metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.5C42, D10.20D12, D10.3Dic6, (C2×F5)⋊Dic3, (C6×F5)⋊1C4, C30.5(C4⋊C4), C6.10(C4×F5), (C6×D5).3Q8, (C2×Dic3)⋊2F5, (C6×D5).27D4, D10.9(C4×S3), D5.(C4⋊Dic3), C2.1(D6⋊F5), C6.12(C4⋊F5), C2.6(Dic3×F5), D5.1(D6⋊C4), C10.9(D6⋊C4), (C10×Dic3)⋊2C4, (C2×Dic15)⋊3C4, C15⋊(C2.C42), C10.5(C4×Dic3), C6.9(C22⋊F5), (C22×F5).1S3, C22.10(S3×F5), C30.9(C22⋊C4), C2.1(Dic3⋊F5), D5.(C6.D4), C5⋊1(C6.C42), D10.7(C2×Dic3), C3⋊2(D10.3Q8), (C22×D5).72D6, D10.24(C3⋊D4), D5.1(Dic3⋊C4), C10.5(Dic3⋊C4), (C2×C3⋊F5)⋊1C4, (C2×C6×F5).1C2, (C3×D5).(C4⋊C4), (C2×C30).1(C2×C4), (C2×C10).3(C4×S3), (C2×C6).11(C2×F5), (C22×C3⋊F5).1C2, (C6×D5).13(C2×C4), (C3×D5).(C22⋊C4), (C2×D5×Dic3).10C2, (D5×C2×C6).63C22, SmallGroup(480,243)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.20D12
G = < a,b,c,d | a10=b2=c12=1, d2=a4b, bab=a-1, cac-1=dad-1=a3, cbc-1=dbd-1=a2b, dcd-1=a-1bc-1 >
Subgroups: 788 in 152 conjugacy classes, 56 normal (50 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, F5, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C3×D5, C30, C2.C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C22×Dic3, C22×C12, C5×Dic3, Dic15, C3×F5, C3⋊F5, C6×D5, C2×C30, C2×C4×D5, C22×F5, C22×F5, C6.C42, D5×Dic3, C10×Dic3, C2×Dic15, C6×F5, C6×F5, C2×C3⋊F5, C2×C3⋊F5, D5×C2×C6, D10.3Q8, C2×D5×Dic3, C2×C6×F5, C22×C3⋊F5, D10.20D12
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, C4×F5, C4⋊F5, C22⋊F5, C6.C42, S3×F5, D10.3Q8, Dic3×F5, D6⋊F5, Dic3⋊F5, D10.20D12
►On 120 points
Generators in S120
(1 68 50 71 59 11 53 65 56 62)(2 72 54 63 51 12 57 69 60 66)(3 64 58 67 55 10 49 61 52 70)(4 119 26 110 35 9 29 116 32 113)(5 111 30 114 27 7 33 120 36 117)(6 115 34 118 31 8 25 112 28 109)(13 103 90 73 40 24 46 79 96 97)(14 74 47 98 91 19 85 104 41 80)(15 99 86 81 48 20 42 75 92 105)(16 82 43 106 87 21 93 100 37 76)(17 107 94 77 44 22 38 83 88 101)(18 78 39 102 95 23 89 108 45 84)
(1 35)(2 27)(3 31)(4 59)(5 51)(6 55)(7 66)(8 70)(9 62)(10 109)(11 113)(12 117)(13 76)(14 101)(15 84)(16 97)(17 80)(18 105)(19 44)(20 95)(21 40)(22 91)(23 48)(24 87)(25 52)(26 50)(28 49)(29 56)(30 54)(32 53)(33 60)(34 58)(36 57)(37 103)(38 98)(39 75)(41 107)(42 102)(43 79)(45 99)(46 106)(47 83)(61 112)(63 111)(64 118)(65 116)(67 115)(68 110)(69 120)(71 119)(72 114)(73 93)(74 88)(77 85)(78 92)(81 89)(82 96)(86 108)(90 100)(94 104)
(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 16 4 24)(2 18 5 20)(3 14 6 22)(7 15 12 23)(8 17 10 19)(9 13 11 21)(25 83 55 41)(26 97 56 93)(27 81 57 39)(28 107 58 91)(29 79 59 37)(30 105 60 89)(31 77 49 47)(32 103 50 87)(33 75 51 45)(34 101 52 85)(35 73 53 43)(36 99 54 95)(38 61 80 118)(40 71 82 116)(42 69 84 114)(44 67 74 112)(46 65 76 110)(48 63 78 120)(62 106 119 90)(64 104 109 88)(66 102 111 86)(68 100 113 96)(70 98 115 94)(72 108 117 92)
G:=sub<Sym(120)| (1,68,50,71,59,11,53,65,56,62)(2,72,54,63,51,12,57,69,60,66)(3,64,58,67,55,10,49,61,52,70)(4,119,26,110,35,9,29,116,32,113)(5,111,30,114,27,7,33,120,36,117)(6,115,34,118,31,8,25,112,28,109)(13,103,90,73,40,24,46,79,96,97)(14,74,47,98,91,19,85,104,41,80)(15,99,86,81,48,20,42,75,92,105)(16,82,43,106,87,21,93,100,37,76)(17,107,94,77,44,22,38,83,88,101)(18,78,39,102,95,23,89,108,45,84), (1,35)(2,27)(3,31)(4,59)(5,51)(6,55)(7,66)(8,70)(9,62)(10,109)(11,113)(12,117)(13,76)(14,101)(15,84)(16,97)(17,80)(18,105)(19,44)(20,95)(21,40)(22,91)(23,48)(24,87)(25,52)(26,50)(28,49)(29,56)(30,54)(32,53)(33,60)(34,58)(36,57)(37,103)(38,98)(39,75)(41,107)(42,102)(43,79)(45,99)(46,106)(47,83)(61,112)(63,111)(64,118)(65,116)(67,115)(68,110)(69,120)(71,119)(72,114)(73,93)(74,88)(77,85)(78,92)(81,89)(82,96)(86,108)(90,100)(94,104), (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), (1,16,4,24)(2,18,5,20)(3,14,6,22)(7,15,12,23)(8,17,10,19)(9,13,11,21)(25,83,55,41)(26,97,56,93)(27,81,57,39)(28,107,58,91)(29,79,59,37)(30,105,60,89)(31,77,49,47)(32,103,50,87)(33,75,51,45)(34,101,52,85)(35,73,53,43)(36,99,54,95)(38,61,80,118)(40,71,82,116)(42,69,84,114)(44,67,74,112)(46,65,76,110)(48,63,78,120)(62,106,119,90)(64,104,109,88)(66,102,111,86)(68,100,113,96)(70,98,115,94)(72,108,117,92)>;
G:=Group( (1,68,50,71,59,11,53,65,56,62)(2,72,54,63,51,12,57,69,60,66)(3,64,58,67,55,10,49,61,52,70)(4,119,26,110,35,9,29,116,32,113)(5,111,30,114,27,7,33,120,36,117)(6,115,34,118,31,8,25,112,28,109)(13,103,90,73,40,24,46,79,96,97)(14,74,47,98,91,19,85,104,41,80)(15,99,86,81,48,20,42,75,92,105)(16,82,43,106,87,21,93,100,37,76)(17,107,94,77,44,22,38,83,88,101)(18,78,39,102,95,23,89,108,45,84), (1,35)(2,27)(3,31)(4,59)(5,51)(6,55)(7,66)(8,70)(9,62)(10,109)(11,113)(12,117)(13,76)(14,101)(15,84)(16,97)(17,80)(18,105)(19,44)(20,95)(21,40)(22,91)(23,48)(24,87)(25,52)(26,50)(28,49)(29,56)(30,54)(32,53)(33,60)(34,58)(36,57)(37,103)(38,98)(39,75)(41,107)(42,102)(43,79)(45,99)(46,106)(47,83)(61,112)(63,111)(64,118)(65,116)(67,115)(68,110)(69,120)(71,119)(72,114)(73,93)(74,88)(77,85)(78,92)(81,89)(82,96)(86,108)(90,100)(94,104), (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), (1,16,4,24)(2,18,5,20)(3,14,6,22)(7,15,12,23)(8,17,10,19)(9,13,11,21)(25,83,55,41)(26,97,56,93)(27,81,57,39)(28,107,58,91)(29,79,59,37)(30,105,60,89)(31,77,49,47)(32,103,50,87)(33,75,51,45)(34,101,52,85)(35,73,53,43)(36,99,54,95)(38,61,80,118)(40,71,82,116)(42,69,84,114)(44,67,74,112)(46,65,76,110)(48,63,78,120)(62,106,119,90)(64,104,109,88)(66,102,111,86)(68,100,113,96)(70,98,115,94)(72,108,117,92) );
G=PermutationGroup([[(1,68,50,71,59,11,53,65,56,62),(2,72,54,63,51,12,57,69,60,66),(3,64,58,67,55,10,49,61,52,70),(4,119,26,110,35,9,29,116,32,113),(5,111,30,114,27,7,33,120,36,117),(6,115,34,118,31,8,25,112,28,109),(13,103,90,73,40,24,46,79,96,97),(14,74,47,98,91,19,85,104,41,80),(15,99,86,81,48,20,42,75,92,105),(16,82,43,106,87,21,93,100,37,76),(17,107,94,77,44,22,38,83,88,101),(18,78,39,102,95,23,89,108,45,84)], [(1,35),(2,27),(3,31),(4,59),(5,51),(6,55),(7,66),(8,70),(9,62),(10,109),(11,113),(12,117),(13,76),(14,101),(15,84),(16,97),(17,80),(18,105),(19,44),(20,95),(21,40),(22,91),(23,48),(24,87),(25,52),(26,50),(28,49),(29,56),(30,54),(32,53),(33,60),(34,58),(36,57),(37,103),(38,98),(39,75),(41,107),(42,102),(43,79),(45,99),(46,106),(47,83),(61,112),(63,111),(64,118),(65,116),(67,115),(68,110),(69,120),(71,119),(72,114),(73,93),(74,88),(77,85),(78,92),(81,89),(82,96),(86,108),(90,100),(94,104)], [(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)], [(1,16,4,24),(2,18,5,20),(3,14,6,22),(7,15,12,23),(8,17,10,19),(9,13,11,21),(25,83,55,41),(26,97,56,93),(27,81,57,39),(28,107,58,91),(29,79,59,37),(30,105,60,89),(31,77,49,47),(32,103,50,87),(33,75,51,45),(34,101,52,85),(35,73,53,43),(36,99,54,95),(38,61,80,118),(40,71,82,116),(42,69,84,114),(44,67,74,112),(46,65,76,110),(48,63,78,120),(62,106,119,90),(64,104,109,88),(66,102,111,86),(68,100,113,96),(70,98,115,94),(72,108,117,92)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 12A | ··· | 12H | 15 | 20A | 20B | 20C | 20D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | ··· | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 10 | ··· | 10 | 8 | 12 | 12 | 12 | 12 | 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 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | - | - | + | - | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | C4×S3 | F5 | C2×F5 | C4×F5 | C4⋊F5 | C22⋊F5 | S3×F5 | Dic3×F5 | D6⋊F5 | Dic3⋊F5 |
| kernel | D10.20D12 | C2×D5×Dic3 | C2×C6×F5 | C22×C3⋊F5 | C10×Dic3 | C2×Dic15 | C6×F5 | C2×C3⋊F5 | C22×F5 | C6×D5 | C6×D5 | C2×F5 | C22×D5 | D10 | D10 | D10 | D10 | C2×C10 | C2×Dic3 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 |
Matrix representation of D10.20D12 ►in GL8(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 29 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 |
G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,1,0,60,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,60],[29,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0] >;
D10.20D12 in GAP, Magma, Sage, TeX
D_{10}._{20}D_{12} % in TeX
G:=Group("D10.20D12"); // GroupNames label
G:=SmallGroup(480,243);
// by ID
G=gap.SmallGroup(480,243);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^4*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations