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## G = D10.20D12order 480 = 25·3·5

### 9th non-split extension by D10 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D10.20D12
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — D5×C2×C6 — C2×C6×F5 — D10.20D12
 Lower central C15 — C30 — D10.20D12
 Upper central C1 — C22

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

Smallest permutation representation of 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`

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