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## G = C24.32D6order 288 = 25·32

### 32nd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24.32D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — Q8×C3⋊S3 — C24.32D6
 Lower central C32 — C3×C6 — C3×C12 — C24.32D6
 Upper central C1 — C2 — C4 — SD16

Generators and relations for C24.32D6
G = < a,b,c | a24=1, b6=c2=a12, bab-1=a19, cac-1=a-1, cbc-1=b5 >

Subgroups: 684 in 180 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C4 [×4], C22 [×2], S3 [×4], C6 [×4], C6 [×4], C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], C32, Dic3 [×12], C12 [×4], C12 [×4], D6 [×4], C2×C6 [×4], M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], Dic6 [×12], C4×S3 [×8], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×4], C3×Q8 [×4], C8.C22, C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C3×C12, C2×C3⋊S3, C62, C8⋊S3 [×4], Dic12 [×4], D4.S3 [×4], C3⋊Q16 [×4], C3×SD16 [×4], D42S3 [×4], S3×Q8 [×4], C324C8, C3×C24, C324Q8 [×2], C324Q8, C4×C3⋊S3, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, Q8×C32, D4.D6 [×4], C24⋊S3, C325Q16, C329SD16, C327Q16, C32×SD16, C12.D6, Q8×C3⋊S3, C24.32D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C8.C22, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, D4.D6 [×4], D4×C3⋊S3, C24.32D6

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

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

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

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

39 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 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H 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 12 12 24 ··· 24 size 1 1 4 18 2 2 2 2 2 4 18 36 36 2 2 2 2 8 8 8 8 4 36 4 4 4 4 8 8 8 8 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C8.C22 S3×D4 D4.D6 kernel C24.32D6 C24⋊S3 C32⋊5Q16 C32⋊9SD16 C32⋊7Q16 C32×SD16 C12.D6 Q8×C3⋊S3 C3×SD16 C3⋊Dic3 C2×C3⋊S3 C24 C3×D4 C3×Q8 C32 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 1 4 4 4 1 4 8

Matrix representation of C24.32D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 39 39 34 0 0 34 68 39 5 0 0 34 39 34 39 0 0 34 68 34 68
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 65 0 64 0 0 0 0 65 0 64 0 0 64 0 8 0 0 0 0 64 0 8
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 65 0 64 0 0 0 8 8 9 9 0 0 64 0 8 0 0 0 9 9 65 65

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,34,34,0,0,39,68,39,68,0,0,39,39,34,34,0,0,34,5,39,68],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,65,0,64,0,0,0,0,65,0,64,0,0,64,0,8,0,0,0,0,64,0,8],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,65,8,64,9,0,0,0,8,0,9,0,0,64,9,8,65,0,0,0,9,0,65] >;`

C24.32D6 in GAP, Magma, Sage, TeX

`C_{24}._{32}D_6`
`% in TeX`

`G:=Group("C24.32D6");`
`// GroupNames label`

`G:=SmallGroup(288,772);`
`// by ID`

`G=gap.SmallGroup(288,772);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,346,185,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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