direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×M5(2), C4.C72, C144⋊7C2, C16⋊3C18, C48.7C6, C72.6C4, C36.4C8, C8.2C36, C22.C72, C12.5C24, C24.10C12, C72.29C22, C8.8(C2×C18), (C2×C4).5C36, (C2×C6).3C24, (C2×C18).1C8, C2.3(C2×C72), (C2×C8).8C18, C3.(C3×M5(2)), (C2×C24).29C6, C18.13(C2×C8), C4.11(C2×C36), (C2×C72).18C2, C24.41(C2×C6), (C2×C36).14C4, C6.13(C2×C24), C36.49(C2×C4), (C3×M5(2)).C3, (C2×C12).19C12, C12.60(C2×C12), SmallGroup(288,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×M5(2)
G = < a,b,c | a9=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C9×M5(2)
►On 144 points
Generators in S144
(1 129 120 77 45 27 60 85 102)(2 130 121 78 46 28 61 86 103)(3 131 122 79 47 29 62 87 104)(4 132 123 80 48 30 63 88 105)(5 133 124 65 33 31 64 89 106)(6 134 125 66 34 32 49 90 107)(7 135 126 67 35 17 50 91 108)(8 136 127 68 36 18 51 92 109)(9 137 128 69 37 19 52 93 110)(10 138 113 70 38 20 53 94 111)(11 139 114 71 39 21 54 95 112)(12 140 115 72 40 22 55 96 97)(13 141 116 73 41 23 56 81 98)(14 142 117 74 42 24 57 82 99)(15 143 118 75 43 25 58 83 100)(16 144 119 76 44 26 59 84 101)
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(66 74)(68 76)(70 78)(72 80)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)(113 121)(115 123)(117 125)(119 127)(130 138)(132 140)(134 142)(136 144)
G:=sub<Sym(144)| (1,129,120,77,45,27,60,85,102)(2,130,121,78,46,28,61,86,103)(3,131,122,79,47,29,62,87,104)(4,132,123,80,48,30,63,88,105)(5,133,124,65,33,31,64,89,106)(6,134,125,66,34,32,49,90,107)(7,135,126,67,35,17,50,91,108)(8,136,127,68,36,18,51,92,109)(9,137,128,69,37,19,52,93,110)(10,138,113,70,38,20,53,94,111)(11,139,114,71,39,21,54,95,112)(12,140,115,72,40,22,55,96,97)(13,141,116,73,41,23,56,81,98)(14,142,117,74,42,24,57,82,99)(15,143,118,75,43,25,58,83,100)(16,144,119,76,44,26,59,84,101), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(66,74)(68,76)(70,78)(72,80)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111)(113,121)(115,123)(117,125)(119,127)(130,138)(132,140)(134,142)(136,144)>;
G:=Group( (1,129,120,77,45,27,60,85,102)(2,130,121,78,46,28,61,86,103)(3,131,122,79,47,29,62,87,104)(4,132,123,80,48,30,63,88,105)(5,133,124,65,33,31,64,89,106)(6,134,125,66,34,32,49,90,107)(7,135,126,67,35,17,50,91,108)(8,136,127,68,36,18,51,92,109)(9,137,128,69,37,19,52,93,110)(10,138,113,70,38,20,53,94,111)(11,139,114,71,39,21,54,95,112)(12,140,115,72,40,22,55,96,97)(13,141,116,73,41,23,56,81,98)(14,142,117,74,42,24,57,82,99)(15,143,118,75,43,25,58,83,100)(16,144,119,76,44,26,59,84,101), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(66,74)(68,76)(70,78)(72,80)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111)(113,121)(115,123)(117,125)(119,127)(130,138)(132,140)(134,142)(136,144) );
G=PermutationGroup([[(1,129,120,77,45,27,60,85,102),(2,130,121,78,46,28,61,86,103),(3,131,122,79,47,29,62,87,104),(4,132,123,80,48,30,63,88,105),(5,133,124,65,33,31,64,89,106),(6,134,125,66,34,32,49,90,107),(7,135,126,67,35,17,50,91,108),(8,136,127,68,36,18,51,92,109),(9,137,128,69,37,19,52,93,110),(10,138,113,70,38,20,53,94,111),(11,139,114,71,39,21,54,95,112),(12,140,115,72,40,22,55,96,97),(13,141,116,73,41,23,56,81,98),(14,142,117,74,42,24,57,82,99),(15,143,118,75,43,25,58,83,100),(16,144,119,76,44,26,59,84,101)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(66,74),(68,76),(70,78),(72,80),(82,90),(84,92),(86,94),(88,96),(97,105),(99,107),(101,109),(103,111),(113,121),(115,123),(117,125),(119,127),(130,138),(132,140),(134,142),(136,144)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 18A | ··· | 18F | 18G | ··· | 18L | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 36A | ··· | 36L | 36M | ··· | 36R | 48A | ··· | 48P | 72A | ··· | 72X | 72Y | ··· | 72AJ | 144A | ··· | 144AV |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C9 | C12 | C12 | C18 | C18 | C24 | C24 | C36 | C36 | C72 | C72 | M5(2) | C3×M5(2) | C9×M5(2) |
| kernel | C9×M5(2) | C144 | C2×C72 | C3×M5(2) | C72 | C2×C36 | C48 | C2×C24 | C36 | C2×C18 | M5(2) | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 6 | 4 | 4 | 12 | 6 | 8 | 8 | 12 | 12 | 24 | 24 | 4 | 8 | 24 |
Matrix representation of C9×M5(2) ►in GL2(𝔽433) generated by
| 256 | 0 |
| 0 | 256 |
| 345 | 431 |
| 334 | 88 |
| 1 | 0 |
| 345 | 432 |
G:=sub<GL(2,GF(433))| [256,0,0,256],[345,334,431,88],[1,345,0,432] >;
C9×M5(2) in GAP, Magma, Sage, TeX
C_9\times M_5(2)
% in TeX
G:=Group("C9xM5(2)"); // GroupNames label
G:=SmallGroup(288,60);
// by ID
G=gap.SmallGroup(288,60);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,84,2045,142,192,124]);
// Polycyclic
G:=Group<a,b,c|a^9=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations
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