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## G = C12.59D12order 288 = 25·32

### 26th non-split extension by C12 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C12.59D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C12.58D6 — C12.59D12
 Lower central C32 — C3×C6 — C3×C12 — C12.59D12
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for C12.59D12
G = < a,b,c | a12=1, b12=a6, c2=a3, ab=ba, cac-1=a5, cbc-1=b11 >

Subgroups: 220 in 90 conjugacy classes, 57 normal (19 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, M4(2) [×2], C3×C6, C3×C6, C3⋊C8 [×8], C24 [×8], C2×C12 [×4], C8.C4, C3×C12 [×2], C62, C4.Dic3 [×8], C2×C24 [×4], C324C8 [×2], C3×C24 [×2], C6×C12, C24.C4 [×4], C12.58D6 [×2], C6×C24, C12.59D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], C8.C4, C3⋊Dic3 [×2], C2×C3⋊S3, C4⋊Dic3 [×4], C324Q8, C12⋊S3, C2×C3⋊Dic3, C24.C4 [×4], C12⋊Dic3, C12.59D12

Smallest permutation representation of C12.59D12
On 144 points
Generators in S144
```(1 33 125 19 27 143 13 45 137 7 39 131)(2 34 126 20 28 144 14 46 138 8 40 132)(3 35 127 21 29 121 15 47 139 9 41 133)(4 36 128 22 30 122 16 48 140 10 42 134)(5 37 129 23 31 123 17 25 141 11 43 135)(6 38 130 24 32 124 18 26 142 12 44 136)(49 89 118 55 95 100 61 77 106 67 83 112)(50 90 119 56 96 101 62 78 107 68 84 113)(51 91 120 57 73 102 63 79 108 69 85 114)(52 92 97 58 74 103 64 80 109 70 86 115)(53 93 98 59 75 104 65 81 110 71 87 116)(54 94 99 60 76 105 66 82 111 72 88 117)
(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)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 104 19 110 13 116 7 98)(2 115 20 97 14 103 8 109)(3 102 21 108 15 114 9 120)(4 113 22 119 16 101 10 107)(5 100 23 106 17 112 11 118)(6 111 24 117 18 99 12 105)(25 95 43 77 37 83 31 89)(26 82 44 88 38 94 32 76)(27 93 45 75 39 81 33 87)(28 80 46 86 40 92 34 74)(29 91 47 73 41 79 35 85)(30 78 48 84 42 90 36 96)(49 129 55 123 61 141 67 135)(50 140 56 134 62 128 68 122)(51 127 57 121 63 139 69 133)(52 138 58 132 64 126 70 144)(53 125 59 143 65 137 71 131)(54 136 60 130 66 124 72 142)```

`G:=sub<Sym(144)| (1,33,125,19,27,143,13,45,137,7,39,131)(2,34,126,20,28,144,14,46,138,8,40,132)(3,35,127,21,29,121,15,47,139,9,41,133)(4,36,128,22,30,122,16,48,140,10,42,134)(5,37,129,23,31,123,17,25,141,11,43,135)(6,38,130,24,32,124,18,26,142,12,44,136)(49,89,118,55,95,100,61,77,106,67,83,112)(50,90,119,56,96,101,62,78,107,68,84,113)(51,91,120,57,73,102,63,79,108,69,85,114)(52,92,97,58,74,103,64,80,109,70,86,115)(53,93,98,59,75,104,65,81,110,71,87,116)(54,94,99,60,76,105,66,82,111,72,88,117), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,104,19,110,13,116,7,98)(2,115,20,97,14,103,8,109)(3,102,21,108,15,114,9,120)(4,113,22,119,16,101,10,107)(5,100,23,106,17,112,11,118)(6,111,24,117,18,99,12,105)(25,95,43,77,37,83,31,89)(26,82,44,88,38,94,32,76)(27,93,45,75,39,81,33,87)(28,80,46,86,40,92,34,74)(29,91,47,73,41,79,35,85)(30,78,48,84,42,90,36,96)(49,129,55,123,61,141,67,135)(50,140,56,134,62,128,68,122)(51,127,57,121,63,139,69,133)(52,138,58,132,64,126,70,144)(53,125,59,143,65,137,71,131)(54,136,60,130,66,124,72,142)>;`

`G:=Group( (1,33,125,19,27,143,13,45,137,7,39,131)(2,34,126,20,28,144,14,46,138,8,40,132)(3,35,127,21,29,121,15,47,139,9,41,133)(4,36,128,22,30,122,16,48,140,10,42,134)(5,37,129,23,31,123,17,25,141,11,43,135)(6,38,130,24,32,124,18,26,142,12,44,136)(49,89,118,55,95,100,61,77,106,67,83,112)(50,90,119,56,96,101,62,78,107,68,84,113)(51,91,120,57,73,102,63,79,108,69,85,114)(52,92,97,58,74,103,64,80,109,70,86,115)(53,93,98,59,75,104,65,81,110,71,87,116)(54,94,99,60,76,105,66,82,111,72,88,117), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,104,19,110,13,116,7,98)(2,115,20,97,14,103,8,109)(3,102,21,108,15,114,9,120)(4,113,22,119,16,101,10,107)(5,100,23,106,17,112,11,118)(6,111,24,117,18,99,12,105)(25,95,43,77,37,83,31,89)(26,82,44,88,38,94,32,76)(27,93,45,75,39,81,33,87)(28,80,46,86,40,92,34,74)(29,91,47,73,41,79,35,85)(30,78,48,84,42,90,36,96)(49,129,55,123,61,141,67,135)(50,140,56,134,62,128,68,122)(51,127,57,121,63,139,69,133)(52,138,58,132,64,126,70,144)(53,125,59,143,65,137,71,131)(54,136,60,130,66,124,72,142) );`

`G=PermutationGroup([(1,33,125,19,27,143,13,45,137,7,39,131),(2,34,126,20,28,144,14,46,138,8,40,132),(3,35,127,21,29,121,15,47,139,9,41,133),(4,36,128,22,30,122,16,48,140,10,42,134),(5,37,129,23,31,123,17,25,141,11,43,135),(6,38,130,24,32,124,18,26,142,12,44,136),(49,89,118,55,95,100,61,77,106,67,83,112),(50,90,119,56,96,101,62,78,107,68,84,113),(51,91,120,57,73,102,63,79,108,69,85,114),(52,92,97,58,74,103,64,80,109,70,86,115),(53,93,98,59,75,104,65,81,110,71,87,116),(54,94,99,60,76,105,66,82,111,72,88,117)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,104,19,110,13,116,7,98),(2,115,20,97,14,103,8,109),(3,102,21,108,15,114,9,120),(4,113,22,119,16,101,10,107),(5,100,23,106,17,112,11,118),(6,111,24,117,18,99,12,105),(25,95,43,77,37,83,31,89),(26,82,44,88,38,94,32,76),(27,93,45,75,39,81,33,87),(28,80,46,86,40,92,34,74),(29,91,47,73,41,79,35,85),(30,78,48,84,42,90,36,96),(49,129,55,123,61,141,67,135),(50,140,56,134,62,128,68,122),(51,127,57,121,63,139,69,133),(52,138,58,132,64,126,70,144),(53,125,59,143,65,137,71,131),(54,136,60,130,66,124,72,142)])`

78 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A ··· 6L 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12P 24A ··· 24AF order 1 2 2 3 3 3 3 4 4 4 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 2 2 2 2 2 1 1 2 2 ··· 2 2 2 2 2 36 36 36 36 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - - + + - image C1 C2 C2 C4 S3 D4 Q8 Dic3 D6 D12 Dic6 C8.C4 C24.C4 kernel C12.59D12 C12.58D6 C6×C24 C3×C24 C2×C24 C3×C12 C62 C24 C2×C12 C12 C2×C6 C32 C3 # reps 1 2 1 4 4 1 1 8 4 8 8 4 32

Matrix representation of C12.59D12 in GL4(𝔽73) generated by

 24 0 0 0 0 3 0 0 0 0 70 0 0 0 0 49
,
 17 0 0 0 0 30 0 0 0 0 52 0 0 0 0 7
,
 0 30 0 0 52 0 0 0 0 0 0 7 0 0 17 0
`G:=sub<GL(4,GF(73))| [24,0,0,0,0,3,0,0,0,0,70,0,0,0,0,49],[17,0,0,0,0,30,0,0,0,0,52,0,0,0,0,7],[0,52,0,0,30,0,0,0,0,0,0,17,0,0,7,0] >;`

C12.59D12 in GAP, Magma, Sage, TeX

`C_{12}._{59}D_{12}`
`% in TeX`

`G:=Group("C12.59D12");`
`// GroupNames label`

`G:=SmallGroup(288,294);`
`// by ID`

`G=gap.SmallGroup(288,294);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,100,675,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^3,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^11>;`
`// generators/relations`

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