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## G = D12⋊14D10order 480 = 25·3·5

### 8th semidirect product of D12 and D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D12⋊14D10
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — D5×D12 — D12⋊14D10
 Lower central C15 — C30 — D12⋊14D10
 Upper central C1 — C2 — D4

Generators and relations for D1214D10
G = < a,b,c,d | a12=b2=c10=d2=1, bab=a-1, cac-1=a7, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >

Subgroups: 1884 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×5], C22 [×2], C22 [×13], C5, S3 [×6], C6, C6 [×3], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], D5 [×4], C10, C10 [×5], Dic3, Dic3, C12, C12 [×3], D6, D6 [×2], D6 [×9], C2×C6 [×2], C2×C6, C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5 [×2], Dic5, C20, C20, D10, D10 [×7], C2×C10 [×2], C2×C10 [×5], Dic6, C4×S3, C4×S3 [×5], D12, D12 [×8], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C22×S3 [×2], C22×S3 [×4], C5×S3 [×3], C3×D5, D15 [×3], C30, C30 [×2], 2+ 1+4, Dic10, Dic10, C4×D5, C4×D5 [×3], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4, C5×D4 [×3], C22×D5 [×4], C22×C10 [×2], C2×D12 [×3], C4○D12 [×3], S3×D4, S3×D4 [×5], Q83S3 [×2], C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5 [×2], Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, S3×C10 [×2], S3×C10 [×2], D30, D30 [×2], D30 [×2], C2×C30 [×2], C4○D20 [×2], D4×D5 [×4], D42D5, D42D5 [×3], C2×C5⋊D4 [×4], D4×C10, D4○D12, S3×Dic5 [×2], D30.C2 [×2], C15⋊D4, C3⋊D20, C5⋊D12, C5⋊D12 [×6], C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], S3×C20, C5×D12, C5×C3⋊D4 [×2], C4×D15, D60, C157D4 [×2], D4×C15, C2×S3×D5 [×2], S3×C2×C10 [×2], C22×D15 [×2], D46D10, D12⋊D5, D60⋊C2, D6.D10, D5×D12, Dic3.D10 [×2], C2×C5⋊D12 [×2], S3×C5⋊D4 [×2], D10⋊D6 [×2], C3×D42D5, C5×S3×D4, D4×D15, D1214D10
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5 [×3], D46D10, C22×S3×D5, D1214D10

Smallest permutation representation of D1214D10
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 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 82)(74 81)(75 80)(76 79)(77 78)(83 84)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 102)(98 101)(99 100)(103 108)(104 107)(105 106)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)
(1 19 41 58 91)(2 14 42 53 92 8 20 48 59 86)(3 21 43 60 93)(4 16 44 55 94 10 22 38 49 88)(5 23 45 50 95)(6 18 46 57 96 12 24 40 51 90)(7 13 47 52 85)(9 15 37 54 87)(11 17 39 56 89)(25 102 80 61 115 31 108 74 67 109)(26 97 81 68 116)(27 104 82 63 117 33 98 76 69 111)(28 99 83 70 118)(29 106 84 65 119 35 100 78 71 113)(30 101 73 72 120)(32 103 75 62 110)(34 105 77 64 112)(36 107 79 66 114)
(1 75)(2 80)(3 73)(4 78)(5 83)(6 76)(7 81)(8 74)(9 79)(10 84)(11 77)(12 82)(13 97)(14 102)(15 107)(16 100)(17 105)(18 98)(19 103)(20 108)(21 101)(22 106)(23 99)(24 104)(25 42)(26 47)(27 40)(28 45)(29 38)(30 43)(31 48)(32 41)(33 46)(34 39)(35 44)(36 37)(49 113)(50 118)(51 111)(52 116)(53 109)(54 114)(55 119)(56 112)(57 117)(58 110)(59 115)(60 120)(61 86)(62 91)(63 96)(64 89)(65 94)(66 87)(67 92)(68 85)(69 90)(70 95)(71 88)(72 93)```

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 30A 30B 30C 30D 30E 30F 60A 60B order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12 15 15 20 20 20 20 30 30 30 30 30 30 60 60 size 1 1 2 2 6 6 6 10 30 30 30 2 2 6 10 10 10 30 2 2 2 4 4 20 2 2 4 4 4 4 6 6 6 6 12 12 12 12 4 10 10 20 20 4 4 4 4 12 12 4 4 8 8 8 8 8 8

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 D10 D10 D10 D10 D10 2+ 1+4 S3×D5 D4○D12 C2×S3×D5 C2×S3×D5 D4⋊6D10 D12⋊14D10 kernel D12⋊14D10 D12⋊D5 D60⋊C2 D6.D10 D5×D12 Dic3.D10 C2×C5⋊D12 S3×C5⋊D4 D10⋊D6 C3×D4⋊2D5 C5×S3×D4 D4×D15 D4⋊2D5 S3×D4 Dic10 C4×D5 C2×Dic5 C5⋊D4 C5×D4 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 C15 D4 C5 C4 C22 C3 C1 # reps 1 1 1 1 1 2 2 2 2 1 1 1 1 2 1 1 2 2 1 2 2 4 2 4 1 2 2 2 4 4 2

Matrix representation of D1214D10 in GL8(𝔽61)

 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 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 1 0 0 0 0 0 0 60 0
,
 1 0 1 0 0 0 0 0 0 1 0 1 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 60 44 0 0 0 0 0 0 17 44 0 0 0 0 0 0 0 0 60 44 0 0 0 0 0 0 17 44 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 34
,
 30 47 0 0 0 0 0 0 25 31 0 0 0 0 0 0 31 14 31 14 0 0 0 0 36 30 36 30 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 34 0 0 0 0 0 0 9 0 0 0 0 0 0 52 0 0 0

`G:=sub<GL(8,GF(61))| [1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[60,17,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,60,17,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,34],[30,25,31,36,0,0,0,0,47,31,14,30,0,0,0,0,0,0,31,36,0,0,0,0,0,0,14,30,0,0,0,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,9,0,0,0,0,0,0,34,0,0,0,0,0,0,27,0,0,0] >;`

D1214D10 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{14}D_{10}`
`% in TeX`

`G:=Group("D12:14D10");`
`// GroupNames label`

`G:=SmallGroup(480,1103);`
`// by ID`

`G=gap.SmallGroup(480,1103);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,185,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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