Copied to
clipboard

## G = D12⋊D5order 240 = 24·3·5

### 3rd semidirect product of D12 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D12⋊D5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — D12⋊D5
 Lower central C15 — C30 — D12⋊D5
 Upper central C1 — C2 — C4

Generators and relations for D12⋊D5
G = < a,b,c,d | a12=b2=c5=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 344 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, D6, C15, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C4×S3, D12, D12, C3×Q8, C5×S3, D15, C30, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, Q83S3, C3×Dic5, Dic15, C60, S3×C10, D30, D42D5, S3×Dic5, C5⋊D12, C3×Dic10, C5×D12, C4×D15, D12⋊D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, Q83S3, S3×D5, D42D5, C2×S3×D5, D12⋊D5

Smallest permutation representation of D12⋊D5
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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 96)(10 95)(11 94)(12 93)(13 98)(14 97)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 120)(45 119)(46 118)(47 117)(48 116)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 84)
(1 46 35 73 23)(2 47 36 74 24)(3 48 25 75 13)(4 37 26 76 14)(5 38 27 77 15)(6 39 28 78 16)(7 40 29 79 17)(8 41 30 80 18)(9 42 31 81 19)(10 43 32 82 20)(11 44 33 83 21)(12 45 34 84 22)(49 102 94 120 62)(50 103 95 109 63)(51 104 96 110 64)(52 105 85 111 65)(53 106 86 112 66)(54 107 87 113 67)(55 108 88 114 68)(56 97 89 115 69)(57 98 90 116 70)(58 99 91 117 71)(59 100 92 118 72)(60 101 93 119 61)
(1 23)(2 16)(3 21)(4 14)(5 19)(6 24)(7 17)(8 22)(9 15)(10 20)(11 13)(12 18)(25 33)(27 31)(28 36)(30 34)(37 76)(38 81)(39 74)(40 79)(41 84)(42 77)(43 82)(44 75)(45 80)(46 73)(47 78)(48 83)(49 118)(50 111)(51 116)(52 109)(53 114)(54 119)(55 112)(56 117)(57 110)(58 115)(59 120)(60 113)(61 67)(62 72)(63 65)(64 70)(66 68)(69 71)(85 103)(86 108)(87 101)(88 106)(89 99)(90 104)(91 97)(92 102)(93 107)(94 100)(95 105)(96 98)```

`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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,96)(10,95)(11,94)(12,93)(13,98)(14,97)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,120)(45,119)(46,118)(47,117)(48,116)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84), (1,46,35,73,23)(2,47,36,74,24)(3,48,25,75,13)(4,37,26,76,14)(5,38,27,77,15)(6,39,28,78,16)(7,40,29,79,17)(8,41,30,80,18)(9,42,31,81,19)(10,43,32,82,20)(11,44,33,83,21)(12,45,34,84,22)(49,102,94,120,62)(50,103,95,109,63)(51,104,96,110,64)(52,105,85,111,65)(53,106,86,112,66)(54,107,87,113,67)(55,108,88,114,68)(56,97,89,115,69)(57,98,90,116,70)(58,99,91,117,71)(59,100,92,118,72)(60,101,93,119,61), (1,23)(2,16)(3,21)(4,14)(5,19)(6,24)(7,17)(8,22)(9,15)(10,20)(11,13)(12,18)(25,33)(27,31)(28,36)(30,34)(37,76)(38,81)(39,74)(40,79)(41,84)(42,77)(43,82)(44,75)(45,80)(46,73)(47,78)(48,83)(49,118)(50,111)(51,116)(52,109)(53,114)(54,119)(55,112)(56,117)(57,110)(58,115)(59,120)(60,113)(61,67)(62,72)(63,65)(64,70)(66,68)(69,71)(85,103)(86,108)(87,101)(88,106)(89,99)(90,104)(91,97)(92,102)(93,107)(94,100)(95,105)(96,98)>;`

`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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,96)(10,95)(11,94)(12,93)(13,98)(14,97)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,120)(45,119)(46,118)(47,117)(48,116)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84), (1,46,35,73,23)(2,47,36,74,24)(3,48,25,75,13)(4,37,26,76,14)(5,38,27,77,15)(6,39,28,78,16)(7,40,29,79,17)(8,41,30,80,18)(9,42,31,81,19)(10,43,32,82,20)(11,44,33,83,21)(12,45,34,84,22)(49,102,94,120,62)(50,103,95,109,63)(51,104,96,110,64)(52,105,85,111,65)(53,106,86,112,66)(54,107,87,113,67)(55,108,88,114,68)(56,97,89,115,69)(57,98,90,116,70)(58,99,91,117,71)(59,100,92,118,72)(60,101,93,119,61), (1,23)(2,16)(3,21)(4,14)(5,19)(6,24)(7,17)(8,22)(9,15)(10,20)(11,13)(12,18)(25,33)(27,31)(28,36)(30,34)(37,76)(38,81)(39,74)(40,79)(41,84)(42,77)(43,82)(44,75)(45,80)(46,73)(47,78)(48,83)(49,118)(50,111)(51,116)(52,109)(53,114)(54,119)(55,112)(56,117)(57,110)(58,115)(59,120)(60,113)(61,67)(62,72)(63,65)(64,70)(66,68)(69,71)(85,103)(86,108)(87,101)(88,106)(89,99)(90,104)(91,97)(92,102)(93,107)(94,100)(95,105)(96,98) );`

`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)], [(1,92),(2,91),(3,90),(4,89),(5,88),(6,87),(7,86),(8,85),(9,96),(10,95),(11,94),(12,93),(13,98),(14,97),(15,108),(16,107),(17,106),(18,105),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,72),(36,71),(37,115),(38,114),(39,113),(40,112),(41,111),(42,110),(43,109),(44,120),(45,119),(46,118),(47,117),(48,116),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,84)], [(1,46,35,73,23),(2,47,36,74,24),(3,48,25,75,13),(4,37,26,76,14),(5,38,27,77,15),(6,39,28,78,16),(7,40,29,79,17),(8,41,30,80,18),(9,42,31,81,19),(10,43,32,82,20),(11,44,33,83,21),(12,45,34,84,22),(49,102,94,120,62),(50,103,95,109,63),(51,104,96,110,64),(52,105,85,111,65),(53,106,86,112,66),(54,107,87,113,67),(55,108,88,114,68),(56,97,89,115,69),(57,98,90,116,70),(58,99,91,117,71),(59,100,92,118,72),(60,101,93,119,61)], [(1,23),(2,16),(3,21),(4,14),(5,19),(6,24),(7,17),(8,22),(9,15),(10,20),(11,13),(12,18),(25,33),(27,31),(28,36),(30,34),(37,76),(38,81),(39,74),(40,79),(41,84),(42,77),(43,82),(44,75),(45,80),(46,73),(47,78),(48,83),(49,118),(50,111),(51,116),(52,109),(53,114),(54,119),(55,112),(56,117),(57,110),(58,115),(59,120),(60,113),(61,67),(62,72),(63,65),(64,70),(66,68),(69,71),(85,103),(86,108),(87,101),(88,106),(89,99),(90,104),(91,97),(92,102),(93,107),(94,100),(95,105),(96,98)]])`

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6 10A 10B 10C 10D 10E 10F 12A 12B 12C 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 10 10 10 10 10 10 12 12 12 15 15 20 20 30 30 60 60 60 60 size 1 1 6 6 30 2 2 10 10 15 15 2 2 2 2 2 12 12 12 12 4 20 20 4 4 4 4 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 Q8⋊3S3 S3×D5 D4⋊2D5 C2×S3×D5 D12⋊D5 kernel D12⋊D5 S3×Dic5 C5⋊D12 C3×Dic10 C5×D12 C4×D15 Dic10 D12 Dic5 C20 C15 C12 D6 C5 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 2 1 2 2 4 1 2 2 2 4

Matrix representation of D12⋊D5 in GL6(𝔽61)

 60 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 50 0 0 0 0 0 21 11
,
 1 1 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 20 50 0 0 0 0 3 41
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 60 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 48 60

`G:=sub<GL(6,GF(61))| [60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,21,0,0,0,0,0,11],[1,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,20,3,0,0,0,0,50,41],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,48,0,0,0,0,0,60] >;`

D12⋊D5 in GAP, Magma, Sage, TeX

`D_{12}\rtimes D_5`
`% in TeX`

`G:=Group("D12:D5");`
`// GroupNames label`

`G:=SmallGroup(240,129);`
`// by ID`

`G=gap.SmallGroup(240,129);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,218,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽