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## G = D10.D12order 480 = 25·3·5

### 3rd non-split extension by D10 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D10.D12
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C2×C6 — C3×C22⋊F5 — D10.D12
 Lower central C15 — C30 — C2×C30 — D10.D12
 Upper central C1 — C2 — C22

Generators and relations for D10.D12
G = < a,b,c,d | a10=b2=c12=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a-1bc-1 >

Subgroups: 708 in 104 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, F5, D10, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C30, C23⋊C4, C2×Dic5, C5⋊D4, C2×F5, C22×D5, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, Dic15, C3×F5, C3⋊F5, C6×D5, C6×D5, S3×C10, C2×C30, C22⋊F5, C22⋊F5, C2×C5⋊D4, C23.6D6, C15⋊D4, C2×Dic15, C6×F5, C2×C3⋊F5, D5×C2×C6, S3×C2×C10, C23⋊F5, C3×C22⋊F5, D10.D6, C2×C15⋊D4, D10.D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C23⋊C4, C2×F5, D6⋊C4, C22⋊F5, C23.6D6, S3×F5, C23⋊F5, D6⋊F5, D10.D12

Smallest permutation representation of D10.D12
On 120 points
Generators in S120
```(1 58 71 93 104 21 43 32 79 116)(2 94 44 117 72 22 80 59 105 33)(3 118 81 34 45 23 106 95 61 60)(4 35 107 49 82 24 62 119 46 96)(5 50 63 85 108 13 47 36 83 120)(6 86 48 109 64 14 84 51 97 25)(7 110 73 26 37 15 98 87 65 52)(8 27 99 53 74 16 66 111 38 88)(9 54 67 89 100 17 39 28 75 112)(10 90 40 113 68 18 76 55 101 29)(11 114 77 30 41 19 102 91 69 56)(12 31 103 57 78 20 70 115 42 92)
(1 116)(2 72)(3 60)(4 82)(5 120)(6 64)(7 52)(8 74)(9 112)(10 68)(11 56)(12 78)(13 108)(14 25)(15 37)(16 88)(17 100)(18 29)(19 41)(20 92)(21 104)(22 33)(23 45)(24 96)(26 98)(27 53)(28 67)(30 102)(31 57)(32 71)(34 106)(35 49)(36 63)(38 66)(39 89)(42 70)(43 93)(46 62)(47 85)(50 83)(54 75)(58 79)(61 118)(65 110)(69 114)(73 87)(76 101)(77 91)(80 105)(81 95)(84 97)(86 109)(90 113)(94 117)
(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 9)(2 16 22 8)(3 15)(4 6 24 14)(7 23)(10 20 18 12)(11 19)(17 21)(25 46 84 119)(26 106 87 45)(27 59 74 105)(28 116 89 58)(29 42 76 115)(30 102 91 41)(31 55 78 101)(32 112 93 54)(33 38 80 111)(34 98 95 37)(35 51 82 97)(36 120 85 50)(39 71 100 79)(40 92 113 70)(43 67 104 75)(44 88 117 66)(47 63 108 83)(48 96 109 62)(49 86 107 64)(52 61 110 81)(53 94 99 72)(56 69 114 77)(57 90 103 68)(60 65 118 73)```

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 8 8 8 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C4 C4 S3 D4 D6 D12 C3⋊D4 C4×S3 F5 C23⋊C4 C2×F5 C22⋊F5 C23.6D6 C23⋊F5 S3×F5 D6⋊F5 D10.D12 kernel D10.D12 C3×C22⋊F5 D10.D6 C2×C15⋊D4 C2×Dic15 S3×C2×C10 C22⋊F5 C6×D5 C22×D5 D10 D10 C2×C10 C22×S3 C15 C2×C6 C6 C5 C3 C22 C2 C1 # reps 1 1 1 1 2 2 1 2 1 2 2 2 1 1 1 2 2 4 1 1 2

Matrix representation of D10.D12 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
,
 0 11 0 0 0 0 50 11 0 0 0 0 0 0 37 7 37 51 0 0 13 58 27 27 0 0 3 34 34 3 0 0 10 10 24 54
,
 50 0 0 0 0 0 50 11 0 0 0 0 0 0 1 0 60 0 0 0 0 0 60 1 0 0 0 1 60 0 0 0 0 0 60 0

`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],[0,50,0,0,0,0,11,11,0,0,0,0,0,0,37,13,3,10,0,0,7,58,34,10,0,0,37,27,34,24,0,0,51,27,3,54],[50,50,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60,0,0,0,1,0,0] >;`

D10.D12 in GAP, Magma, Sage, TeX

`D_{10}.D_{12}`
`% in TeX`

`G:=Group("D10.D12");`
`// GroupNames label`

`G:=SmallGroup(480,248);`
`// by ID`

`G=gap.SmallGroup(480,248);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,675,1356,9414,4724]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;`
`// generators/relations`

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