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## G = D12.D5order 240 = 24·3·5

### 1st non-split extension by D12 of D5 acting via D5/C5=C2

Aliases: C20.5D6, C30.8D4, C155SD16, D12.1D5, C10.7D12, Dic308C2, C12.23D10, C60.16C22, C52C82S3, C4.9(S3×D5), C53(C24⋊C2), C31(D4.D5), (C5×D12).1C2, C6.2(C5⋊D4), C2.5(C5⋊D12), (C3×C52C8)⋊2C2, SmallGroup(240,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D12.D5
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — D12.D5
 Lower central C15 — C30 — C60 — D12.D5
 Upper central C1 — C2 — C4

Generators and relations for D12.D5
G = < a,b,c,d | a12=b2=c5=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

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

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

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

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

D12.D5 is a maximal subgroup of
D5×C24⋊C2  D24⋊D5  Dic60⋊C2  D247D5  C20.60D12  D6036C22  D12.33D10  D30.8D4  S3×D4.D5  D2010D6  D30.11D4  D15⋊SD16  Dic10.26D6  D20.27D6  D30.44D4
D12.D5 is a maximal quotient of
C10.D24  Dic3015C4  C60.7Q8

33 conjugacy classes

 class 1 2A 2B 3 4A 4B 5A 5B 6 8A 8B 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 20A 20B 24A 24B 24C 24D 30A 30B 60A 60B 60C 60D order 1 2 2 3 4 4 5 5 6 8 8 10 10 10 10 10 10 12 12 15 15 20 20 24 24 24 24 30 30 60 60 60 60 size 1 1 12 2 2 60 2 2 2 10 10 2 2 12 12 12 12 2 2 4 4 4 4 10 10 10 10 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + - + - image C1 C2 C2 C2 S3 D4 D5 D6 SD16 D10 D12 C5⋊D4 C24⋊C2 S3×D5 D4.D5 C5⋊D12 D12.D5 kernel D12.D5 C3×C5⋊2C8 C5×D12 Dic30 C5⋊2C8 C30 D12 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 2 4 4 2 2 2 4

Matrix representation of D12.D5 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 43 99 0 0 142 142
,
 1 0 0 0 0 1 0 0 0 0 99 99 0 0 198 142
,
 0 1 0 0 240 51 0 0 0 0 1 0 0 0 0 1
,
 173 160 0 0 66 68 0 0 0 0 66 147 0 0 94 213
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,43,142,0,0,99,142],[1,0,0,0,0,1,0,0,0,0,99,198,0,0,99,142],[0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[173,66,0,0,160,68,0,0,0,0,66,94,0,0,147,213] >;`

D12.D5 in GAP, Magma, Sage, TeX

`D_{12}.D_5`
`% in TeX`

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

`G:=SmallGroup(240,20);`
`// by ID`

`G=gap.SmallGroup(240,20);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,169,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^5=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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