direct product, non-abelian, soluble
Aliases: C9×CSU2(𝔽3), C18.8S4, SL2(𝔽3).C18, Q8.(S3×C9), C2.2(C9×S4), C6.16(C3×S4), (Q8×C9).4S3, (C3×CSU2(𝔽3)).C3, (C9×SL2(𝔽3)).2C2, (C3×SL2(𝔽3)).5C6, C3.4(C3×CSU2(𝔽3)), (C3×Q8).8(C3×S3), SmallGroup(432,240)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C9×SL2(𝔽3) — C9×CSU2(𝔽3) |
| SL2(𝔽3) — C9×CSU2(𝔽3) |
Generators and relations for C9×CSU2(𝔽3)
G = < a,b,c,d,e | a9=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >
Smallest permutation representation of C9×CSU2(𝔽3)
►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)
(1 114 101 94)(2 115 102 95)(3 116 103 96)(4 117 104 97)(5 109 105 98)(6 110 106 99)(7 111 107 91)(8 112 108 92)(9 113 100 93)(10 139 21 47)(11 140 22 48)(12 141 23 49)(13 142 24 50)(14 143 25 51)(15 144 26 52)(16 136 27 53)(17 137 19 54)(18 138 20 46)(28 58 72 38)(29 59 64 39)(30 60 65 40)(31 61 66 41)(32 62 67 42)(33 63 68 43)(34 55 69 44)(35 56 70 45)(36 57 71 37)(73 120 133 85)(74 121 134 86)(75 122 135 87)(76 123 127 88)(77 124 128 89)(78 125 129 90)(79 126 130 82)(80 118 131 83)(81 119 132 84)
(1 81 101 132)(2 73 102 133)(3 74 103 134)(4 75 104 135)(5 76 105 127)(6 77 106 128)(7 78 107 129)(8 79 108 130)(9 80 100 131)(10 30 21 65)(11 31 22 66)(12 32 23 67)(13 33 24 68)(14 34 25 69)(15 35 26 70)(16 36 27 71)(17 28 19 72)(18 29 20 64)(37 53 57 136)(38 54 58 137)(39 46 59 138)(40 47 60 139)(41 48 61 140)(42 49 62 141)(43 50 63 142)(44 51 55 143)(45 52 56 144)(82 92 126 112)(83 93 118 113)(84 94 119 114)(85 95 120 115)(86 96 121 116)(87 97 122 117)(88 98 123 109)(89 99 124 110)(90 91 125 111)
(10 60 30)(11 61 31)(12 62 32)(13 63 33)(14 55 34)(15 56 35)(16 57 36)(17 58 28)(18 59 29)(19 38 72)(20 39 64)(21 40 65)(22 41 66)(23 42 67)(24 43 68)(25 44 69)(26 45 70)(27 37 71)(73 85 115)(74 86 116)(75 87 117)(76 88 109)(77 89 110)(78 90 111)(79 82 112)(80 83 113)(81 84 114)(91 129 125)(92 130 126)(93 131 118)(94 132 119)(95 133 120)(96 134 121)(97 135 122)(98 127 123)(99 128 124)
(1 46 101 138)(2 47 102 139)(3 48 103 140)(4 49 104 141)(5 50 105 142)(6 51 106 143)(7 52 107 144)(8 53 108 136)(9 54 100 137)(10 73 21 133)(11 74 22 134)(12 75 23 135)(13 76 24 127)(14 77 25 128)(15 78 26 129)(16 79 27 130)(17 80 19 131)(18 81 20 132)(28 83 72 118)(29 84 64 119)(30 85 65 120)(31 86 66 121)(32 87 67 122)(33 88 68 123)(34 89 69 124)(35 90 70 125)(36 82 71 126)(37 92 57 112)(38 93 58 113)(39 94 59 114)(40 95 60 115)(41 96 61 116)(42 97 62 117)(43 98 63 109)(44 99 55 110)(45 91 56 111)
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), (1,114,101,94)(2,115,102,95)(3,116,103,96)(4,117,104,97)(5,109,105,98)(6,110,106,99)(7,111,107,91)(8,112,108,92)(9,113,100,93)(10,139,21,47)(11,140,22,48)(12,141,23,49)(13,142,24,50)(14,143,25,51)(15,144,26,52)(16,136,27,53)(17,137,19,54)(18,138,20,46)(28,58,72,38)(29,59,64,39)(30,60,65,40)(31,61,66,41)(32,62,67,42)(33,63,68,43)(34,55,69,44)(35,56,70,45)(36,57,71,37)(73,120,133,85)(74,121,134,86)(75,122,135,87)(76,123,127,88)(77,124,128,89)(78,125,129,90)(79,126,130,82)(80,118,131,83)(81,119,132,84), (1,81,101,132)(2,73,102,133)(3,74,103,134)(4,75,104,135)(5,76,105,127)(6,77,106,128)(7,78,107,129)(8,79,108,130)(9,80,100,131)(10,30,21,65)(11,31,22,66)(12,32,23,67)(13,33,24,68)(14,34,25,69)(15,35,26,70)(16,36,27,71)(17,28,19,72)(18,29,20,64)(37,53,57,136)(38,54,58,137)(39,46,59,138)(40,47,60,139)(41,48,61,140)(42,49,62,141)(43,50,63,142)(44,51,55,143)(45,52,56,144)(82,92,126,112)(83,93,118,113)(84,94,119,114)(85,95,120,115)(86,96,121,116)(87,97,122,117)(88,98,123,109)(89,99,124,110)(90,91,125,111), (10,60,30)(11,61,31)(12,62,32)(13,63,33)(14,55,34)(15,56,35)(16,57,36)(17,58,28)(18,59,29)(19,38,72)(20,39,64)(21,40,65)(22,41,66)(23,42,67)(24,43,68)(25,44,69)(26,45,70)(27,37,71)(73,85,115)(74,86,116)(75,87,117)(76,88,109)(77,89,110)(78,90,111)(79,82,112)(80,83,113)(81,84,114)(91,129,125)(92,130,126)(93,131,118)(94,132,119)(95,133,120)(96,134,121)(97,135,122)(98,127,123)(99,128,124), (1,46,101,138)(2,47,102,139)(3,48,103,140)(4,49,104,141)(5,50,105,142)(6,51,106,143)(7,52,107,144)(8,53,108,136)(9,54,100,137)(10,73,21,133)(11,74,22,134)(12,75,23,135)(13,76,24,127)(14,77,25,128)(15,78,26,129)(16,79,27,130)(17,80,19,131)(18,81,20,132)(28,83,72,118)(29,84,64,119)(30,85,65,120)(31,86,66,121)(32,87,67,122)(33,88,68,123)(34,89,69,124)(35,90,70,125)(36,82,71,126)(37,92,57,112)(38,93,58,113)(39,94,59,114)(40,95,60,115)(41,96,61,116)(42,97,62,117)(43,98,63,109)(44,99,55,110)(45,91,56,111)>;
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), (1,114,101,94)(2,115,102,95)(3,116,103,96)(4,117,104,97)(5,109,105,98)(6,110,106,99)(7,111,107,91)(8,112,108,92)(9,113,100,93)(10,139,21,47)(11,140,22,48)(12,141,23,49)(13,142,24,50)(14,143,25,51)(15,144,26,52)(16,136,27,53)(17,137,19,54)(18,138,20,46)(28,58,72,38)(29,59,64,39)(30,60,65,40)(31,61,66,41)(32,62,67,42)(33,63,68,43)(34,55,69,44)(35,56,70,45)(36,57,71,37)(73,120,133,85)(74,121,134,86)(75,122,135,87)(76,123,127,88)(77,124,128,89)(78,125,129,90)(79,126,130,82)(80,118,131,83)(81,119,132,84), (1,81,101,132)(2,73,102,133)(3,74,103,134)(4,75,104,135)(5,76,105,127)(6,77,106,128)(7,78,107,129)(8,79,108,130)(9,80,100,131)(10,30,21,65)(11,31,22,66)(12,32,23,67)(13,33,24,68)(14,34,25,69)(15,35,26,70)(16,36,27,71)(17,28,19,72)(18,29,20,64)(37,53,57,136)(38,54,58,137)(39,46,59,138)(40,47,60,139)(41,48,61,140)(42,49,62,141)(43,50,63,142)(44,51,55,143)(45,52,56,144)(82,92,126,112)(83,93,118,113)(84,94,119,114)(85,95,120,115)(86,96,121,116)(87,97,122,117)(88,98,123,109)(89,99,124,110)(90,91,125,111), (10,60,30)(11,61,31)(12,62,32)(13,63,33)(14,55,34)(15,56,35)(16,57,36)(17,58,28)(18,59,29)(19,38,72)(20,39,64)(21,40,65)(22,41,66)(23,42,67)(24,43,68)(25,44,69)(26,45,70)(27,37,71)(73,85,115)(74,86,116)(75,87,117)(76,88,109)(77,89,110)(78,90,111)(79,82,112)(80,83,113)(81,84,114)(91,129,125)(92,130,126)(93,131,118)(94,132,119)(95,133,120)(96,134,121)(97,135,122)(98,127,123)(99,128,124), (1,46,101,138)(2,47,102,139)(3,48,103,140)(4,49,104,141)(5,50,105,142)(6,51,106,143)(7,52,107,144)(8,53,108,136)(9,54,100,137)(10,73,21,133)(11,74,22,134)(12,75,23,135)(13,76,24,127)(14,77,25,128)(15,78,26,129)(16,79,27,130)(17,80,19,131)(18,81,20,132)(28,83,72,118)(29,84,64,119)(30,85,65,120)(31,86,66,121)(32,87,67,122)(33,88,68,123)(34,89,69,124)(35,90,70,125)(36,82,71,126)(37,92,57,112)(38,93,58,113)(39,94,59,114)(40,95,60,115)(41,96,61,116)(42,97,62,117)(43,98,63,109)(44,99,55,110)(45,91,56,111) );
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)], [(1,114,101,94),(2,115,102,95),(3,116,103,96),(4,117,104,97),(5,109,105,98),(6,110,106,99),(7,111,107,91),(8,112,108,92),(9,113,100,93),(10,139,21,47),(11,140,22,48),(12,141,23,49),(13,142,24,50),(14,143,25,51),(15,144,26,52),(16,136,27,53),(17,137,19,54),(18,138,20,46),(28,58,72,38),(29,59,64,39),(30,60,65,40),(31,61,66,41),(32,62,67,42),(33,63,68,43),(34,55,69,44),(35,56,70,45),(36,57,71,37),(73,120,133,85),(74,121,134,86),(75,122,135,87),(76,123,127,88),(77,124,128,89),(78,125,129,90),(79,126,130,82),(80,118,131,83),(81,119,132,84)], [(1,81,101,132),(2,73,102,133),(3,74,103,134),(4,75,104,135),(5,76,105,127),(6,77,106,128),(7,78,107,129),(8,79,108,130),(9,80,100,131),(10,30,21,65),(11,31,22,66),(12,32,23,67),(13,33,24,68),(14,34,25,69),(15,35,26,70),(16,36,27,71),(17,28,19,72),(18,29,20,64),(37,53,57,136),(38,54,58,137),(39,46,59,138),(40,47,60,139),(41,48,61,140),(42,49,62,141),(43,50,63,142),(44,51,55,143),(45,52,56,144),(82,92,126,112),(83,93,118,113),(84,94,119,114),(85,95,120,115),(86,96,121,116),(87,97,122,117),(88,98,123,109),(89,99,124,110),(90,91,125,111)], [(10,60,30),(11,61,31),(12,62,32),(13,63,33),(14,55,34),(15,56,35),(16,57,36),(17,58,28),(18,59,29),(19,38,72),(20,39,64),(21,40,65),(22,41,66),(23,42,67),(24,43,68),(25,44,69),(26,45,70),(27,37,71),(73,85,115),(74,86,116),(75,87,117),(76,88,109),(77,89,110),(78,90,111),(79,82,112),(80,83,113),(81,84,114),(91,129,125),(92,130,126),(93,131,118),(94,132,119),(95,133,120),(96,134,121),(97,135,122),(98,127,123),(99,128,124)], [(1,46,101,138),(2,47,102,139),(3,48,103,140),(4,49,104,141),(5,50,105,142),(6,51,106,143),(7,52,107,144),(8,53,108,136),(9,54,100,137),(10,73,21,133),(11,74,22,134),(12,75,23,135),(13,76,24,127),(14,77,25,128),(15,78,26,129),(16,79,27,130),(17,80,19,131),(18,81,20,132),(28,83,72,118),(29,84,64,119),(30,85,65,120),(31,86,66,121),(32,87,67,122),(33,88,68,123),(34,89,69,124),(35,90,70,125),(36,82,71,126),(37,92,57,112),(38,93,58,113),(39,94,59,114),(40,95,60,115),(41,96,61,116),(42,97,62,117),(43,98,63,109),(44,99,55,110),(45,91,56,111)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 18G | ··· | 18L | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 72A | ··· | 72L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 6 | 12 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | ||||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | CSU2(𝔽3) | S3×C9 | C3×CSU2(𝔽3) | C9×CSU2(𝔽3) | S4 | C3×S4 | C9×S4 | CSU2(𝔽3) | C3×CSU2(𝔽3) | C9×CSU2(𝔽3) |
| kernel | C9×CSU2(𝔽3) | C9×SL2(𝔽3) | C3×CSU2(𝔽3) | C3×SL2(𝔽3) | CSU2(𝔽3) | SL2(𝔽3) | Q8×C9 | C3×Q8 | C9 | Q8 | C3 | C1 | C18 | C6 | C2 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 2 | 6 | 4 | 12 | 2 | 4 | 12 | 1 | 2 | 6 |
Matrix representation of C9×CSU2(𝔽3) ►in GL2(𝔽73) generated by
| 2 | 0 |
| 0 | 2 |
| 6 | 9 |
| 4 | 67 |
| 63 | 69 |
| 7 | 10 |
| 69 | 6 |
| 10 | 3 |
| 37 | 62 |
| 25 | 36 |
G:=sub<GL(2,GF(73))| [2,0,0,2],[6,4,9,67],[63,7,69,10],[69,10,6,3],[37,25,62,36] >;
C9×CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_9\times {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C9xCSU(2,3)"); // GroupNames label
G:=SmallGroup(432,240);
// by ID
G=gap.SmallGroup(432,240);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,50,1011,3784,655,172,2273,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^4=d^3=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=b^-1,d*b*d^-1=b*c,e*b*e^-1=b^2*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations
Export