metacyclic, supersoluble, monomial
Aliases: C9⋊C48, C18.C24, C72.2C6, C36.2C12, 3- 1+2⋊C16, C9⋊C16⋊C3, C2.(C9⋊C24), C8.2(C9⋊C6), (C3×C24).7S3, C32.(C3⋊C16), C4.2(C9⋊C12), C24.16(C3×S3), (C3×C12).6Dic3, C12.10(C3×Dic3), (C2×3- 1+2).C8, (C4×3- 1+2).2C4, (C8×3- 1+2).2C2, C6.3(C3×C3⋊C8), C3.3(C3×C3⋊C16), (C3×C6).3(C3⋊C8), SmallGroup(432,31)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C48 |
Generators and relations for C9⋊C48
G = < a,b | a9=b48=1, bab-1=a5 >
Smallest permutation representation of C9⋊C48
►On 144 points
Generators in S144
(1 56 121 36 88 137 31 72 105)(2 138 57 32 122 73 37 106 89)(3 74 139 38 58 107 17 90 123)(4 108 75 18 140 91 39 124 59)(5 92 109 40 76 125 19 60 141)(6 126 93 20 110 61 41 142 77)(7 62 127 42 94 143 21 78 111)(8 144 63 22 128 79 43 112 95)(9 80 97 44 64 113 23 96 129)(10 114 81 24 98 49 45 130 65)(11 50 115 46 82 131 25 66 99)(12 132 51 26 116 67 47 100 83)(13 68 133 48 52 101 27 84 117)(14 102 69 28 134 85 33 118 53)(15 86 103 34 70 119 29 54 135)(16 120 87 30 104 55 35 136 71)
(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)
G:=sub<Sym(144)| (1,56,121,36,88,137,31,72,105)(2,138,57,32,122,73,37,106,89)(3,74,139,38,58,107,17,90,123)(4,108,75,18,140,91,39,124,59)(5,92,109,40,76,125,19,60,141)(6,126,93,20,110,61,41,142,77)(7,62,127,42,94,143,21,78,111)(8,144,63,22,128,79,43,112,95)(9,80,97,44,64,113,23,96,129)(10,114,81,24,98,49,45,130,65)(11,50,115,46,82,131,25,66,99)(12,132,51,26,116,67,47,100,83)(13,68,133,48,52,101,27,84,117)(14,102,69,28,134,85,33,118,53)(15,86,103,34,70,119,29,54,135)(16,120,87,30,104,55,35,136,71), (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)>;
G:=Group( (1,56,121,36,88,137,31,72,105)(2,138,57,32,122,73,37,106,89)(3,74,139,38,58,107,17,90,123)(4,108,75,18,140,91,39,124,59)(5,92,109,40,76,125,19,60,141)(6,126,93,20,110,61,41,142,77)(7,62,127,42,94,143,21,78,111)(8,144,63,22,128,79,43,112,95)(9,80,97,44,64,113,23,96,129)(10,114,81,24,98,49,45,130,65)(11,50,115,46,82,131,25,66,99)(12,132,51,26,116,67,47,100,83)(13,68,133,48,52,101,27,84,117)(14,102,69,28,134,85,33,118,53)(15,86,103,34,70,119,29,54,135)(16,120,87,30,104,55,35,136,71), (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) );
G=PermutationGroup([[(1,56,121,36,88,137,31,72,105),(2,138,57,32,122,73,37,106,89),(3,74,139,38,58,107,17,90,123),(4,108,75,18,140,91,39,124,59),(5,92,109,40,76,125,19,60,141),(6,126,93,20,110,61,41,142,77),(7,62,127,42,94,143,21,78,111),(8,144,63,22,128,79,43,112,95),(9,80,97,44,64,113,23,96,129),(10,114,81,24,98,49,45,130,65),(11,50,115,46,82,131,25,66,99),(12,132,51,26,116,67,47,100,83),(13,68,133,48,52,101,27,84,117),(14,102,69,28,134,85,33,118,53),(15,86,103,34,70,119,29,54,135),(16,120,87,30,104,55,35,136,71)], [(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)]])
80 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 24E | ··· | 24L | 36A | ··· | 36F | 48A | ··· | 48P | 72A | ··· | 72L |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 6 | ··· | 6 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3⋊C16 | C3×C3⋊C8 | C3×C3⋊C16 | C9⋊C6 | C9⋊C12 | C9⋊C24 | C9⋊C48 |
| kernel | C9⋊C48 | C8×3- 1+2 | C9⋊C16 | C4×3- 1+2 | C72 | C2×3- 1+2 | C36 | 3- 1+2 | C18 | C9 | C3×C24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of C9⋊C48 ►in GL8(𝔽433)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 432 | 432 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 432 | 431 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 432 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 432 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 432 | 432 | 432 |
| 0 | 0 | 0 | 0 | 0 | 432 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 432 | 0 | 0 |
| 222 | 318 | 0 | 0 | 0 | 0 | 0 | 0 |
| 96 | 211 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 184 | 178 | 0 | 0 | 0 | 0 |
| 0 | 0 | 362 | 249 | 0 | 0 | 0 | 0 |
| 0 | 0 | 362 | 178 | 0 | 0 | 71 | 255 |
| 0 | 0 | 0 | 249 | 0 | 0 | 184 | 362 |
| 0 | 0 | 184 | 0 | 178 | 249 | 0 | 0 |
| 0 | 0 | 362 | 178 | 71 | 255 | 0 | 0 |
G:=sub<GL(8,GF(433))| [0,432,0,0,0,0,0,0,1,432,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,432,1,0,0,0,0,0,0,431,432,432,432,432,432,0,0,0,0,0,432,0,0,0,0,0,0,1,432,0,0],[222,96,0,0,0,0,0,0,318,211,0,0,0,0,0,0,0,0,184,362,362,0,184,362,0,0,178,249,178,249,0,178,0,0,0,0,0,0,178,71,0,0,0,0,0,0,249,255,0,0,0,0,71,184,0,0,0,0,0,0,255,362,0,0] >;
C9⋊C48 in GAP, Magma, Sage, TeX
C_9\rtimes C_{48} % in TeX
G:=Group("C9:C48"); // GroupNames label
G:=SmallGroup(432,31);
// by ID
G=gap.SmallGroup(432,31);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-2,-3,-3,42,58,80,10085,4044,292,14118]);
// Polycyclic
G:=Group<a,b|a^9=b^48=1,b*a*b^-1=a^5>;
// generators/relations
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