Copied to
clipboard

## G = C24.94D6order 288 = 25·32

### 28th non-split extension by C24 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C24.94D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C24.S3 — C24.94D6
 Lower central C32 — C3×C6 — C24.94D6
 Upper central C1 — C8 — C2×C8

Generators and relations for C24.94D6
G = < a,b,c | a24=b6=1, c2=a21, ab=ba, cac-1=a17, cbc-1=a12b-1 >

Subgroups: 132 in 78 conjugacy classes, 57 normal (19 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C16 [×2], C2×C8, C3×C6, C3×C6, C24 [×8], C2×C12 [×4], M5(2), C3×C12 [×2], C62, C3⋊C16 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C12.C8 [×4], C24.S3 [×2], C6×C24, C24.94D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, Dic3 [×8], D6 [×4], C2×C8, C3⋊S3, C3⋊C8 [×8], C2×Dic3 [×4], M5(2), C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×4], C324C8 [×2], C2×C3⋊Dic3, C12.C8 [×4], C2×C324C8, C24.94D6

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A ··· 6L 8A 8B 8C 8D 8E 8F 12A ··· 12P 16A ··· 16H 24A ··· 24AF order 1 2 2 3 3 3 3 4 4 4 6 ··· 6 8 8 8 8 8 8 12 ··· 12 16 ··· 16 24 ··· 24 size 1 1 2 2 2 2 2 1 1 2 2 ··· 2 1 1 1 1 2 2 2 ··· 2 18 ··· 18 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 S3 Dic3 D6 Dic3 C3⋊C8 C3⋊C8 M5(2) C12.C8 kernel C24.94D6 C24.S3 C6×C24 C3×C24 C6×C12 C3×C12 C62 C2×C24 C24 C24 C2×C12 C12 C2×C6 C32 C3 # reps 1 2 1 2 2 4 4 4 4 4 4 8 8 4 32

Matrix representation of C24.94D6 in GL4(𝔽97) generated by

 22 0 0 0 0 22 0 0 0 0 73 0 0 0 0 88
,
 35 0 0 0 2 61 0 0 0 0 61 0 0 0 0 62
,
 77 31 0 0 41 20 0 0 0 0 0 35 0 0 24 0
`G:=sub<GL(4,GF(97))| [22,0,0,0,0,22,0,0,0,0,73,0,0,0,0,88],[35,2,0,0,0,61,0,0,0,0,61,0,0,0,0,62],[77,41,0,0,31,20,0,0,0,0,0,24,0,0,35,0] >;`

C24.94D6 in GAP, Magma, Sage, TeX

`C_{24}._{94}D_6`
`% in TeX`

`G:=Group("C24.94D6");`
`// GroupNames label`

`G:=SmallGroup(288,287);`
`// by ID`

`G=gap.SmallGroup(288,287);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,58,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^6=1,c^2=a^21,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;`
`// generators/relations`

׿
×
𝔽