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## G = D10.4D12order 480 = 25·3·5

### 4th non-split extension by D10 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D10.4D12
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C2×C6 — C3×C22⋊F5 — D10.4D12
 Lower central C15 — C30 — C2×C30 — D10.4D12
 Upper central C1 — C2 — C22

Generators and relations for D10.4D12
G = < a,b,c,d | a10=b2=c12=1, d2=a4b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a-1bc-1 >

Subgroups: 836 in 104 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×5], C5, S3, C6, C6 [×3], C2×C4 [×3], D4 [×2], C23 [×2], D5 [×3], C10, C10, Dic3 [×2], C12, D6 [×2], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×2], C2×D4, C20, F5 [×2], D10 [×2], D10 [×3], C2×C10, C2×Dic3, C2×Dic3, C3⋊D4 [×2], C2×C12, C22×S3, C22×C6, C3×D5 [×2], D15, C30, C30, C23⋊C4, D20 [×2], C2×C20, C2×F5 [×2], C22×D5, C22×D5, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C3×F5, C3⋊F5, C6×D5 [×2], C6×D5, D30 [×2], C2×C30, C22⋊F5, C22⋊F5, C2×D20, C23.6D6, C3⋊D20 [×2], C10×Dic3, C6×F5, C2×C3⋊F5, D5×C2×C6, C22×D15, D10.D4, C3×C22⋊F5, D10.D6, C2×C3⋊D20, D10.4D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C23⋊C4, C2×F5, D6⋊C4, C22⋊F5, C23.6D6, S3×F5, D10.D4, D6⋊F5, D10.4D12

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

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

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

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

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 12A 12B 12C 12D 15 20A 20B 20C 20D 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 12 12 12 12 15 20 20 20 20 30 30 30 size 1 1 2 10 10 60 2 12 20 20 60 60 4 2 4 10 10 20 4 4 4 20 20 20 20 8 12 12 12 12 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 D10.D4 S3×F5 D6⋊F5 D10.4D12 kernel D10.4D12 C3×C22⋊F5 D10.D6 C2×C3⋊D20 C10×Dic3 C22×D15 C22⋊F5 C6×D5 C22×D5 D10 D10 C2×C10 C2×Dic3 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 D10.4D12 in GL8(𝔽61)

 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 0 60 1 0 0 0 0 0 0 60 0 0 0 0 0 1 0 60 0 0 0 0 0 0 1 60 0
,
 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 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
,
 13 0 0 0 0 0 0 0 0 48 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 47 0 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 0 0 14 0 0 0 0 0 0 47 0 0 0 0 0 48 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 14 47 54 0 0 0 0 0 7 47 0 14 0 0 0 0 14 0 47 7 0 0 0 0 0 54 47 14

`G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0],[0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,60],[13,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,0,48,0,0,0,0,0,0,0,0,13,0,0,0,0,0,47,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,14,7,14,0,0,0,0,0,47,47,0,54,0,0,0,0,54,0,47,47,0,0,0,0,0,14,7,14] >;`

D10.4D12 in GAP, Magma, Sage, TeX

`D_{10}._4D_{12}`
`% in TeX`

`G:=Group("D10.4D12");`
`// GroupNames label`

`G:=SmallGroup(480,249);`
`// by ID`

`G=gap.SmallGroup(480,249);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,675,1356,9414,4724]);`
`// 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=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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