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## G = C12.28D10order 240 = 24·3·5

### 7th non-split extension by C12 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C12.28D10
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C3⋊D20 — C12.28D10
 Lower central C15 — C30 — C12.28D10
 Upper central C1 — C2 — C4

Generators and relations for C12.28D10
G = < a,b,c | a12=c2=1, b10=a6, bab-1=cac=a-1, cbc=a6b9 >

Subgroups: 384 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C15, C4○D4, Dic5, C20, C20 [×2], D10, D10 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C3×D5, D15 [×2], C30, C4×D5, C4×D5 [×2], D20 [×3], C5×Q8, C4○D12, C5×Dic3 [×2], C3×Dic5, C60, C6×D5, D30 [×2], Q82D5, D30.C2 [×2], C3⋊D20 [×2], D5×C12, C5×Dic6, D60, C12.28D10
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, C4○D12, S3×D5, Q82D5, C2×S3×D5, C12.28D10

Smallest permutation representation of C12.28D10
On 120 points
Generators in S120
```(1 59 97 38 80 119 11 49 87 28 70 109)(2 110 71 29 88 50 12 120 61 39 98 60)(3 41 99 40 62 101 13 51 89 30 72 111)(4 112 73 31 90 52 14 102 63 21 100 42)(5 43 81 22 64 103 15 53 91 32 74 113)(6 114 75 33 92 54 16 104 65 23 82 44)(7 45 83 24 66 105 17 55 93 34 76 115)(8 116 77 35 94 56 18 106 67 25 84 46)(9 47 85 26 68 107 19 57 95 36 78 117)(10 118 79 37 96 58 20 108 69 27 86 48)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 120)(53 119)(54 118)(55 117)(56 116)(57 115)(58 114)(59 113)(60 112)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 C4○D12 S3×D5 Q8⋊2D5 C2×S3×D5 C12.28D10 kernel C12.28D10 D30.C2 C3⋊D20 D5×C12 C5×Dic6 D60 C4×D5 Dic6 Dic5 C20 D10 C15 Dic3 C12 C5 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 1 1 1 2 4 2 4 2 2 2 4

Matrix representation of C12.28D10 in GL6(𝔽61)

 60 15 0 0 0 0 12 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 3 0 0 0 0 40 1
,
 1 0 0 0 0 0 49 60 0 0 0 0 0 0 44 44 0 0 0 0 17 60 0 0 0 0 0 0 50 33 0 0 0 0 0 11
,
 1 0 0 0 0 0 49 60 0 0 0 0 0 0 44 44 0 0 0 0 60 17 0 0 0 0 0 0 1 0 0 0 0 0 21 60

`G:=sub<GL(6,GF(61))| [60,12,0,0,0,0,15,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,40,0,0,0,0,3,1],[1,49,0,0,0,0,0,60,0,0,0,0,0,0,44,17,0,0,0,0,44,60,0,0,0,0,0,0,50,0,0,0,0,0,33,11],[1,49,0,0,0,0,0,60,0,0,0,0,0,0,44,60,0,0,0,0,44,17,0,0,0,0,0,0,1,21,0,0,0,0,0,60] >;`

C12.28D10 in GAP, Magma, Sage, TeX

`C_{12}._{28}D_{10}`
`% in TeX`

`G:=Group("C12.28D10");`
`// GroupNames label`

`G:=SmallGroup(240,134);`
`// by ID`

`G=gap.SmallGroup(240,134);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,55,116,50,490,6917]);`
`// Polycyclic`

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

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