Copied to
clipboard

## G = C12.31D12order 288 = 25·32

### 31st non-split extension by C12 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.31D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C12.31D12
 Lower central C32 — C62 — C12.31D12
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C12.31D12
G = < a,b,c | a12=b12=1, c2=a6, bab-1=a7, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 860 in 222 conjugacy classes, 79 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4 [×5], C22, C22 [×4], S3 [×8], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C32, Dic3 [×12], C12 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C3⋊S3 [×2], C3×C6 [×3], Dic6 [×8], C4×S3 [×8], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C22⋊Q8, C3⋊Dic3 [×3], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4⋊Dic3 [×8], D6⋊C4 [×8], C3×C4⋊C4 [×4], C2×Dic6 [×4], S3×C2×C4 [×4], C324Q8 [×2], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C22×C3⋊S3, C4.D12 [×4], C12⋊Dic3 [×2], C6.11D12 [×2], C32×C4⋊C4, C2×C324Q8, C2×C4×C3⋊S3, C12.31D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], Q8 [×2], C23, D6 [×12], C2×D4, C2×Q8, C4○D4, C3⋊S3, D12 [×8], C22×S3 [×4], C22⋊Q8, C2×C3⋊S3 [×3], C2×D12 [×4], D42S3 [×4], S3×Q8 [×4], C12⋊S3 [×2], C22×C3⋊S3, C4.D12 [×4], C2×C12⋊S3, C12.D6, Q8×C3⋊S3, C12.31D12

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 12A ··· 12X order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 2 2 2 2 2 2 4 4 18 18 36 36 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 C4○D4 D12 D4⋊2S3 S3×Q8 kernel C12.31D12 C12⋊Dic3 C6.11D12 C32×C4⋊C4 C2×C32⋊4Q8 C2×C4×C3⋊S3 C3×C4⋊C4 C3×C12 C2×C3⋊S3 C2×C12 C3×C6 C12 C6 C6 # reps 1 2 2 1 1 1 4 2 2 12 2 16 4 4

Matrix representation of C12.31D12 in GL6(𝔽13)

 0 12 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 3 6 0 0 0 0 7 10 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 3 6 0 0 0 0 3 10 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0

`G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;`

C12.31D12 in GAP, Magma, Sage, TeX

`C_{12}._{31}D_{12}`
`% in TeX`

`G:=Group("C12.31D12");`
`// GroupNames label`

`G:=SmallGroup(288,754);`
`// by ID`

`G=gap.SmallGroup(288,754);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,142,2693,9414]);`
`// Polycyclic`

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

׿
×
𝔽