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## G = C60.8D4order 480 = 25·3·5

### 8th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.8D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C60.7C4 — C60.8D4
 Lower central C15 — C30 — C2×C30 — C60.8D4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for C60.8D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a29, cbc-1=a15b3 >

Subgroups: 308 in 92 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×3], C4.D4, C52C8 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3 [×2], C6×D4, C60 [×2], C2×C30, C2×C30 [×4], C4.Dic5 [×2], D4×C10, C12.D4, C153C8 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C20.D4, C60.7C4 [×2], D4×C30, C60.8D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, C4.D4, C2×Dic5, C5⋊D4 [×2], C6.D4, Dic15 [×2], D30, C23.D5, C12.D4, C2×Dic15, C157D4 [×2], C20.D4, C30.38D4, C60.8D4

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

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

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

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

81 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 ··· 10F 10G ··· 10N 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 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 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 4 4 2 2 2 2 2 2 2 2 4 4 4 4 60 60 60 60 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + - + + - + image C1 C2 C2 C4 S3 D4 D5 D6 Dic3 D10 Dic5 C3⋊D4 D15 C5⋊D4 D30 Dic15 C15⋊7D4 C4.D4 C12.D4 C20.D4 C60.8D4 kernel C60.8D4 C60.7C4 D4×C30 C22×C30 D4×C10 C60 C6×D4 C2×C20 C22×C10 C2×C12 C22×C6 C20 C2×D4 C12 C2×C4 C23 C4 C15 C5 C3 C1 # reps 1 2 1 4 1 2 2 1 2 2 4 4 4 8 4 8 16 1 2 4 8

Matrix representation of C60.8D4 in GL4(𝔽241) generated by

 0 24 0 0 217 0 0 0 0 0 0 231 0 0 10 0
,
 0 0 1 0 0 0 0 240 0 1 0 0 1 0 0 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 240 0 0 0
`G:=sub<GL(4,GF(241))| [0,217,0,0,24,0,0,0,0,0,0,10,0,0,231,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;`

C60.8D4 in GAP, Magma, Sage, TeX

`C_{60}._8D_4`
`% in TeX`

`G:=Group("C60.8D4");`
`// GroupNames label`

`G:=SmallGroup(480,193);`
`// by ID`

`G=gap.SmallGroup(480,193);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

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