,7,114,109,12,85,100,17,104,119)(3,110,105,8,81,96,13,120,115,18,91,86)(4,97,92,9,116,111,14,87,82,19,106,101)(5,112,107,10,83,98,15,102,117,20,93,88)(21,43,48,36,74,79,31,53,58,26,64,69)(22,80,65,37,59,44,32,70,75,27,49,54)(23,45,50,38,76,61,33,55,60,28,66,71)(24,62,67,39,41,46,34,72,77,29,51,56)(25,47,52,40,78,63,35,57,42,30,68,73), (1,84)(2,109)(3,86)(4,111)(5,88)(6,113)(7,90)(8,115)(9,92)(10,117)(11,94)(12,119)(13,96)(14,101)(15,98)(16,103)(17,100)(18,105)(19,82)(20,107)(21,58)(22,65)(23,60)(24,67)(25,42)(26,69)(27,44)(28,71)(29,46)(30,73)(31,48)(32,75)(33,50)(34,77)(35,52)(36,79)(37,54)(38,61)(39,56)(40,63)(41,51)(43,53)(45,55)(47,57)(49,59)(81,120)(83,102)(85,104)(87,106)(89,108)(91,110)(93,112)(95,114)(97,116)(99,118), (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), (1,33)(2,22)(3,31)(4,40)(5,29)(6,38)(7,27)(8,36)(9,25)(10,34)(11,23)(12,32)(13,21)(14,30)(15,39)(16,28)(17,37)(18,26)(19,35)(20,24)(41,98)(42,87)(43,96)(44,85)(45,94)(46,83)(47,92)(48,81)(49,90)(50,99)(51,88)(52,97)(53,86)(54,95)(55,84)(56,93)(57,82)(58,91)(59,100)(60,89)(61,108)(62,117)(63,106)(64,115)(65,104)(66,113)(67,102)(68,111)(69,120)(70,109)(71,118)(72,107)(73,116)(74,105)(75,114)(76,103)(77,112)(78,101)(79,110)(80,119) );
G=PermutationGroup([[(1,108,103,6,99,94,11,118,113,16,89,84),(2,95,90,7,114,109,12,85,100,17,104,119),(3,110,105,8,81,96,13,120,115,18,91,86),(4,97,92,9,116,111,14,87,82,19,106,101),(5,112,107,10,83,98,15,102,117,20,93,88),(21,43,48,36,74,79,31,53,58,26,64,69),(22,80,65,37,59,44,32,70,75,27,49,54),(23,45,50,38,76,61,33,55,60,28,66,71),(24,62,67,39,41,46,34,72,77,29,51,56),(25,47,52,40,78,63,35,57,42,30,68,73)], [(1,84),(2,109),(3,86),(4,111),(5,88),(6,113),(7,90),(8,115),(9,92),(10,117),(11,94),(12,119),(13,96),(14,101),(15,98),(16,103),(17,100),(18,105),(19,82),(20,107),(21,58),(22,65),(23,60),(24,67),(25,42),(26,69),(27,44),(28,71),(29,46),(30,73),(31,48),(32,75),(33,50),(34,77),(35,52),(36,79),(37,54),(38,61),(39,56),(40,63),(41,51),(43,53),(45,55),(47,57),(49,59),(81,120),(83,102),(85,104),(87,106),(89,108),(91,110),(93,112),(95,114),(97,116),(99,118)], [(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)], [(1,33),(2,22),(3,31),(4,40),(5,29),(6,38),(7,27),(8,36),(9,25),(10,34),(11,23),(12,32),(13,21),(14,30),(15,39),(16,28),(17,37),(18,26),(19,35),(20,24),(41,98),(42,87),(43,96),(44,85),(45,94),(46,83),(47,92),(48,81),(49,90),(50,99),(51,88),(52,97),(53,86),(54,95),(55,84),(56,93),(57,82),(58,91),(59,100),(60,89),(61,108),(62,117),(63,106),(64,115),(65,104),(66,113),(67,102),(68,111),(69,120),(70,109),(71,118),(72,107),(73,116),(74,105),(75,114),(76,103),(77,112),(78,101),(79,110),(80,119)]])
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