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## G = C9×CSU2(𝔽3)  order 432 = 24·33

### Direct product of C9 and CSU2(𝔽3)

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

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C9×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C9×SL2(𝔽3) — C9×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C9×CSU2(𝔽3)
 Upper central C1 — C18

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

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