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## G = D12⋊5D5order 240 = 24·3·5

### The semidirect product of D12 and D5 acting through Inn(D12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D12⋊5D5
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — D12⋊5D5
 Lower central C15 — C30 — D12⋊5D5
 Upper central C1 — C2 — C4

Generators and relations for D125D5
G = < a,b,c,d | a12=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 320 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, D10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, C30, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C4○D12, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D42D5, S3×Dic5, C15⋊D4, D5×C12, C5×D12, Dic30, D125D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, C4○D12, S3×D5, D42D5, C2×S3×D5, D125D5

Smallest permutation representation of D125D5
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 87)(2 86)(3 85)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 101)(26 100)(27 99)(28 98)(29 97)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(49 110)(50 109)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 84)(70 83)(71 82)(72 81)
(1 22 55 27 79)(2 23 56 28 80)(3 24 57 29 81)(4 13 58 30 82)(5 14 59 31 83)(6 15 60 32 84)(7 16 49 33 73)(8 17 50 34 74)(9 18 51 35 75)(10 19 52 36 76)(11 20 53 25 77)(12 21 54 26 78)(37 112 107 70 95)(38 113 108 71 96)(39 114 97 72 85)(40 115 98 61 86)(41 116 99 62 87)(42 117 100 63 88)(43 118 101 64 89)(44 119 102 65 90)(45 120 103 66 91)(46 109 104 67 92)(47 110 105 68 93)(48 111 106 69 94)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 97)(46 98)(47 99)(48 100)(61 92)(62 93)(63 94)(64 95)(65 96)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 C4○D12 S3×D5 D4⋊2D5 C2×S3×D5 D12⋊5D5 kernel D12⋊5D5 S3×Dic5 C15⋊D4 D5×C12 C5×D12 Dic30 C4×D5 D12 Dic5 C20 D10 C15 C12 D6 C5 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 1 1 1 2 2 4 4 2 2 2 4

Matrix representation of D125D5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 32 0 0 0 0 21
,
 1 0 0 0 0 1 0 0 0 0 0 21 0 0 32 0
,
 0 1 0 0 60 43 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 60
`G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,21],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,21,0],[0,60,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,60] >;`

D125D5 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_5D_5`
`% in TeX`

`G:=Group("D12:5D5");`
`// GroupNames label`

`G:=SmallGroup(240,133);`
`// by ID`

`G=gap.SmallGroup(240,133);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,116,50,490,6917]);`
`// 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,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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