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G = D10.10D12order 480 = 25·3·5

6th non-split extension by D10 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D10.10D12
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — D5×C2×C6 — C22×C3⋊F5 — D10.10D12
 Lower central C15 — C30 — D10.10D12
 Upper central C1 — C22 — C2×C4

Generators and relations for D10.10D12
G = < a,b,c,d | a10=b2=c12=1, d2=a4b, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a5c-1 >

Subgroups: 812 in 152 conjugacy classes, 57 normal (35 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2.C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C22×Dic3, C22×C12, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C2×C4×D5, C22×F5, C6.C42, D5×C12, C6×Dic5, C2×C60, C2×C3⋊F5, C2×C3⋊F5, D5×C2×C6, D10.3Q8, D5×C2×C12, C22×C3⋊F5, D10.10D12
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, C3⋊F5, C4×F5, C4⋊F5, C22⋊F5, C6.C42, C2×C3⋊F5, D10.3Q8, C4×C3⋊F5, C60⋊C4, D10.D6, D10.10D12

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - - - + - + + + + image C1 C2 C2 C4 C4 C4 S3 D4 Q8 Dic3 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 F5 C2×F5 C3⋊F5 C4×F5 C4⋊F5 C22⋊F5 C2×C3⋊F5 C4×C3⋊F5 C60⋊C4 D10.D6 kernel D10.10D12 D5×C2×C12 C22×C3⋊F5 C6×Dic5 C2×C60 C2×C3⋊F5 C2×C4×D5 C6×D5 C6×D5 C2×Dic5 C2×C20 C22×D5 D10 D10 D10 D10 C2×C12 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 2 2 2 8 1 3 1 1 1 1 2 4 2 4 1 1 2 2 2 2 2 4 4 4

Matrix representation of D10.10D12 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
,
 44 7 0 0 0 0 35 25 0 0 0 0 0 0 8 56 0 5 0 0 0 3 56 5 0 0 5 56 3 0 0 0 5 0 56 8
,
 30 16 0 0 0 0 39 31 0 0 0 0 0 0 3 56 5 58 0 0 0 56 8 53 0 0 56 0 5 53 0 0 56 3 0 58

`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],[44,35,0,0,0,0,7,25,0,0,0,0,0,0,8,0,5,5,0,0,56,3,56,0,0,0,0,56,3,56,0,0,5,5,0,8],[30,39,0,0,0,0,16,31,0,0,0,0,0,0,3,0,56,56,0,0,56,56,0,3,0,0,5,8,5,0,0,0,58,53,53,58] >;`

D10.10D12 in GAP, Magma, Sage, TeX

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

`G:=Group("D10.10D12");`
`// GroupNames label`

`G:=SmallGroup(480,311);`
`// by ID`

`G=gap.SmallGroup(480,311);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,2693,14118,4724]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^4*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^5*c^-1>;`
`// generators/relations`

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