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## G = C30.3C42order 480 = 25·3·5

### 3rd non-split extension by C30 of C42 acting via C42/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.3C42
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C12×F5 — C30.3C42
 Lower central C15 — C30 — C30.3C42
 Upper central C1 — C4

Generators and relations for C30.3C42
G = < a,b,c | a30=b4=1, c4=a15, bab-1=a13, cac-1=a11, cbc-1=a15b >

Subgroups: 308 in 80 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C3⋊C8, C2×C12, C3×D5, C30, C8⋊C4, C52C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×C12, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C42.S3, C5×C3⋊C8, C153C8, C15⋊C8, D5×C12, C6×F5, C8⋊F5, D5×C3⋊C8, C12×F5, C60.C4, C30.3C42
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), F5, C4×S3, C2×Dic3, C8⋊C4, C2×F5, C4.Dic3, C4×Dic3, C4×F5, C42.S3, S3×F5, C8⋊F5, Dic3×F5, C30.3C42

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 8 8 type + + + + + + - + + + - image C1 C2 C2 C2 C4 C4 C4 C4 S3 D6 Dic3 M4(2) C4×S3 C4×S3 C4.Dic3 F5 C2×F5 C4×F5 C8⋊F5 S3×F5 Dic3×F5 C30.3C42 kernel C30.3C42 D5×C3⋊C8 C12×F5 C60.C4 C5×C3⋊C8 C15⋊3C8 C15⋊C8 C6×F5 C4×F5 C4×D5 C2×F5 C3×D5 Dic5 C20 D5 C3⋊C8 C12 C6 C3 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 1 1 2 4 2 2 8 1 1 2 4 1 1 2

Matrix representation of C30.3C42 in GL6(𝔽241)

 0 240 0 0 0 0 1 240 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 1 1 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 240 0 0 0 0 0 240 1 0 0 0 0 0 0 213 0 185 185 0 0 56 28 56 0 0 0 0 56 28 56 0 0 185 185 0 213

`G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,240,0,0,0,0,0,0,0,0,0,1,0,0,240,0,0,1,0,0,0,240,0,1,0,0,0,0,240,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,213,56,0,185,0,0,0,28,56,185,0,0,185,56,28,0,0,0,185,0,56,213] >;`

C30.3C42 in GAP, Magma, Sage, TeX

`C_{30}._3C_4^2`
`% in TeX`

`G:=Group("C30.3C4^2");`
`// GroupNames label`

`G:=SmallGroup(480,225);`
`// by ID`

`G=gap.SmallGroup(480,225);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,80,1356,9414,4724]);`
`// Polycyclic`

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

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