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## G = C12.60D12order 288 = 25·32

### 27th non-split extension by C12 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.60D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C4×C3⋊S3 — C12.60D12
 Lower central C32 — C3×C6 — C12.60D12
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C12.60D12
G = < a,b,c | a12=1, b12=a6, c2=a9, ab=ba, cac-1=a5, cbc-1=a3b11 >

Subgroups: 572 in 150 conjugacy classes, 61 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4, C22, C22 [×4], S3 [×8], C6 [×12], C8 [×2], C2×C4, C2×C4 [×3], C23, C32, Dic3 [×4], C12 [×8], D6 [×16], C2×C6 [×4], C2×C8, C2×C8, C22×C4, C3⋊S3 [×2], C3×C6 [×3], C3⋊C8 [×4], C24 [×4], C4×S3 [×8], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C22⋊C8, C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C2×C3⋊C8 [×4], C2×C24 [×4], S3×C2×C4 [×4], C324C8, C3×C24, C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊C8 [×4], C2×C324C8, C6×C24, C2×C4×C3⋊S3, C12.60D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C2×C8, M4(2), C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C22⋊C8, C2×C3⋊S3, S3×C8 [×4], C8⋊S3 [×4], D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, D6⋊C8 [×4], C8×C3⋊S3, C24⋊S3, C6.11D12, C12.60D12

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6L 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 18 18 2 2 2 2 1 1 1 1 18 18 2 ··· 2 2 2 2 2 18 18 18 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D4 D6 M4(2) D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 kernel C12.60D12 C2×C32⋊4C8 C6×C24 C2×C4×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C2×C3⋊S3 C2×C24 C3×C12 C2×C12 C3×C6 C12 C12 C2×C6 C6 C6 # reps 1 1 1 1 2 2 8 4 2 4 2 8 8 8 16 16

Matrix representation of C12.60D12 in GL4(𝔽73) generated by

 1 1 0 0 72 0 0 0 0 0 27 0 0 0 0 27
,
 43 30 0 0 43 13 0 0 0 0 0 10 0 0 63 10
,
 43 30 0 0 60 30 0 0 0 0 10 63 0 0 0 63
`G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,27,0,0,0,0,27],[43,43,0,0,30,13,0,0,0,0,0,63,0,0,10,10],[43,60,0,0,30,30,0,0,0,0,10,0,0,0,63,63] >;`

C12.60D12 in GAP, Magma, Sage, TeX

`C_{12}._{60}D_{12}`
`% in TeX`

`G:=Group("C12.60D12");`
`// GroupNames label`

`G:=SmallGroup(288,295);`
`// by ID`

`G=gap.SmallGroup(288,295);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;`
`// generators/relations`

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