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## G = C12.20D12order 288 = 25·32

### 20th non-split extension by C12 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.20D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C32⋊4Q8 — C12.20D12
 Lower central C32 — C3×C6 — C62 — C12.20D12
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C12.20D12
G = < a,b,c | a12=1, b12=a6, c2=a3, bab-1=a7, cac-1=a5, cbc-1=a3b11 >

Subgroups: 396 in 114 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C2×C6, M4(2), M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C4.10D4, C3⋊Dic3, C3×C12, C62, C4.Dic3, C3×M4(2), C2×Dic6, C324C8, C3×C24, C324Q8, C2×C3⋊Dic3, C6×C12, C12.47D4, C12.58D6, C32×M4(2), C2×C324Q8, C12.20D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C4.10D4, C2×C3⋊S3, D6⋊C4, C4×C3⋊S3, C12⋊S3, C327D4, C12.47D4, C6.11D12, C12.20D12

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

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 S3 D4 D6 D12 C3⋊D4 C4×S3 C4.10D4 C12.47D4 kernel C12.20D12 C12.58D6 C32×M4(2) C2×C32⋊4Q8 C2×C3⋊Dic3 C3×M4(2) C3×C12 C2×C12 C12 C12 C2×C6 C32 C3 # reps 1 1 1 1 4 4 2 4 8 8 8 1 8

Matrix representation of C12.20D12 in GL6(𝔽73)

 36 26 0 0 0 0 47 36 0 0 0 0 0 0 66 7 0 0 0 0 66 59 0 0 0 0 60 60 7 66 0 0 13 0 7 14
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 27 27 72 1 0 0 46 0 72 71 0 0 42 17 46 46 0 0 56 25 27 0
,
 2 56 0 0 0 0 56 71 0 0 0 0 0 0 11 13 40 49 0 0 2 62 9 33 0 0 27 17 11 13 0 0 63 46 2 62

`G:=sub<GL(6,GF(73))| [36,47,0,0,0,0,26,36,0,0,0,0,0,0,66,66,60,13,0,0,7,59,60,0,0,0,0,0,7,7,0,0,0,0,66,14],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,27,46,42,56,0,0,27,0,17,25,0,0,72,72,46,27,0,0,1,71,46,0],[2,56,0,0,0,0,56,71,0,0,0,0,0,0,11,2,27,63,0,0,13,62,17,46,0,0,40,9,11,2,0,0,49,33,13,62] >;`

C12.20D12 in GAP, Magma, Sage, TeX

`C_{12}._{20}D_{12}`
`% in TeX`

`G:=Group("C12.20D12");`
`// GroupNames label`

`G:=SmallGroup(288,299);`
`// by ID`

`G=gap.SmallGroup(288,299);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^3,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;`
`// generators/relations`

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