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## G = C60.28D4order 480 = 25·3·5

### 28th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.28D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×D20 — C60.28D4
 Lower central C15 — C30 — C2×C30 — C60.28D4
 Upper central C1 — C2 — C2×C4

Generators and relations for C60.28D4
G = < a,b,c | a12=1, b20=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b19 >

Subgroups: 476 in 92 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, D10, C2×C10, C3⋊C8, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C4.D4, C52C8, C40, D20, C2×C20, C22×D5, C4.Dic3, C4.Dic3, C6×D4, C60, C6×D5, C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.D4, C5×C3⋊C8, C153C8, C3×D20, C2×C60, D5×C2×C6, C20.46D4, C5×C4.Dic3, C60.7C4, C6×D20, C60.28D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C4.D4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C12.D4, D5×Dic3, C15⋊D4, C3⋊D20, C20.46D4, D10⋊Dic3, C60.28D4

Smallest permutation representation of C60.28D4
On 120 points
Generators in S120
```(1 93 49 31 83 79 21 113 69 11 103 59)(2 60 104 12 70 114 22 80 84 32 50 94)(3 95 51 33 85 41 23 115 71 13 105 61)(4 62 106 14 72 116 24 42 86 34 52 96)(5 97 53 35 87 43 25 117 73 15 107 63)(6 64 108 16 74 118 26 44 88 36 54 98)(7 99 55 37 89 45 27 119 75 17 109 65)(8 66 110 18 76 120 28 46 90 38 56 100)(9 101 57 39 91 47 29 81 77 19 111 67)(10 68 112 20 78 82 30 48 92 40 58 102)
(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 30 11 20 21 10 31 40)(2 39 32 9 22 19 12 29)(3 28 13 18 23 8 33 38)(4 37 34 7 24 17 14 27)(5 26 15 16 25 6 35 36)(41 46 51 76 61 66 71 56)(42 55 72 65 62 75 52 45)(43 44 53 74 63 64 73 54)(47 80 57 70 67 60 77 50)(48 49 78 59 68 69 58 79)(81 114 91 104 101 94 111 84)(82 83 112 93 102 103 92 113)(85 110 95 100 105 90 115 120)(86 119 116 89 106 99 96 109)(87 108 97 98 107 88 117 118)```

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 15A 15B 20A 20B 20C 20D 20E 20F 30A ··· 30F 40A ··· 40H 60A ··· 60H order 1 2 2 2 2 3 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 10 10 10 10 12 12 15 15 20 20 20 20 20 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 20 20 2 2 2 2 2 2 2 2 20 20 20 20 12 12 60 60 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4 12 ··· 12 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + - + - + image C1 C2 C2 C2 C4 S3 D4 D5 D6 Dic3 D10 C3⋊D4 D20 C5⋊D4 C4×D5 C4.D4 S3×D5 C12.D4 C15⋊D4 C3⋊D20 D5×Dic3 C20.46D4 C60.28D4 kernel C60.28D4 C5×C4.Dic3 C60.7C4 C6×D20 D5×C2×C6 C2×D20 C60 C4.Dic3 C2×C20 C22×D5 C2×C12 C20 C12 C12 C2×C6 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 2 2 1 2 2 4 4 4 4 1 2 2 2 2 2 4 8

Matrix representation of C60.28D4 in GL4(𝔽241) generated by

 178 45 0 0 196 63 0 0 0 0 222 48 0 0 193 19
,
 0 0 0 1 0 0 240 51 238 197 0 0 44 163 0 0
,
 0 0 0 1 0 0 1 0 238 197 0 0 44 3 0 0
`G:=sub<GL(4,GF(241))| [178,196,0,0,45,63,0,0,0,0,222,193,0,0,48,19],[0,0,238,44,0,0,197,163,0,240,0,0,1,51,0,0],[0,0,238,44,0,0,197,3,0,1,0,0,1,0,0,0] >;`

C60.28D4 in GAP, Magma, Sage, TeX

`C_{60}._{28}D_4`
`% in TeX`

`G:=Group("C60.28D4");`
`// GroupNames label`

`G:=SmallGroup(480,34);`
`// by ID`

`G=gap.SmallGroup(480,34);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,100,346,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^20=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^19>;`
`// generators/relations`

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