metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.10D12, C30.12C42, D10.7Dic6, (C2×C60)⋊6C4, (C2×C12)⋊4F5, C6.5(C4×F5), C6.7(C4⋊F5), (C6×D5).9Q8, (C6×D5).75D4, (C2×C20)⋊2Dic3, C30.14(C4⋊C4), D5.2(D6⋊C4), (C6×Dic5)⋊12C4, D10.14(C4×S3), C2.3(C60⋊C4), (C2×Dic5)⋊6Dic3, C10.7(C4⋊Dic3), C5⋊2(C6.C42), C3⋊1(D10.3Q8), C10.12(C4×Dic3), C6.19(C22⋊F5), D10.32(C3⋊D4), C30.19(C22⋊C4), D5.2(Dic3⋊C4), C15⋊2(C2.C42), (C22×D5).100D6, C10.4(C6.D4), C2.2(D10.D6), (C2×C3⋊F5)⋊3C4, C2.5(C4×C3⋊F5), (C2×C4)⋊2(C3⋊F5), (C2×C4×D5).9S3, (D5×C2×C12).25C2, (C2×C6).39(C2×F5), (C2×C30).33(C2×C4), (C22×C3⋊F5).4C2, C22.14(C2×C3⋊F5), (C3×D5).2(C4⋊C4), (C6×D5).41(C2×C4), (C2×C10).9(C2×Dic3), (D5×C2×C6).142C22, (C3×D5).2(C22⋊C4), SmallGroup(480,311)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.10D12
G = < a,b,c,d | a10=b2=c12=1, d2=a4b, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a5c-1 >
Subgroups: 812 in 152 conjugacy classes, 57 normal (35 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2.C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C22×Dic3, C22×C12, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C2×C4×D5, C22×F5, C6.C42, D5×C12, C6×Dic5, C2×C60, C2×C3⋊F5, C2×C3⋊F5, D5×C2×C6, D10.3Q8, D5×C2×C12, C22×C3⋊F5, D10.10D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, F5, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C2×F5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3⋊F5, C4×F5, C4⋊F5, C22⋊F5, C6.C42, C2×C3⋊F5, D10.3Q8, C4×C3⋊F5, C60⋊C4, D10.D6, D10.10D12
►On 120 points
Generators in S120
(1 98 115 77 24 54 66 85 26 42)(2 99 116 78 13 55 67 86 27 43)(3 100 117 79 14 56 68 87 28 44)(4 101 118 80 15 57 69 88 29 45)(5 102 119 81 16 58 70 89 30 46)(6 103 120 82 17 59 71 90 31 47)(7 104 109 83 18 60 72 91 32 48)(8 105 110 84 19 49 61 92 33 37)(9 106 111 73 20 50 62 93 34 38)(10 107 112 74 21 51 63 94 35 39)(11 108 113 75 22 52 64 95 36 40)(12 97 114 76 23 53 65 96 25 41)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(61 84)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)
(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 53 54 12)(2 11 55 52)(3 51 56 10)(4 9 57 50)(5 49 58 8)(6 7 59 60)(13 113 99 75)(14 94 100 35)(15 111 101 73)(16 92 102 33)(17 109 103 83)(18 90 104 31)(19 119 105 81)(20 88 106 29)(21 117 107 79)(22 86 108 27)(23 115 97 77)(24 96 98 25)(26 41 85 65)(28 39 87 63)(30 37 89 61)(32 47 91 71)(34 45 93 69)(36 43 95 67)(38 118 62 80)(40 116 64 78)(42 114 66 76)(44 112 68 74)(46 110 70 84)(48 120 72 82)
G:=sub<Sym(120)| (1,98,115,77,24,54,66,85,26,42)(2,99,116,78,13,55,67,86,27,43)(3,100,117,79,14,56,68,87,28,44)(4,101,118,80,15,57,69,88,29,45)(5,102,119,81,16,58,70,89,30,46)(6,103,120,82,17,59,71,90,31,47)(7,104,109,83,18,60,72,91,32,48)(8,105,110,84,19,49,61,92,33,37)(9,106,111,73,20,50,62,93,34,38)(10,107,112,74,21,51,63,94,35,39)(11,108,113,75,22,52,64,95,36,40)(12,97,114,76,23,53,65,96,25,41), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(61,84)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114), (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), (1,53,54,12)(2,11,55,52)(3,51,56,10)(4,9,57,50)(5,49,58,8)(6,7,59,60)(13,113,99,75)(14,94,100,35)(15,111,101,73)(16,92,102,33)(17,109,103,83)(18,90,104,31)(19,119,105,81)(20,88,106,29)(21,117,107,79)(22,86,108,27)(23,115,97,77)(24,96,98,25)(26,41,85,65)(28,39,87,63)(30,37,89,61)(32,47,91,71)(34,45,93,69)(36,43,95,67)(38,118,62,80)(40,116,64,78)(42,114,66,76)(44,112,68,74)(46,110,70,84)(48,120,72,82)>;
G:=Group( (1,98,115,77,24,54,66,85,26,42)(2,99,116,78,13,55,67,86,27,43)(3,100,117,79,14,56,68,87,28,44)(4,101,118,80,15,57,69,88,29,45)(5,102,119,81,16,58,70,89,30,46)(6,103,120,82,17,59,71,90,31,47)(7,104,109,83,18,60,72,91,32,48)(8,105,110,84,19,49,61,92,33,37)(9,106,111,73,20,50,62,93,34,38)(10,107,112,74,21,51,63,94,35,39)(11,108,113,75,22,52,64,95,36,40)(12,97,114,76,23,53,65,96,25,41), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(61,84)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114), (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), (1,53,54,12)(2,11,55,52)(3,51,56,10)(4,9,57,50)(5,49,58,8)(6,7,59,60)(13,113,99,75)(14,94,100,35)(15,111,101,73)(16,92,102,33)(17,109,103,83)(18,90,104,31)(19,119,105,81)(20,88,106,29)(21,117,107,79)(22,86,108,27)(23,115,97,77)(24,96,98,25)(26,41,85,65)(28,39,87,63)(30,37,89,61)(32,47,91,71)(34,45,93,69)(36,43,95,67)(38,118,62,80)(40,116,64,78)(42,114,66,76)(44,112,68,74)(46,110,70,84)(48,120,72,82) );
G=PermutationGroup([[(1,98,115,77,24,54,66,85,26,42),(2,99,116,78,13,55,67,86,27,43),(3,100,117,79,14,56,68,87,28,44),(4,101,118,80,15,57,69,88,29,45),(5,102,119,81,16,58,70,89,30,46),(6,103,120,82,17,59,71,90,31,47),(7,104,109,83,18,60,72,91,32,48),(8,105,110,84,19,49,61,92,33,37),(9,106,111,73,20,50,62,93,34,38),(10,107,112,74,21,51,63,94,35,39),(11,108,113,75,22,52,64,95,36,40),(12,97,114,76,23,53,65,96,25,41)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,37),(9,38),(10,39),(11,40),(12,41),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(61,84),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(91,109),(92,110),(93,111),(94,112),(95,113),(96,114)], [(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)], [(1,53,54,12),(2,11,55,52),(3,51,56,10),(4,9,57,50),(5,49,58,8),(6,7,59,60),(13,113,99,75),(14,94,100,35),(15,111,101,73),(16,92,102,33),(17,109,103,83),(18,90,104,31),(19,119,105,81),(20,88,106,29),(21,117,107,79),(22,86,108,27),(23,115,97,77),(24,96,98,25),(26,41,85,65),(28,39,87,63),(30,37,89,61),(32,47,91,71),(34,45,93,69),(36,43,95,67),(38,118,62,80),(40,116,64,78),(42,114,66,76),(44,112,68,74),(46,110,70,84),(48,120,72,82)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 10 | 10 | 30 | ··· | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | - | + | - | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | Dic3 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | F5 | C2×F5 | C3⋊F5 | C4×F5 | C4⋊F5 | C22⋊F5 | C2×C3⋊F5 | C4×C3⋊F5 | C60⋊C4 | D10.D6 |
| kernel | D10.10D12 | D5×C2×C12 | C22×C3⋊F5 | C6×Dic5 | C2×C60 | C2×C3⋊F5 | C2×C4×D5 | C6×D5 | C6×D5 | C2×Dic5 | C2×C20 | C22×D5 | D10 | D10 | D10 | D10 | C2×C12 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 3 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D10.10D12 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 44 | 7 | 0 | 0 | 0 | 0 |
| 35 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 56 | 0 | 5 |
| 0 | 0 | 0 | 3 | 56 | 5 |
| 0 | 0 | 5 | 56 | 3 | 0 |
| 0 | 0 | 5 | 0 | 56 | 8 |
| 30 | 16 | 0 | 0 | 0 | 0 |
| 39 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 56 | 5 | 58 |
| 0 | 0 | 0 | 56 | 8 | 53 |
| 0 | 0 | 56 | 0 | 5 | 53 |
| 0 | 0 | 56 | 3 | 0 | 58 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0],[44,35,0,0,0,0,7,25,0,0,0,0,0,0,8,0,5,5,0,0,56,3,56,0,0,0,0,56,3,56,0,0,5,5,0,8],[30,39,0,0,0,0,16,31,0,0,0,0,0,0,3,0,56,56,0,0,56,56,0,3,0,0,5,8,5,0,0,0,58,53,53,58] >;
D10.10D12 in GAP, Magma, Sage, TeX
D_{10}._{10}D_{12} % in TeX
G:=Group("D10.10D12"); // GroupNames label
G:=SmallGroup(480,311);
// by ID
G=gap.SmallGroup(480,311);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^4*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^5*c^-1>;
// generators/relations