Copied to
clipboard

## G = C20.5D12order 480 = 25·3·5

### 5th non-split extension by C20 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C20.5D12
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C20.5D12
 Lower central C15 — C30 — C2×C30 — C20.5D12
 Upper central C1 — C2 — C2×C4

Generators and relations for C20.5D12
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a49, cbc-1=a15b3 >

Subgroups: 380 in 92 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], D6 [×4], C2×C6, C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], C5×S3 [×2], C30, C30, C4.D4, C52C8 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], S3×C10 [×4], C2×C30, C4.Dic5, C4.Dic5, D4×C10, C12.46D4, C3×C52C8, C153C8, C5×D12 [×2], C2×C60, S3×C2×C10 [×2], C20.D4, C3×C4.Dic5, C60.7C4, C10×D12, C20.5D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C4.D4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, C12.46D4, S3×Dic5, C15⋊D4, C5⋊D12, C20.D4, D6⋊Dic5, C20.5D12

Smallest permutation representation of C20.5D12
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 84 46 99 31 114 16 69)(2 83 47 98 32 113 17 68)(3 82 48 97 33 112 18 67)(4 81 49 96 34 111 19 66)(5 80 50 95 35 110 20 65)(6 79 51 94 36 109 21 64)(7 78 52 93 37 108 22 63)(8 77 53 92 38 107 23 62)(9 76 54 91 39 106 24 61)(10 75 55 90 40 105 25 120)(11 74 56 89 41 104 26 119)(12 73 57 88 42 103 27 118)(13 72 58 87 43 102 28 117)(14 71 59 86 44 101 29 116)(15 70 60 85 45 100 30 115)
(1 114 46 99 31 84 16 69)(2 103 47 88 32 73 17 118)(3 92 48 77 33 62 18 107)(4 81 49 66 34 111 19 96)(5 70 50 115 35 100 20 85)(6 119 51 104 36 89 21 74)(7 108 52 93 37 78 22 63)(8 97 53 82 38 67 23 112)(9 86 54 71 39 116 24 101)(10 75 55 120 40 105 25 90)(11 64 56 109 41 94 26 79)(12 113 57 98 42 83 27 68)(13 102 58 87 43 72 28 117)(14 91 59 76 44 61 29 106)(15 80 60 65 45 110 30 95)```

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 5A 5B 6A 6B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 12C 15A 15B 20A 20B 20C 20D 24A 24B 24C 24D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 3 4 4 5 5 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 15 15 20 20 20 20 24 24 24 24 30 ··· 30 60 ··· 60 size 1 1 2 12 12 2 2 2 2 2 2 4 20 20 60 60 2 ··· 2 12 ··· 12 2 2 4 4 4 4 4 4 4 20 20 20 20 4 ··· 4 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 D10 Dic5 D12 C3⋊D4 C4×S3 C5⋊D4 C4.D4 S3×D5 C12.46D4 C15⋊D4 C5⋊D12 S3×Dic5 C20.D4 C20.5D12 kernel C20.5D12 C3×C4.Dic5 C60.7C4 C10×D12 S3×C2×C10 C4.Dic5 C60 C2×D12 C2×C20 C2×C12 C22×S3 C20 C20 C2×C10 C12 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 2 2 1 2 4 2 2 2 8 1 2 2 2 2 2 4 8

Matrix representation of C20.5D12 in GL8(𝔽241)

 205 0 0 0 0 0 0 0 24 87 0 0 0 0 0 0 0 0 226 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 54 191 0 0 0 0 0 0 121 187 0 0 0 0 0 0 192 73 185 144 0 0 0 0 184 99 87 56
,
 18 32 0 0 0 0 0 0 133 223 0 0 0 0 0 0 0 0 0 177 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 202 0 3 0 0 0 0 0 149 0 77 240 0 0 0 0 234 144 39 0 0 0 0 0 36 56 76 0
,
 223 209 0 0 0 0 0 0 108 18 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 177 0 0 0 0 0 0 0 0 202 0 3 0 0 0 0 0 162 0 103 1 0 0 0 0 234 144 39 0 0 0 0 0 171 56 76 0

`G:=sub<GL(8,GF(241))| [205,24,0,0,0,0,0,0,0,87,0,0,0,0,0,0,0,0,226,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,54,121,192,184,0,0,0,0,191,187,73,99,0,0,0,0,0,0,185,87,0,0,0,0,0,0,144,56],[18,133,0,0,0,0,0,0,32,223,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,202,149,234,36,0,0,0,0,0,0,144,56,0,0,0,0,3,77,39,76,0,0,0,0,0,240,0,0],[223,108,0,0,0,0,0,0,209,18,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,202,162,234,171,0,0,0,0,0,0,144,56,0,0,0,0,3,103,39,76,0,0,0,0,0,1,0,0] >;`

C20.5D12 in GAP, Magma, Sage, TeX

`C_{20}._5D_{12}`
`% in TeX`

`G:=Group("C20.5D12");`
`// GroupNames label`

`G:=SmallGroup(480,35);`
`// by ID`

`G=gap.SmallGroup(480,35);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,1356,18822]);`
`// 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^49,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

׿
×
𝔽