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## G = C30.C42order 480 = 25·3·5

### 2nd non-split extension by C30 of C42 acting via C42/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C30.C42
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D5×C12 — C3×D5⋊C8 — C30.C42
 Lower central C15 — C30.C42
 Upper central C1 — C4

Generators and relations for C30.C42
G = < a,b,c | a30=1, b4=c4=a15, bab-1=a13, cac-1=a11, bc=cb >

Subgroups: 356 in 88 conjugacy classes, 40 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C3⋊C8, C24, C2×Dic3, C2×C12, C3×D5, C30, C4×C8, C52C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C8×Dic3, C5×C3⋊C8, C153C8, C3×C5⋊C8, D5×C12, C2×C3⋊F5, C8×F5, D5×C3⋊C8, C3×D5⋊C8, C4×C3⋊F5, C30.C42
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, F5, C4×S3, C2×Dic3, C4×C8, C2×F5, S3×C8, C4×Dic3, C4×F5, C8×Dic3, S3×F5, C8×F5, Dic3×F5, C30.C42

Smallest permutation representation of C30.C42
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 91 55 68 16 106 40 83)(2 98 44 81 17 113 59 66)(3 105 33 64 18 120 48 79)(4 112 52 77 19 97 37 62)(5 119 41 90 20 104 56 75)(6 96 60 73 21 111 45 88)(7 103 49 86 22 118 34 71)(8 110 38 69 23 95 53 84)(9 117 57 82 24 102 42 67)(10 94 46 65 25 109 31 80)(11 101 35 78 26 116 50 63)(12 108 54 61 27 93 39 76)(13 115 43 74 28 100 58 89)(14 92 32 87 29 107 47 72)(15 99 51 70 30 114 36 85)
(1 106 55 83 16 91 40 68)(2 117 56 64 17 102 41 79)(3 98 57 75 18 113 42 90)(4 109 58 86 19 94 43 71)(5 120 59 67 20 105 44 82)(6 101 60 78 21 116 45 63)(7 112 31 89 22 97 46 74)(8 93 32 70 23 108 47 85)(9 104 33 81 24 119 48 66)(10 115 34 62 25 100 49 77)(11 96 35 73 26 111 50 88)(12 107 36 84 27 92 51 69)(13 118 37 65 28 103 52 80)(14 99 38 76 29 114 53 61)(15 110 39 87 30 95 54 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), (1,91,55,68,16,106,40,83)(2,98,44,81,17,113,59,66)(3,105,33,64,18,120,48,79)(4,112,52,77,19,97,37,62)(5,119,41,90,20,104,56,75)(6,96,60,73,21,111,45,88)(7,103,49,86,22,118,34,71)(8,110,38,69,23,95,53,84)(9,117,57,82,24,102,42,67)(10,94,46,65,25,109,31,80)(11,101,35,78,26,116,50,63)(12,108,54,61,27,93,39,76)(13,115,43,74,28,100,58,89)(14,92,32,87,29,107,47,72)(15,99,51,70,30,114,36,85), (1,106,55,83,16,91,40,68)(2,117,56,64,17,102,41,79)(3,98,57,75,18,113,42,90)(4,109,58,86,19,94,43,71)(5,120,59,67,20,105,44,82)(6,101,60,78,21,116,45,63)(7,112,31,89,22,97,46,74)(8,93,32,70,23,108,47,85)(9,104,33,81,24,119,48,66)(10,115,34,62,25,100,49,77)(11,96,35,73,26,111,50,88)(12,107,36,84,27,92,51,69)(13,118,37,65,28,103,52,80)(14,99,38,76,29,114,53,61)(15,110,39,87,30,95,54,72)>;`

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E ··· 4L 5 6A 6B 6C 8A 8B 8C 8D 8E ··· 8L 8M 8N 8O 8P 10 12A 12B 12C 12D 15 20A 20B 24A ··· 24H 30 40A 40B 40C 40D 60A 60B order 1 2 2 2 3 4 4 4 4 4 ··· 4 5 6 6 6 8 8 8 8 8 ··· 8 8 8 8 8 10 12 12 12 12 15 20 20 24 ··· 24 30 40 40 40 40 60 60 size 1 1 5 5 2 1 1 5 5 15 ··· 15 4 2 10 10 3 3 3 3 5 ··· 5 15 15 15 15 4 2 2 10 10 8 4 4 10 ··· 10 8 12 12 12 12 8 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 8 type + + + + + - + + + + - image C1 C2 C2 C2 C4 C4 C4 C4 C8 S3 Dic3 D6 C4×S3 C4×S3 S3×C8 F5 C2×F5 C4×F5 C8×F5 S3×F5 Dic3×F5 C30.C42 kernel C30.C42 D5×C3⋊C8 C3×D5⋊C8 C4×C3⋊F5 C5×C3⋊C8 C15⋊3C8 C3×C5⋊C8 C2×C3⋊F5 C3⋊F5 D5⋊C8 C5⋊C8 C4×D5 C20 D10 D5 C3⋊C8 C12 C6 C3 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 2 1 2 2 8 1 1 2 4 1 1 2

Matrix representation of C30.C42 in GL6(𝔽241)

 0 1 0 0 0 0 240 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 211 0 0 0 0 0 0 211 0 0 0 0 0 0 0 0 64 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 64 0 0
,
 0 233 0 0 0 0 233 0 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64

`G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[211,0,0,0,0,0,0,211,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,64,0,0,64,0,0,0,0,0,0,0,64,0],[0,233,0,0,0,0,233,0,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64] >;`

C30.C42 in GAP, Magma, Sage, TeX

`C_{30}.C_4^2`
`% in TeX`

`G:=Group("C30.C4^2");`
`// GroupNames label`

`G:=SmallGroup(480,224);`
`// by ID`

`G=gap.SmallGroup(480,224);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,100,80,1356,9414,4724]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=1,b^4=c^4=a^15,b*a*b^-1=a^13,c*a*c^-1=a^11,b*c=c*b>;`
`// generators/relations`

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