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## G = D12.9D10order 480 = 25·3·5

### 9th non-split extension by D12 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D12.9D10
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — D60⋊C2 — D12.9D10
 Lower central C15 — C30 — C60 — D12.9D10
 Upper central C1 — C2 — C4 — D4

Generators and relations for D12.9D10
G = < a,b,c,d | a12=b2=d2=1, c10=a6, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a7b, dcd=c9 >

Subgroups: 732 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, D15, C30, C30, C8⋊C22, C52C8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×C10, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C5×Dic3, C3×Dic5, C60, S3×C10, S3×C10, D30, C2×C30, C4.Dic5, D4⋊D5, D4.D5, D4.D5, C4○D20, D4×C10, Q83D6, C3×C52C8, C153C8, D30.C2, C5⋊D12, C3×Dic10, S3×C20, C5×D12, C5×C3⋊D4, D60, D4×C15, S3×C2×C10, D4.D10, D6.Dic5, C5⋊D24, C20.D6, C3×D4.D5, D4⋊D15, D60⋊C2, C5×S3×D4, D12.9D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q83D6, C2×S3×D5, D4.D10, S3×C5⋊D4, D12.9D10

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 12A 12B 15A 15B 20A 20B 20C 20D 24A 24B 30A 30B 30C 30D 30E 30F 60A 60B order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 15 15 20 20 20 20 24 24 30 30 30 30 30 30 60 60 size 1 1 4 6 12 60 2 2 6 20 2 2 2 8 20 60 2 2 4 4 4 4 6 6 6 6 12 12 12 12 4 40 4 4 4 4 12 12 20 20 4 4 8 8 8 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 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 S3×D4 S3×D5 Q8⋊3D6 C2×S3×D5 D4.D10 S3×C5⋊D4 D12.9D10 kernel D12.9D10 D6.Dic5 C5⋊D24 C20.D6 C3×D4.D5 D4⋊D15 D60⋊C2 C5×S3×D4 D4.D5 C5×Dic3 S3×C10 S3×D4 C5⋊2C8 Dic10 C5×D4 C4×S3 D12 C3×D4 Dic3 D6 C15 C10 D4 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 4 4 1 1 2 2 2 4 4 2

Matrix representation of D12.9D10 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 2 2 0 0 240 0 239 0 0 0 240 240 240 240 0 0 1 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 2 2 0 0 0 240 0 239 0 0 0 0 240 240 0 0 0 0 0 1
,
 51 51 0 0 0 0 190 1 0 0 0 0 0 0 1 0 2 0 0 0 240 240 239 239 0 0 240 0 240 0 0 0 1 1 1 1
,
 231 59 0 0 0 0 31 10 0 0 0 0 0 0 9 18 9 18 0 0 9 232 9 232 0 0 116 232 232 223 0 0 116 125 232 9

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,240,240,1,0,0,1,0,240,0,0,0,2,239,240,1,0,0,2,0,240,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,240,0,0,0,0,2,0,240,0,0,0,2,239,240,1],[51,190,0,0,0,0,51,1,0,0,0,0,0,0,1,240,240,1,0,0,0,240,0,1,0,0,2,239,240,1,0,0,0,239,0,1],[231,31,0,0,0,0,59,10,0,0,0,0,0,0,9,9,116,116,0,0,18,232,232,125,0,0,9,9,232,232,0,0,18,232,223,9] >;`

D12.9D10 in GAP, Magma, Sage, TeX

`D_{12}._9D_{10}`
`% in TeX`

`G:=Group("D12.9D10");`
`// GroupNames label`

`G:=SmallGroup(480,572);`
`// by ID`

`G=gap.SmallGroup(480,572);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,675,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^10=a^6,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^7*b,d*c*d=c^9>;`
`// generators/relations`

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