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## G = C24.5D6order 288 = 25·32

### 5th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24.5D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C12.59D6 — C24.5D6
 Lower central C32 — C3×C6 — C3×C12 — C24.5D6
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C24.5D6
G = < a,b,c | a24=1, b6=c2=a12, bab-1=a13, cac-1=a-1, cbc-1=b5 >

Subgroups: 748 in 180 conjugacy classes, 65 normal (19 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, S3 [×4], C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C32, Dic3 [×12], C12 [×8], D6 [×4], C2×C6 [×4], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C24 [×8], Dic6 [×16], C4×S3 [×4], D12 [×4], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×4], C8.C22, C3⋊Dic3 [×3], C3×C12 [×2], C2×C3⋊S3, C62, C24⋊C2 [×8], Dic12 [×8], C3×M4(2) [×4], C2×Dic6 [×4], C4○D12 [×4], C3×C24 [×2], C324Q8, C324Q8 [×2], C324Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, C6×C12, C8.D6 [×4], C242S3 [×2], C325Q16 [×2], C32×M4(2), C2×C324Q8, C12.59D6, C24.5D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C8.C22, C2×C3⋊S3 [×3], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C8.D6 [×4], C2×C12⋊S3, C24.5D6

Smallest permutation representation of C24.5D6
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)
(1 144 80 32 109 53 13 132 92 44 97 65)(2 133 81 45 110 66 14 121 93 33 98 54)(3 122 82 34 111 55 15 134 94 46 99 67)(4 135 83 47 112 68 16 123 95 35 100 56)(5 124 84 36 113 57 17 136 96 48 101 69)(6 137 85 25 114 70 18 125 73 37 102 58)(7 126 86 38 115 59 19 138 74 26 103 71)(8 139 87 27 116 72 20 127 75 39 104 60)(9 128 88 40 117 61 21 140 76 28 105 49)(10 141 89 29 118 50 22 129 77 41 106 62)(11 130 90 42 119 63 23 142 78 30 107 51)(12 143 91 31 120 52 24 131 79 43 108 64)
(1 53 13 65)(2 52 14 64)(3 51 15 63)(4 50 16 62)(5 49 17 61)(6 72 18 60)(7 71 19 59)(8 70 20 58)(9 69 21 57)(10 68 22 56)(11 67 23 55)(12 66 24 54)(25 75 37 87)(26 74 38 86)(27 73 39 85)(28 96 40 84)(29 95 41 83)(30 94 42 82)(31 93 43 81)(32 92 44 80)(33 91 45 79)(34 90 46 78)(35 89 47 77)(36 88 48 76)(97 132 109 144)(98 131 110 143)(99 130 111 142)(100 129 112 141)(101 128 113 140)(102 127 114 139)(103 126 115 138)(104 125 116 137)(105 124 117 136)(106 123 118 135)(107 122 119 134)(108 121 120 133)```

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D12 D12 C8.C22 C8.D6 kernel C24.5D6 C24⋊2S3 C32⋊5Q16 C32×M4(2) C2×C32⋊4Q8 C12.59D6 C3×M4(2) C3×C12 C62 C24 C2×C12 C12 C2×C6 C32 C3 # reps 1 2 2 1 1 1 4 1 1 8 4 8 8 1 8

Matrix representation of C24.5D6 in GL6(𝔽73)

 72 72 0 0 0 0 1 0 0 0 0 0 0 0 71 38 0 26 0 0 35 33 47 47 0 0 56 37 2 35 0 0 36 19 38 40
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 7 45 43 40 0 0 28 35 33 3 0 0 57 21 66 28 0 0 52 36 45 38
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 7 45 43 40 0 0 38 66 70 30 0 0 34 52 2 3 0 0 18 39 1 71

`G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,71,35,56,36,0,0,38,33,37,19,0,0,0,47,2,38,0,0,26,47,35,40],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,28,57,52,0,0,45,35,21,36,0,0,43,33,66,45,0,0,40,3,28,38],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,7,38,34,18,0,0,45,66,52,39,0,0,43,70,2,1,0,0,40,30,3,71] >;`

C24.5D6 in GAP, Magma, Sage, TeX

`C_{24}._5D_6`
`% in TeX`

`G:=Group("C24.5D6");`
`// GroupNames label`

`G:=SmallGroup(288,766);`
`// by ID`

`G=gap.SmallGroup(288,766);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,675,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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